Recent zbMATH articles in MSC 37https://www.zbmath.org/atom/cc/372021-03-30T15:24:00+00:00WerkzeugThe quantum variance of the modular surface.https://www.zbmath.org/1455.110762021-03-30T15:24:00+00:00"Sarnak, P."https://www.zbmath.org/authors/?q=ai:sarnak.peter|sarnak.peter-c"Zhao, P."https://www.zbmath.org/authors/?q=ai:zhao.pengli|zhao.pinghui|zhao.pei|zhao.pengcheng|zhao.peilin|zhao.panlei|zhao.pinjiao|zhao.pingping|zhao.peipei|zhao.pingxi|zhao.puying|zhao.paihao|zhao.pengjun|zhao.pengda|zhao.panpan|zhao.pinghua|zhao.peiyi|zhao.peng|zhao.pindong|zhao.peixin|zhao.pingfu|zhao.ping|zhao.peichen|zhao.peiwu|zhao.pingya|zhao.pengnian|zhao.peitao|zhao.pengtao|zhao.peiying|zhao.peibiao|zhao.pingzhong|zhao.pan|zhao.peihao|zhao.pengfei|zhao.peigen|zhao.peiji|zhao.penglei|zhao.pinxuan|zhao.puningRoughly speaking, the main result of the paper under review is that the ``quantum variance'' and the classical variance of observables of the Laplacian on the modular surface are simultaneously diagonalised by the irreducible representations of the modular quotient and on each of these subspaces the two are equal up to a constant factor which is equal to the central value of an automorphic \(L\)-function.
Let \(G=\mathrm{PSL}(2,\mathbb{R}),\Gamma=\mathrm{PSL}(2,\mathbb{Z})\) and consider the hyperbolic surface \(X=\Gamma\backslash\mathbb{H}\) (``the modular surface'') whose (diagonalisable) eigenfunctions under the Laplacian \(\Delta\) on \(X\) will be denoted by \(\phi_j,j=1,2,\dots\) These so-called ``Hecke-Maaß forms'' are real-valued and satisfy
\[
\Delta\phi_j+\lambda_j\phi_j=0,\quad T_n\phi_j=\lambda_j(n)\phi_j,
\]
where \(T_n\) is the \(n\)th Hecke operator. They are normalised by \(\int_X \phi_j(z)^2dA(z)=1\) where \(dA\) is the normalised hyperbolic area form. The Poincaré upper half plane \(\mathbb{H}\) may be identified with \(G/K\) where \(K=\mathrm{SO}(2)/(\pm I) \) and \(\Gamma\backslash G\) may then be identified with the unit tangent space for the geodesic flow on \(X.\) The objects of study in this paper are the Wigner distributions \(d\omega_j\), which are quadratic functionals of the Hecke-Maaß\, forms \(\phi_j\) and are given by
\[
d\omega_j=\phi_j(z)\sum_{k\in\mathbb{Z}} \phi_{j,k}(z) e^{-2ik\theta}d\omega,
\]
where \(d\omega=y^{-2}dxdy \,d\theta/2\pi\) and the \(\phi_{j,k}\) are the shifted Maaß\, cusp forms of weight \(k\), normalised such that \(\|\phi_{j,k}\|_2=1\) by raising and lowering operators, \(E_+\) and \(E_-\). Recall that \textit{E. Lindenstrauss} [in: Current developments in mathematics, 2004. Somerville, MA: International Press. 111--139 (2006; Zbl 1186.58022)] and \textit{K. Soundararajan} [Ann. Math. (2) 172, No. 2, 1529--1538 (2010; Zbl 1209.58019)] have shown the ``QUE'' property, which is that for an ``observable'' \(\psi\in C(\Gamma\backslash G)\) we have
\[
\omega_j(\psi)\to \frac{1}{\mathrm{vol}(\Gamma\backslash G)}\int_{\Gamma\backslash G}\psi(g)dg, \quad \text{ as } j\to\infty ,
\]
where \(dg\) is the normalised Haar measure. The generalised Lindelöf hypothesis implies (see [\textit{T. Watson}, Central value of Rankin triple \(L\)-function for unramified Maass cusp forms. Princeton: Princeton University (PhD Thesis) (2004)] and [\textit{D. Jakobson}, Ann. Inst. Fourier 44, No. 5, 1477--1504 (1994; Zbl 0820.11040)]) that if the observable \(\psi\) is of mean \(0\) over \(\Gamma\backslash G\) (i.e., if the above integral vanishes), then \(\omega_j(\psi)\ll t_j^{-1/2+\varepsilon}\) for any \(\varepsilon >0.\) Here, as usual, we have written \(\lambda_j=1/4+t_j^2. \) The variance sums \(S_\psi(T):=\sum_{t_j\leqslant T} |\omega_j(\psi)|^2\) were introduced by \textit{S. Zelditch} [Duke Math. J. 55, 919--941 (1987; Zbl 0643.58029)], who showed \(S_\psi(T)\ll T^2\log T\). The first result proved in the paper under review concerns the variance sums, weighted by the ``harmonic'' weights \(L(1,\mathrm{sym}^2\phi_j)\), which satisfy \(t_j^{-\varepsilon}\ll_\varepsilon L(1,\mathrm{sym}^2\phi_j) \ll_\varepsilon t_j^{\varepsilon}\) for \(\varepsilon>0.\)
Theorem 1. Let \(A_0(\Gamma\backslash G)\) denote the space of smooth right \(K\)-finite functions on \(\Gamma\backslash G\) which are of mean \(0\) and of rapid decay. Then there is a sesquilinear form \(Q\) (called the ``Quantum Variance'')on \(A_0(\Gamma\backslash G)\times A_0(\Gamma\backslash G)\) such that
\[
\lim_{T\to \infty} \frac{1}{T}\sum_{t_j\leqslant T}L(1,\mathrm{sym}^2 \phi_j)\omega_j(\psi_1)\overline{\omega}_j(\psi_2)=Q(\psi_1,\psi_2).
\]
To describe the main result of this paper, we introduce some more notation. Recall that the fluctuations of an observable \(\psi\in C_0(\Gamma\backslash G)\) under the classical motion \(\mathcal{G}_t\) by geodesics was determined by \textit{M. Ratner} [Ergodic Theory Dyn. Syst. 7, 267--288 (1987; Zbl 0623.22008); Isr. J. Math. 16, 181--197 (1974; Zbl 0283.58010)] and it asserts that as \(T\to\infty\), the random variable \(T^{-1/2}\int_0^T \psi(\mathcal{G}_t(g))dt\) on \(\Gamma\backslash G\) becomes Gaussian with mean \(0 \) and variance \(V\) given by
\[
V(\psi_1,\psi_2)=\int_{-\infty}^{\infty}\int_{\Gamma\backslash G}\psi_1\left(g\begin{pmatrix}e^{t/2} & 0 \\ 0 & e^{-t/2} \end{pmatrix}\right)\overline{\psi_2(g)}dgdt.
\]
Let \(w\) denote the involution of \(\Gamma\backslash G\) given by \(\Gamma g\mapsto \Gamma g\begin{pmatrix} 0 & 1\\ -1 & 0
\end{pmatrix}\) and let \(r\) denote the symmetry on \(X\) induced by the isometry \(z\mapsto -\bar{z}\) of \(\mathbb{H}.\) Then the symmetrised form of the classical variance \(V\) is defined by
\[
V^{\mathrm{sym}}(\psi_1,\psi_2)=V(\psi_1^{\mathrm{sym}},\psi_2^{\mathrm{sym}})
\]
where \(\psi^{\mathrm{sym}}:=(\psi+w\psi+r\psi+wr\psi)/4.\) We also write
\begin{align*}
L_{\mathrm{cusp}}^2(\Gamma\backslash G)&=\bigoplus_{j=1}^\infty W_{\pi_j}=\sum_{j=1}^{\infty}W_{\pi_j^0}\oplus \sum_{k\geqslant 12}\sum_{j=1}^{d_k}\left(W_{\pi_j^k}\oplus W_{\pi_j^{-k}}\right)\\
&:=\sum_{j=1}^{\infty}U_{\pi_j^0}\oplus\sum_{k\geqslant 12}\sum_{j=1}^{d_k}U_{\pi_j^k},
\end{align*}
where \(W_{\pi_j}\) are irreducible cuspidal automorphic representations invariant under the Hecke algebra, the \(\pi_j^{\pm k}\) (\(k\) even ) are the holomorphic and antiholomorphic discrete series and \(\pi_j^0\) are the spherical representations. Also, \(d_k\) is the dimension of the space of holomorphic and antiholomorphic forms of weight \(k\) and \(d_k\) is either \([k/12]\) or \([k/12]+1\) depending on whether \(k/2\) is congruent or not to \(1\) modulo \(6\). With these preliminaries, the main result of the paper is that the quantum variance \(Q\) and the symmetrised classical variance \(V^{\mathrm{sym}}\) are both diagonalised by the above orthogonal decomposition and are equal up to a factor of \(L(1/2,\pi)\) on each irreducible subspace \(U_{\pi_j^k}\). Precisely we have
\[
Q| U_{\pi_j^k}=L(1/2, {\pi_j^k})V^{\mathrm{sym}}| U_{\pi_j^k}.
\]
The notation here is that \(L(s,\pi)\) denotes the finite part and \(\Lambda(s,\pi)\) is the completed \(L\)-function.
To conclude, the paper is beautifully written and the brief outline given of the proof is quite clear, so we have chosen not to include it in our review.
Reviewer: Ramdin Mawia (Kolkata)Conservation laws and line soliton solutions of a family of modified KP equations.https://www.zbmath.org/1455.350432021-03-30T15:24:00+00:00"Anco, Stephen C."https://www.zbmath.org/authors/?q=ai:anco.stephen-c"Gandarias, Maria Luz"https://www.zbmath.org/authors/?q=ai:gandarias.maria-luz"Recio, Elena"https://www.zbmath.org/authors/?q=ai:recio.elenaSummary: A family of modified Kadomtsev-Petviashvili equations (mKP) in \(2+1\) dimensions is studied. This family includes the integrable mKP equation when the coefficients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable case.Transverse homoclinic orbit bifurcated from a homoclinic manifold by the higher order Melnikov integrals.https://www.zbmath.org/1455.340382021-03-30T15:24:00+00:00"Long, Bin"https://www.zbmath.org/authors/?q=ai:long.bin"Zhu, Changrong"https://www.zbmath.org/authors/?q=ai:zhu.changrongSummary: Consider an autonomous ordinary differential equation in \(\mathbb{R}^n\) that has a \(d\) dimensional homoclinic solution manifold \(W^H\). Suppose the homoclinic manifold can be locally parametrized by \((\alpha,\theta) \in \mathbb{R}^{d-1}\times \mathbb{R}\). We study the bifurcation of the homoclinic solution manifold \(W^H\) under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed \((\alpha_0,\theta_0)\) on \(W^H\), if the Melnikov function have a simple zeros, then the perturbed system can have transverse homoclinic solutions near \(W^H\).Periodic solutions to a forced Kepler problem in the plane.https://www.zbmath.org/1455.370502021-03-30T15:24:00+00:00"Boscaggin, Alberto"https://www.zbmath.org/authors/?q=ai:boscaggin.alberto"Dambrosio, Walter"https://www.zbmath.org/authors/?q=ai:dambrosio.walter"Papini, Duccio"https://www.zbmath.org/authors/?q=ai:papini.duccioThe authors show that the forced Kepler problem defined by
\[\ddot x = - \frac{x}{|x|^3} + \nabla_x U(t,x), \qquad x \in \mathbb{R}^2\,,\]
where \(U(t,x)\) is \(T\)-periodic in the first variable, has generalized \(T\)-periodic solutions. The proof follows three steps: first, the action is minimized with nontrivial winding number around the origin. Second, the behaviour of this minimum is studied near the possible collisions corresponding to a \(T\)-periodic solution. Third, it is shown that the directions of ingoing and outgoing collisions coincide.
Reviewer: Wolfgang G. Hollik (Karlsruhe)A linear estimation to the number of zeros for abelian integrals in a kind of quadratic reversible centers of genus one.https://www.zbmath.org/1455.340292021-03-30T15:24:00+00:00"Hong, Lijun"https://www.zbmath.org/authors/?q=ai:hong.lijun"Hong, Xiaochun"https://www.zbmath.org/authors/?q=ai:hong.xiaochun"Lu, Junliang"https://www.zbmath.org/authors/?q=ai:lu.junliangSummary: In this paper, using the method of Picard-Fuchs equation and Riccati equation, we consider the number of zeros for abelian integrals in a kind of quadratic reversible centers of genus one under arbitrary polynomial perturbations of degree \(n\), and obtain that the upper bound of the number is \(2\left[{(n+1)}/{2}\right]+ \left[{n}/{2}\right]+2 (n\geq 1)\), which linearly depends on \(n\).Model error in the LANS-alpha and NS-alpha deconvolution models of turbulence.https://www.zbmath.org/1455.652272021-03-30T15:24:00+00:00"Olson, Eric"https://www.zbmath.org/authors/?q=ai:olson.eric-jSummary: This paper reports on a computational study of the model error in the LANS-alpha and NS-alpha deconvolution models of homogeneous isotropic turbulence. Computations are also performed for a new turbulence model obtained as a rescaled limit of the deconvolution model. The technique used is to plug a solution obtained from direct numerical simulation of the incompressible Navier-Stokes equations into the competing turbulence models and to then compute the time evolution of the resulting residual. All computations have been done in two dimensions rather than three for convenience and efficiency. When the effective averaging length scale in any of the models is \(\alpha_0= 0.01\) the time evolution of the root-mean-squared residual error grows as \(\sqrt{t}\). This growth rate similar to what would happen if the model error were given by a stochastic force. When \(\alpha_0= 0.20\) the residual error grows linearly. Linear growth suggests that the model error possesses a systematic bias. Finally, for \(\alpha_0= 0.04\) the residual error in LANS-alpha model exhibited linear growth; however, for this value of \(\alpha_0\) the higher-order alpha models that were tested did not.High-rate secure key distribution based on private chaos synchronization and alternating step algorithms.https://www.zbmath.org/1455.941662021-03-30T15:24:00+00:00"Jiang, Ning"https://www.zbmath.org/authors/?q=ai:jiang.ning"Zhao, Xiaoyan"https://www.zbmath.org/authors/?q=ai:zhao.xiaoyan"Zhao, Anke"https://www.zbmath.org/authors/?q=ai:zhao.anke"Wang, Hui"https://www.zbmath.org/authors/?q=ai:wang.hui.5|wang.hui.6"Qiu, Kun"https://www.zbmath.org/authors/?q=ai:qiu.kun"Tang, Jianming"https://www.zbmath.org/authors/?q=ai:tang.jianmingLocally discrete expanding groups of analytic diffeomorphisms of the circle.https://www.zbmath.org/1455.370292021-03-30T15:24:00+00:00"Deroin, Bertrand"https://www.zbmath.org/authors/?q=ai:deroin.bertrandSummary: We show that a finitely subgroup of \(\operatorname{Diff}^\omega(\mathbf{S}^1)\) that is expanding and locally discrete in the analytic category is analytically conjugated to a uniform lattice in \(\widetilde{\mathrm{PGL}}_2^k(\mathbf{R})\) acting on the \(k\)th covering of \(\mathbf{R}P^1\) for a certain integer \(k>0\).Dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate.https://www.zbmath.org/1455.370712021-03-30T15:24:00+00:00"Cao, Wenjie"https://www.zbmath.org/authors/?q=ai:cao.wenjie"Pan, Tao"https://www.zbmath.org/authors/?q=ai:pan.taoSummary: In this paper, the dynamics of an impulsive stochastic SIR epidemic model with saturated incidence rate are analyzed. The existence and uniqueness of the global positive solution is proved by constructing the equivalent system without pulses. The threshold which determines the extinction and persistence of the disease is obtained. The global attraction of disease-free periodic solution is addressed. Sufficient condition for the existence of a positive periodic solution is established. These results are supported by computer simulations.One-way hash function based on delay-induced hyperchaos.https://www.zbmath.org/1455.941902021-03-30T15:24:00+00:00"Ren, Hai-Peng"https://www.zbmath.org/authors/?q=ai:ren.haipeng"Zhao, Chao-Feng"https://www.zbmath.org/authors/?q=ai:zhao.chaofeng"Grebogi, Celso"https://www.zbmath.org/authors/?q=ai:grebogi.celsoOn topological classification of Morse-Smale diffeomorphisms on the sphere \(S^n\) \((n>3)\).https://www.zbmath.org/1455.370192021-03-30T15:24:00+00:00"Grines, V."https://www.zbmath.org/authors/?q=ai:grines.vyacheslav-z"Gurevich, E."https://www.zbmath.org/authors/?q=ai:gurevich.e-a|gurevich.elena-ya"Pochinka, O."https://www.zbmath.org/authors/?q=ai:pochinka.olga-v"Malyshev, D."https://www.zbmath.org/authors/?q=ai:malyshev.dmitry-sThe paper is about the topological classification of dynamical systems, a domain which takes its origin in several contributions by A. Andronov, L. Pontryagin, E. Leontovich, A. Mayer, and M. Peixoto.
Consider the class \(G(S^n)\) of orientation-preserving Morse-Smale diffeomorphisms of the sphere \(S^n\), \(n>3\), with no intersection between the invariant manifolds of different saddle periodic points.
For any such diffeomorphism \(f\) a coloured graph \(\Gamma _f\) is defined which describes the mutual arrangement of the invariant manifolds of the saddle periodic points of the diffeomorphism \(f\).
The graph \(\Gamma _f\) is enriched by an automorphism \(P_f\) induced by the dynamics of \(f\), and the notion of isomorphism between two coloured graphs is defined.
The paper shows that two diffeomorphisms \(f\), \(f'\in G(S^n)\) are topologically conjugated if and only if the graphs \(\Gamma _f\), \(\Gamma _{f'}\) are isomorphic. It also establishes the existence of a linear-time algorithm which distinguishes coloured graphs of diffeomorphisms from the class \(G(S^n)\).
Reviewer: Vladimir P. Kostov (Nice)A theorem of Tits type for automorphism groups of projective varieties in arbitrary characteristic. With an appendix by Tomohide Terasoma.https://www.zbmath.org/1455.140822021-03-30T15:24:00+00:00"Hu, Fei"https://www.zbmath.org/authors/?q=ai:hu.fei.1|hu.feiSummary: We prove a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by \textit{J. Keum} et al. [Math. Res. Lett. 16, No. 1, 133--148 (2009; Zbl 1172.14025)] for complex projective varieties.Hölder regularity and exponential decay of correlations for a class of piecewise partially hyperbolic maps.https://www.zbmath.org/1455.370312021-03-30T15:24:00+00:00"Bilbao, Rafael A."https://www.zbmath.org/authors/?q=ai:bilbao.rafael-a"Bioni, Ricardo"https://www.zbmath.org/authors/?q=ai:bioni.ricardo"Lucena, Rafael"https://www.zbmath.org/authors/?q=ai:lucena.rafaelSinai billiard maps with Ruelle resonances.https://www.zbmath.org/1455.370282021-03-30T15:24:00+00:00"Thomine, Damien"https://www.zbmath.org/authors/?q=ai:thomine.damienNumerical computations of geometric ergodicity for stochastic dynamics.https://www.zbmath.org/1455.370672021-03-30T15:24:00+00:00"Li, Yao"https://www.zbmath.org/authors/?q=ai:li.yao"Wang, Shirou"https://www.zbmath.org/authors/?q=ai:wang.shirouUnstable metric pressure of partially hyperbolic diffeomorphisms with sub-additive potentials.https://www.zbmath.org/1455.370332021-03-30T15:24:00+00:00"Zhang, Wenda"https://www.zbmath.org/authors/?q=ai:zhang.wenda"Li, Zhiqiang"https://www.zbmath.org/authors/?q=ai:li.zhiqiang.1"Zhou, Yunhua"https://www.zbmath.org/authors/?q=ai:zhou.yunhuaApproach in theory of nonlinear evolution equations: the Vakhnenko-Parkes equation.https://www.zbmath.org/1455.352172021-03-30T15:24:00+00:00"Vakhnenko, V. O."https://www.zbmath.org/authors/?q=ai:vakhnenko.vyacheslav-o"Parkes, E. J."https://www.zbmath.org/authors/?q=ai:parkes.e-johnThe paper studies various methods for examining the properties and solutions of nonlinear evolution equations using the Vakhnenko equation (VE)
\[\frac{\partial}{\partial x}\left(\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}\right)u+u=0,\]
as an example. The above equation arises in modelling the propagation of high-frequency waves in a relaxing medium, and has periodic and solitary traveling wave solutions some of which are loop-like in nature. The equation can be rewritten in an alternative form, known as the Vakhnenko-Parkes equation (VPE), by a change of independent variables. The VPE has an \(N\)-soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises \(N\)-loop-like solitons. Aspects of the inverse scattering transform (IST) method, as applied originally to the KdV equation, are used to find one- and two-soliton solutions to the VPE even though the VPE's spectral equation is third-order and not second-order. A Bäcklund transformation for the VPE is used to construct conservation laws. The standard IST method for third-order spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to \(N\)-soliton solutions and M-mode periodic solutions, respectively. Interactions between these types of solutions are investigated.
Reviewer: Solomon Manukure (Austin)Exponential law for random maps on compact manifolds.https://www.zbmath.org/1455.370062021-03-30T15:24:00+00:00"Haydn, Nicolai T. A."https://www.zbmath.org/authors/?q=ai:haydn.nicolai-t-a"Rousseau, Jérôme"https://www.zbmath.org/authors/?q=ai:rousseau.jerome"Yang, Fan"https://www.zbmath.org/authors/?q=ai:yang.fan.6Typical properties of interval maps preserving the Lebesgue measure.https://www.zbmath.org/1455.370372021-03-30T15:24:00+00:00"Bobok, Jozef"https://www.zbmath.org/authors/?q=ai:bobok.jozef"Troubetzkoy, Serge"https://www.zbmath.org/authors/?q=ai:troubetzkoy.serge-eugeneThe authors prove that, in the class of continuous closed-unit-interval self-maps preserving the Lebesgue measure, each of the following properties is typical (i.e., satisfied by maps forming a subclass which is dense with respect to the uniform metric): being weakly mixing with respect to the Lebesgue measure, being locally eventually onto, satisfying the periodic specification property, having a knot point at almost every point, mapping a set of Lebesgue measure zero to the closed unit interval, and having infinite topological entropy.
Reviewer: Jonathan Hoseana (Bandung)Computability at zero temperature.https://www.zbmath.org/1455.370322021-03-30T15:24:00+00:00"Burr, Michael"https://www.zbmath.org/authors/?q=ai:burr.michael-a"Wolf, Christian"https://www.zbmath.org/authors/?q=ai:wolf.christianThe authors give some answers concerning the computability of basic thermodynamic invariants at zero temperature. They investigate the computability of the residual entropy on the space of continuous potentials for subshifts of finite type (SFTs).
Let \(f : X\rightarrow X\) be a subshift of finite type over an alphabet with \(d\) elements and let \(\mathcal{M}\) be the set of \(f\)-invariant Borel probability measures on \(X\) endowed with the weak* topology. Define \(\mathcal{M}_{\max}(\phi) = \{\mu\in \mathcal{M}: \mu(\phi) = b_{\phi}\}\), where \(\phi:X\rightarrow \mathbb{R}\) is a continuous function. The residual entropy of the potential \(\phi\) is defined by \(h_{\infty,\phi}= \sup\{h_{\mu}(f): \mu\in \mathcal{M}_{\max}(\phi)\}\), where \(h_{\mu}(f)\) is the measure entropy of \(f\) with respect to the measure \(\mu\). The authors prove that the function \(\phi\rightarrow h_{\infty,\phi}\) is upper semi-computable, but not computable on \(C(X,\mathbb{R})\). Moreover, the map \(\phi\rightarrow h_{\infty,\phi}\) is continuous at \(\phi_0\) if and only if \(h_{\infty,\phi_0} = 0\). Therefore, they prove that the residual entropy is semi-computable, but not computable. Then they study locally constant potentials for which the zero-temperature measure is known to exist. Let LC\((X,\mathbb{R}) =\bigcup_{k\in \mathbb{N}}\)LC\(_{k}(X,\mathbb{R})\) denote the space of locally constant potentials, where LC\(_{k}(X,\mathbb{R})\) denotes the space of potentials that are constant on cylinders of length \(k\). The authors prove that if \(f : X\rightarrow X\) is a transitive SFT with positive topological entropy, then the set \(\mathcal{O}\) has no interior points in LC\((X,\mathbb{R})\), where \(\mathcal{O}\) is the set of locally constant potentials that are uniquely maximizing.
Reviewer: Hasan Akin (Gaziantep)Completely degenerate responsive tori in Hamiltonian systems.https://www.zbmath.org/1455.370482021-03-30T15:24:00+00:00"Si, Wen"https://www.zbmath.org/authors/?q=ai:si.wen"Yi, Yingfei"https://www.zbmath.org/authors/?q=ai:yi.yingfeiSpecification properties for non-autonomous discrete systems.https://www.zbmath.org/1455.370182021-03-30T15:24:00+00:00"Salman, Mohammad"https://www.zbmath.org/authors/?q=ai:salman.mohammad"Das, Ruchi"https://www.zbmath.org/authors/?q=ai:das.ruchiIf \((X,d)\) is a compact metric space and \(f_n:X\to X\) is a continuous map for each \(n\in \mathbb{N}\), a nonautonomous discrete dynamical system is defined by \(x_{n+1} = f_n(x_n),~ n \geq 1,\) and denoted by \((X,f_{1,\infty })\), where \(f_{1,\infty }=\{f_n:n\in \mathbb{N} \}\). When \(f_n = f\), for each \(n \in \mathbb{N}\), the system becomes an autonomous system.
The authors introduce the notions of strong specification property and quasi-weak specification property for nonautonomous discrete dynamical systems (recall that \(f_1^j=f_j\circ \cdots \circ f_2 \circ f_1 \)):
1) A nonautonomous system \((X, f_{1,\infty })\) has the strong
specification property (SSP) if for every \(\epsilon > 0\), there exists \(M(\epsilon )\) such that,
for every choice of points \(x_1,x_2,\dots , x_k \in X\) and any sequence \(a_1\leq b_1 < \cdots < a_k \leq b_k\) of non-negative integers with \(a_j - b_{j-1} > M(\epsilon)\), \(2 \leq j \leq k\), and any \(p > M(\epsilon) + b_k - a_1\), there exists a periodic point \(z \in X\) with period \(p\) satisfying
\(d(f_1^j(z), f_1^j(x_i))
< \epsilon \), for all \(a_i \leq j \leq b_i\), \(1 \leq i \leq k\).
2) A nonautonomous system \((X, f_{1,\infty })\) has the quasi-weak specification property (QSP) if for every \(\epsilon > 0\), there exists a positive integer \(M(\epsilon )\) such that for all \(x_1,x_2\in X\) and for every \(n \geq M(\epsilon)\), there is a point \(z \in X \) such that \(d(z, x_1) < \epsilon\) and \(d(f_1^n(z), f_1^n(x_2))
< \epsilon \). Observe that SSP \(\Rightarrow \) QSP.
The authors show that SSP and QSP are dynamical properties which are preserved under finite products. They also prove that a \(k\)-periodic nonautonomous system on intervals having the weak specification is Devaney chaotic. Moreover, it is shown that if the system has strong specification then the result is true in general. Specification properties of induced systems on hyperspaces and probability measures spaces are also studied. Some examples are considered.
Reviewer: Daniel Jardon (Ciudad de México)Topological entropy of switched linear systems: general matrices and matrices with commutation relations.https://www.zbmath.org/1455.930312021-03-30T15:24:00+00:00"Yang, Guosong"https://www.zbmath.org/authors/?q=ai:yang.guosong"Schmidt, A. James"https://www.zbmath.org/authors/?q=ai:schmidt.a-james"Liberzon, Daniel"https://www.zbmath.org/authors/?q=ai:liberzon.daniel"Hespanha, João P."https://www.zbmath.org/authors/?q=ai:hespanha.joao-pedroSummary: This paper studies a notion of topological entropy for switched systems, formulated in terms of the minimal number of trajectories needed to approximate all trajectories with a finite precision. For general switched linear systems, we prove that the topological entropy is independent of the set of initial states. We construct an upper bound for the topological entropy in terms of an average of the measures of system matrices of individual modes, weighted by their corresponding active times, and a lower bound in terms of an active-time-weighted average of their traces. For switched linear systems with scalar-valued state and those with pairwise commuting matrices, we establish formulae for the topological entropy in terms of active-time-weighted averages of the eigenvalues of system matrices of individual modes. For the more general case with simultaneously triangularizable matrices, we construct upper bounds for the topological entropy that only depend on the eigenvalues, their order in a simultaneous triangularization, and the active times. In each case above, we also establish upper bounds that are more conservative but require less information on the system matrices or on the switching, with their relations illustrated by numerical examples. Stability conditions inspired by the upper bounds for the topological entropy are presented as well.WAP systems and labeled subshifts.https://www.zbmath.org/1455.370022021-03-30T15:24:00+00:00"Akin, Ethan"https://www.zbmath.org/authors/?q=ai:akin.ethan"Glasner, Eli"https://www.zbmath.org/authors/?q=ai:glasner.eliThe survey deals with subshift systems in symbolic dynamics. Consider a symbolical dynamical system: \(S: A^{\mathbb Z}\to A^{\mathbb Z}\), \(S\) defining a shift transformation. Typically, a subshift \(B\) (i.e., a closed \(S\)-invariant subset of \(A^{\mathbb Z}\)) is uncountable. If \(B\) is countable, some nice theory can be built.
The main purpose of this work is to present a robust method to construct subshifts which are then used to build WAP (weakly almost periodic) systems with various properties. As an application of this strategy many examples are explicitly given and discussed.
Reviewer: Alexei Kanel-Belov (Ramat-Gan)Transfer operator approach to ray-tracing in circular domains.https://www.zbmath.org/1455.370682021-03-30T15:24:00+00:00"Slipantschuk, J."https://www.zbmath.org/authors/?q=ai:slipantschuk.julia"Richter, Martin"https://www.zbmath.org/authors/?q=ai:richter.martin"Chappell, David J."https://www.zbmath.org/authors/?q=ai:chappell.david-j"Tanner, Gregor"https://www.zbmath.org/authors/?q=ai:tanner.gregor-k"Just, W."https://www.zbmath.org/authors/?q=ai:just.wolfram"Bandtlow, O. F."https://www.zbmath.org/authors/?q=ai:bandtlow.oscar-fOn the linearization of vector fields on a torus with prescribed frequency.https://www.zbmath.org/1455.370492021-03-30T15:24:00+00:00"Zhang, Dongfeng"https://www.zbmath.org/authors/?q=ai:zhang.dongfeng"Xu, Xindong"https://www.zbmath.org/authors/?q=ai:xu.xindongThe authors consider a vector field of the form \(X = N + P,\) where \(N = \omega(\xi)\) is a vector on the torus \(\mathbb{T}^n = \mathbb{R}^n/2\pi\mathbb{Z}^n\), describing uniform rotation motions with frequency \(\omega (\xi) = (\omega_1(\xi),\dots,\omega_n(\xi)),\) \(P(\theta, \xi)\) is a small perturbation, and the parameters \(\xi\) vary on some bounded closed connected domain \(\Pi.\) Let \(\Delta \colon [1,\infty) \to [1,\infty)\) be a continuous increasing unbounded function such that \(\Delta (1) = 1\) and \(\int_1^\infty \frac{\ln \Delta (t)}{t^2}dt < \infty\), \(D(s) = \{\theta \in \mathbb{C}^n/2\pi \mathbb{Z}^n \colon |Im \theta_j| \leq s, j - 1,\dots,n \},\) and \(\Pi_h = \{\xi \in \mathbb{C}^n \colon \mathrm{dist} (\xi, \Pi) \leq h \}.\)
The main result is as follows. Suppose that \(X\) is real analytic on \(D(s) \times \Pi_h\) and for \(\omega_0 = \omega (\xi_0)\) the following non-resonant condition
\[
|\langle k, \omega_0\rangle | \geq \frac{\alpha}{\Delta (|k|)} \quad \forall \,0 \neq k \in \mathbb{Z}^n
\]
is satisfied with \(\alpha >0\) and the Brouwer degree of \(\omega (\xi)\) at \(\xi_0\) on \(\Pi\) is non-zero. Then for a sufficiently small \(\|P\|_{D(s)\times\Pi_h} \neq 0\) there exists a real analytic diffeomorphism \(\Phi_{\omega_0}\) such that \(\Phi_{\omega_0}^\star(N+P) = N_\star\), and at least one \(\xi_\star \in \Pi\) such that the conjugated vector field \(N_\star\) at \(\xi = \xi_\star\) has a linear flow with \(\omega_0\) as its frequency.
Reviewer: Valerii V. Obukhovskij (Voronezh)Preservation of bifurcations of Hamiltonian boundary value problems under discretisation.https://www.zbmath.org/1455.651142021-03-30T15:24:00+00:00"McLachlan, Robert I."https://www.zbmath.org/authors/?q=ai:mclachlan.robert-i"Offen, Christian"https://www.zbmath.org/authors/?q=ai:offen.christianSummary: We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that non-symplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic or non-symplectic integrators, but in some circumstances symplecticity greatly reduces the error.Rauzy substitution and local structure of torus tilings.https://www.zbmath.org/1455.370172021-03-30T15:24:00+00:00"Zhukova, Alla Adol'fovna"https://www.zbmath.org/authors/?q=ai:zhukova.alla-adolfovna"Shutov, Anton Vladimirovich"https://www.zbmath.org/authors/?q=ai:shutov.anton-vSummary: For any irrational \(\alpha\) we can consider tilings of the segment \([0;1]\) by the points \(\{i\alpha\}\) with \(0\leq i<n\). These tilings have some interesting properties, including well-known three lengths and three gaps theorems. In particular, these tilings contain segments of either two, or three different lengths. In the case of two lengths, the corresponding tilings are known as generalized Fibonacci tilings. They are closely connected with the combinatorics of words, one-dimensional quasiperiodic tilings, bounded remainder sets, first return maps for irrational circle rotations, etc.
Transferring the general three lengths and three gaps theorems to two-dimensional case, i.e. to the points \((\{i\alpha_1\},\{i\alpha_2\})\) is a well-known open problem. In the present work we consider a special case of this problem. associated with two-dimensional generalizations of Fibonacci tilings. These tilings are obtained using iterations of the geometric version of the famous Rauzy substitution. They arise in the words combinatorics in the study of generalizations of Sturmian sequences, as well as in number theory in the study of toric shifts. Considered tilings consist of rhombuses of three different types. It is proved that there are exactly 9 types of sets of rhombuses adjacent to a given rhombus. Also we obtain a method that allows explicitly determine all neighbours of the given rhombus. The results can be considered as a first step to a multidimensional generalization of the three lengths and three gaps theorems.On uniformly recurrent words generated by shifting segments, including with a change in orientation.https://www.zbmath.org/1455.370112021-03-30T15:24:00+00:00"Kanel'-Belov, Alekseĭ Yakovlev"https://www.zbmath.org/authors/?q=ai:kanel-belov.aleksei-yakovlev"Chernyat'ev, Aleksandr Leonidovich"https://www.zbmath.org/authors/?q=ai:chernyatev.aleksandr-leonidovichSummary: The work is devoted to the review of some problems of symbolic dynamics. The description of uniformly recurrent words associated with the shifting of segments is given. Let \(W\) be an infinite word over finite alphabet \(A\). We get combinatorial criteria of existence of interval exchange transformations that generate the word \(W\).
In this paper we study words generated by general piecewise-continuous transformation of the interval. Further we prove equivalence set words generated by piecewise-continuous transformation and words generated by interval exchange transformation. This method get capability of descriptions of the words generated by arbitrary interval exchange transformation.
This work is targeted to the following
Inverse problem: Which conditions should be imposed on a uniformly recurrent word \(W\) in order that it be generated by a dynamical system of the form \((I,T,U_1,\ldots,U_k)\), where \(I\) is the unit interval and \(T\) is the interval exchange transformation?
The answer to this question is given in terms of the evolution of the \textit{labeled Rauzy graphs} of the word \(W\). The \textit{Rauzy graph} of order \(k\) (the \(k\)-graph) of the word \(W\) is the directed graph whose vertices biuniquely correspond to the factors of length \(k\) of the word \(W\) and there exists an arc from vertex \(A\) to vertex \(B\) if and only if \(W\) has a factor of length \(k+1\) such that its first \(k\) letters make the subword that corresponds to \(A\) and the last \(k\) symbols make the subword that corresponds to \(B\). By the \textit{follower} of the directed \(k\)-graph \(G\) we call the directed graph \(\operatorname{Fol}(G)\) constructed as follows: the vertices of graph \(\operatorname{Fol}(G)\) are in one-to-one correspondence with the arcs of graph \(G\) and there exists an arc from vertex \(A\) to vertex \(B\) if and only if the head of the arc \(A\) in the graph \(G\) is at the notch end of \(B\). The \((k+1)\)-graph is a subgraph of the follower of the \(k\)-graph; it results from the latter by removing some arcs. Vertices which are tails of (or heads of) at least two arcs correspond to \textit{special factors} (see Section 2); vertices which are heads and tails of more than one arc correspond to \textit{bispecial factors}. The sequence of the Rauzy \(k\)-graphs constitutes the \textit{evolution} of the Rauzy graphs of the word \(W\). The Rauzy graph is said to be \textit{labeled} if its arcs are assigned letters \(l\) and \(r\) and some of its vertices (perhaps, none of them) are assigned symbol ``-''.
The \textit{follower} of the labeled Rauzy graph is the directed graph which is the follower of the latter (considered a Rauzy graph with the labeling neglected) and whose arcs are labeled according to the following rule:
\begin{itemize}
\item[1.] Arcs that enter a branching vertex should be labeled by the same symbols as the arcs that enter any left successor of this vertex;
\item[2.] Arcs that go out of a branching vertex should be labeled by the same symbols as the arcs that go out of any right successor of this vertex;
\item[3.] If a vertex is labeled by symbol ``-'', then all its right successors should also be labeled by symbol ``-''.
\end{itemize}
In terms of Rauzy labeled graphs we define the \textit{asymptotically correct} evolution of Rauzy graphs, i.e., we introduce rules of passing from \(k\)-graphs to \((k+1)\)-graphs. Namely, the evolution is said to be \textit{correct} if, for all \(k\geq 1\), the following conditions hold when passing from the \(k\)-graph \(G_k\) to the \((k+1)\)-graph \(G_{k+1} \) :
\begin{itemize}
\item[1.] The degree of any vertex is at most 2, i.e., it is incident to at most two incoming and outgoing arcs;
\item[2.] If the graph contains no vertices corresponding to bispecial factors, then \(G_{n+1}\) coincides with the follower \(D(G_n)\);
\item[3.] If the vertex that corresponds to a bispecial factor is not labeled by symbol ``-'', then the arcs that correspond to forbidden words are chosen among the pairs \(lr\) and \(rl\);
\item[4.] If the vertex is labeled by symbol ``-'', then the arcs to be deleted should be chosen among the pairs \(ll\) or \(rr\).
\end{itemize}
The evolution is said to be \textit{asymptotically correct} if this condition is valid for all \(k\) beginning with a certain \(k=K\). The \textit{oriented} evolution of the graphs means that there are no vertices labeled by symbol ``-''. The main result of this work consists in the description of infinite words generated by interval exchange transformations (and answers a \textit{G. Rouzy} question [``Mots infinis en arithmetique'', Lect. Notes Comput. Sci. 192, 164--171 (1984; \url{doi:10.1007/3-540-15641-0_32})]):
Main theorem. A uniformly recurrent word \(W\)
\begin{itemize}
\item[1.] is generated by an interval exchange transformation if and only if the word is provided with the asymptotically correct evolution of the labeled Rauzy graphs;
\item[2.] is generated by an orientation-preserving interval exchange transformation if and only if the word is provided with the asymptotically correct oriented evolution of the labeled Rauzy graphs.
\end{itemize}Further studies on limit cycle bifurcations for piecewise smooth near-Hamiltonian systems with multiple parameters.https://www.zbmath.org/1455.370452021-03-30T15:24:00+00:00"Han, Maoan"https://www.zbmath.org/authors/?q=ai:han.maoan"Liu, Shanshan"https://www.zbmath.org/authors/?q=ai:liu.shanshanSummary: This paper investigates the limit cycle bifurcations for piecewise smooth near-Hamiltonian systems with multiple parameters. The formulas for the second and third term in expansions of the first order Melnikov function are derived respectively. The main results improve some known conclusions.Sofic mean length.https://www.zbmath.org/1455.160012021-03-30T15:24:00+00:00"Li, Hanfeng"https://www.zbmath.org/authors/?q=ai:li.hanfeng"Liang, Bingbing"https://www.zbmath.org/authors/?q=ai:liang.bingbingLet \(R\) be a ring admitting a length function on \(R\)-modules, that is, a function \(L\) from left \(R\)-modules \(M\) to the extended non-negative reals such that \(L(R)=1\) and if \(N\) is a submodule of \(M\) then \(L(M)=L(N)+L(M/N)\). For a group \(\Gamma\), a mean length function \(mL\) on \(R\Gamma\)-modules is length function which is invariant under the action of \(\Gamma\), that is, \(mL(R\Gamma\otimes_R N)=L(N)\) for all \(R\)-modules \(N\).
A group \(\Gamma\) is sofic if for every finite subset \(F\) of \( \Gamma\) and every \(\varepsilon >0\), there exist \(n\in\mathbb N\) and a map \(\phi\colon F\to\mathrm{Sym}(n)\) whose only fixed point is \(e\), such that for all \(g,\, h\in F\) such that \(gh\in F\), the Hamming distance \(d(\phi(gh),\,\phi(g)\phi (h))<\varepsilon\). This property is a weak finiteness condition on \(\Gamma\) that generalises both amenability and residual finiteness.
This deep and significant paper concerns the construction and algebraic properties of a mean length function on the \(R\Gamma\)-modules for unital rings \(R\) and sofic groups \(\Gamma\). For example, the authors show that if \(R\) is Noetherian and \(\Gamma\) is sofic, them \(R\Gamma\) is stably finite, that is, for all \(n,\ M_n\) is directly finite. They show also that if \(M\) is a \(\mathbb Z\Gamma\) module, the mean topological dimension of the induced \(\Gamma\)-action on the Pontryagin dual of \(M\) coincides with the von Neumann-Lück rank of \(M\).
Reviewer: Phillip Schultz (Perth)On the existence of full dimensional KAM torus for fractional nonlinear Schrödinger equation.https://www.zbmath.org/1455.370612021-03-30T15:24:00+00:00"Wu, Yuan"https://www.zbmath.org/authors/?q=ai:wu.yuan"Yuan, Xiaoping"https://www.zbmath.org/authors/?q=ai:yuan.xiaopingSummary: In this paper, we study fractional nonlinear Schrödinger equation (FNLS) with periodic boundary condition
\[
\mathbf{i}u_t=-(-\Delta)^{s_0} u-V*u-\epsilon f(x)|u|^4u, \quad x\in \mathbb{T}, \quad t\in \mathbb{R}, \quad s_0\in (\frac12,1),\tag{0.1}
\]
where \((-\Delta)^{s_0}\) is the Riesz fractional differentiation defined in [\textit{A. A. Kilbas} et al., Theory and applications of fractional differential equations. Amsterdam: Elsevier (2006; Zbl 1092.45003)] and \(V*\) is the Fourier multiplier defined by \(\widehat{V*u}(n)=V_n\widehat{u}(n)\), \(V_n\in\left[-1,1\right]\), and \(f(x)\) is Gevrey smooth. We prove that for \(0\leq|\epsilon|\ll1\) and appropriate \(V\), the equation (0.1) admits a full dimensional KAM torus in the Gevrey space satisfying \(\frac12e^{-rn^{\theta}}\leq \left|q_n\right|\leq 2e^{-rn^{\theta}}\), \(\theta\in (0,1)\), which generalizes the results given by \textit{J. Bourgain} [J. Funct. Anal. 229, No. 1, 62--94 (2005; Zbl 1088.35061)], \textit{H. Cong} et al. [J. Differ. Equations 264, No. 7, 4504--4563 (2018; Zbl 1382.37079)] and \textit{H. Cong} et al. [Discrete Contin. Dyn. Syst. 39, No. 11, 6599--6630 (2019; Zbl 07107992)] to fractional nonlinear Schrödinger equation.Mahler coefficients of \(1\)-Lipschitz measure-preserving functions on \(\mathbb{Z}_p\).https://www.zbmath.org/1455.111602021-03-30T15:24:00+00:00"Memić, Nacima"https://www.zbmath.org/authors/?q=ai:memic.nacimaData driven governing equations approximation using deep neural networks.https://www.zbmath.org/1455.651252021-03-30T15:24:00+00:00"Qin, Tong"https://www.zbmath.org/authors/?q=ai:qin.tong"Wu, Kailiang"https://www.zbmath.org/authors/?q=ai:wu.kailiang"Xiu, Dongbin"https://www.zbmath.org/authors/?q=ai:xiu.dongbinSummary: We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods.Identifying logarithmic tracts.https://www.zbmath.org/1455.300202021-03-30T15:24:00+00:00"Waterman, James"https://www.zbmath.org/authors/?q=ai:waterman.jamesLet \(D\) be a direct tract of a holomorphic function \(f\). It is proved, if the boundary of \(D\) is an unbounded simple
curve, then \(D\) is a logarithmic tract, i.e., the restriction \(f:D\rightarrow\{z\in \mathbb{C}: |z|>R\}\) is a universal covering, where \(R\) is the boundary value of the direct tract. With the additional assumption that there are no asymptotic paths in a logarithmic tract \(D\) with finite asymptotic values, then the following converse to this assertion is true.
Theorem 1.2. Let \(D\) be a logarithmic tract containing no asymptotic paths
with finite asymptotic values. Then \(D\) is bounded by a single unbounded curve.
Further, if \(D\) is a logarithmic tract with boundary value \(R\), then for all \(R'>R\),
\(\{z\in D:|f(z)|>R'\}\) is a logarithmic tract bounded by a simple curve.
As an application of these results, it is shown that an example of a function with infinitely many direct singularities,
but no logarithmic singularity over any finite value, is in the Eremenko-Lyubich class.
Reviewer: Konstantin Malyutin (Kursk)Painlevé analysis for various nonlinear Schrödinger dynamical equations.https://www.zbmath.org/1455.352312021-03-30T15:24:00+00:00"Ali, Ijaz"https://www.zbmath.org/authors/?q=ai:ali.ijaz"Seadawy, Aly R."https://www.zbmath.org/authors/?q=ai:seadawy.aly-r"Rizvi, Syed Tahir Raza"https://www.zbmath.org/authors/?q=ai:rizvi.syed-tahir-raza"Younis, Muhammad"https://www.zbmath.org/authors/?q=ai:younis.muhammadDegrees of iterates of rational maps on normal projective varieties.https://www.zbmath.org/1455.140282021-03-30T15:24:00+00:00"Dang, Nguyen-Bac"https://www.zbmath.org/authors/?q=ai:dang.nguyen-bacSummary: Let \(X\) be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of \(X\), recovering former results by
\textit{T.-C. Dinh} and \textit{N. Sibony} [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 959--971 (2004; Zbl 1074.53058)], and by
\textit{T. T. Truong} [J. Reine Angew. Math. 758, 139--182 (2020; Zbl 07148358)]. Precisely, we give a new proof of the submultiplicativity properties of these degrees and of their birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by
\textit{J. Xiao} [Ann. Inst. Fourier 65, No. 3, 1367--1379 (2015; Zbl 1333.32025)] and
\textit{D. Popovici} [Math. Ann. 364, No. 1--2, 649--655 (2016; Zbl 1341.32015)], which generalizes
Siu's inequality (see [\textit{S. Trapani}, Math. Z. 219, No. 3, 387--401 (1995; Zbl 0828.14002)] to algebraic cycles of arbitrary codimension. This allows us to show that the degree of a map is controlled up to a uniform constant by the norm of its action by pull-back on the space of numerical classes in \(X\).Painlevé analysis of a nonlinear Schrödinger equation discussing dynamics of solitons in optical fiber.https://www.zbmath.org/1455.352412021-03-30T15:24:00+00:00"Rizvi, Syed T. R."https://www.zbmath.org/authors/?q=ai:rizvi.syed-tahir-raza"Seadawy, Aly R."https://www.zbmath.org/authors/?q=ai:seadawy.aly-r"Ali, Ijaz"https://www.zbmath.org/authors/?q=ai:ali.ijaz"Younis, Muhammad"https://www.zbmath.org/authors/?q=ai:younis.muhammadA characterization of multiplicity-preserving global bifurcations of complex polynomial vector fields.https://www.zbmath.org/1455.370422021-03-30T15:24:00+00:00"Dias, Kealey"https://www.zbmath.org/authors/?q=ai:dias.kealeySummary: For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree \(d\), it is proved that any bifurcation which preserves the multiplicity of equilibrium points admits a decomposition into a finite number of elementary bifurcations, and the elementary bifurcations are characterized.Polarized symplectic structures.https://www.zbmath.org/1455.530892021-03-30T15:24:00+00:00"Awane, Azzouz"https://www.zbmath.org/authors/?q=ai:awane.azzouzSummary: We study various aspects and properties of polarized symplectic manifolds. We give a new proof of the Darboux theorem for symplectic manifolds equipped with Lagrangian foliation using only quadratures.
We give a special interest to the study of Poisson structures subordinate to a real polarization, in this case, the polarized Hamiltonians are locally affine mappings with respect to the affine structure of the Lagrangian foliation. Also, we show that polarized Hamiltonians consist of all real smooth mappings whose the associated vector Hamiltonian field preserves the Lagrangian foliation. We give some examples and properties of these objects.
The study of the integrability of the almost polarized symplectic manifolds are given in this work.The order-\(n\) breather and degenerate breather solutions of the \((2+1)\)-dimensional cmKdV equations.https://www.zbmath.org/1455.352292021-03-30T15:24:00+00:00"Yuan, Feng"https://www.zbmath.org/authors/?q=ai:yuan.fengDiscrete geodesic flows on Stiefel manifolds.https://www.zbmath.org/1455.530942021-03-30T15:24:00+00:00"Jovanović, Božidar"https://www.zbmath.org/authors/?q=ai:jovanovic.bozidar-zarko"Fedorov, Yuri N."https://www.zbmath.org/authors/?q=ai:fedorov.yuri-nSummary: We study integrable discretizations of geodesic flows of Euclidean metrics on the cotangent bundles of the Stiefel manifolds \(V_{n,r}\). In particular, for \(n=3\) and \(r=2\), after the identification \(V_{3,2}\cong\operatorname{SO}(3)\), we obtain a discrete analog of the Euler case of the rigid body motion corresponding to the inertia operator \(I=(1,1,2)\). In addition, billiard-type mappings are considered; one of them turns out to be the ``square root'' of the discrete Neumann system on \(V_{n,r} \).Inverse scattering for nonlocal reverse-space multicomponent nonlinear Schrödinger equations.https://www.zbmath.org/1455.352392021-03-30T15:24:00+00:00"Ma, Wen-Xiu"https://www.zbmath.org/authors/?q=ai:ma.wen-xiu"Huang, Yehui"https://www.zbmath.org/authors/?q=ai:huang.yehui"Wang, Fudong"https://www.zbmath.org/authors/?q=ai:wang.fudongStability of dipole gap solitons in two-dimensional lattice potentials.https://www.zbmath.org/1455.370622021-03-30T15:24:00+00:00"Dror, Nir"https://www.zbmath.org/authors/?q=ai:dror.nir"Malomed, Boris A."https://www.zbmath.org/authors/?q=ai:malomed.boris-aSummary: This chapter presents, for both self-focusing (SF) and self-defocusing (SDF) signs of the nonlinearity, the stability analysis for families of 2D gap solitons (GSs), in the first and second bandgaps, which are different from the ``standard'' extensively studied fundamental solitons and vortices supported by the SDF nonlinearity. The chapter introduces the model. It starts with the well-known 2D Gross-Pitaevskii/nonlinear Schrödinger equation for the mean-field complex wave function in the Bose-Einstein condensate (BEC). The chapter presents families of GSs residing in the first finite bandgap, under the SF nonlinearity. A stable subfamily of dipole solitons is identified. Bound states of the dipoles are also introduced and analyzed. A similar analysis is presented for dipole GSs in the second finite bandgap under the SDF nonlinearity. In addition, a stability region is found for a new GS family, which is sustained in the second bandgap under the SF nonlinearity.
For the entire collection see [Zbl 1280.37002].Using a CAS/DGS to analyze computationally the configuration of planar bar linkage mechanisms based on partial Latin squares.https://www.zbmath.org/1455.050142021-03-30T15:24:00+00:00"Falcón, Raúl M."https://www.zbmath.org/authors/?q=ai:falcon.raul-mSummary: Currently, the study of new isotopism invariants of partial Latin squares constitutes an open and active problem. This paper delves into this topic by analyzing computationally the configuration of planar bar linkage mechanisms for which the array formed by the lengths of their bars constitutes an empty-diagonal symmetric partial Latin square such that each one of its rows and columns has at least two non-empty cells. These assumptions enable one to define a series of algebraic and geometric constraints that can be readily implemented in any Computer Algebra or Dynamic Geometry System. In order to illustrate the different concepts and results introduced throughout the paper, it is explicitly determined and characterized the distribution of planar bar linkage mechanisms based on partial Latin squares of order up to five.Assouad dimension of planar self-affine sets.https://www.zbmath.org/1455.280052021-03-30T15:24:00+00:00"Bárány, Balázs"https://www.zbmath.org/authors/?q=ai:barany.balazs"Käenmäki, Antti"https://www.zbmath.org/authors/?q=ai:kaenmaki.antti"Rossi, Eino"https://www.zbmath.org/authors/?q=ai:rossi.einoSummary: We calculate the Assouad dimension of a planar self-affine set \(X\) satisfying the strong separation condition and the projection condition and show that \(X\) is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set \(X\) adheres to very strong tangential regularity by showing that any two points of \(X\), which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.The computation of overlap coincidence in Taylor-Socolar substitution tiling.https://www.zbmath.org/1455.520212021-03-30T15:24:00+00:00"Akiyama, Shigeki"https://www.zbmath.org/authors/?q=ai:akiyama.shigeki"Lee, Jeong-Yup"https://www.zbmath.org/authors/?q=ai:lee.jeong-yupSummary: Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of the authors [Adv. Math. 226, No. 4, 2855--2883 (2011; Zbl 1219.37013)] to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has a quasicrystalline structure.Harmonic measure: algorithms and applications.https://www.zbmath.org/1455.300152021-03-30T15:24:00+00:00"Bishop, Christopher J."https://www.zbmath.org/authors/?q=ai:bishop.christopher-jThis survey article discusses several results in the field of planar harmonic measure, starting from \textit{N. G. Makarov}'s results [Proc. Lond. Math. Soc. (3) 51, 369--384 (1985; Zbl 0573.30029)] to recent applications involving 4-manifolds, dessins d'enfants and transcendental dynamics. Various areas from analysis, topology and algebra that are influenced by harmonic measure are illustrated.
For the entire collection see [Zbl 1437.00045].
Reviewer: Marius Ghergu (Dublin)Scalar field model applied to the lamellar to the inverse hexagonal phase transition in lipid systems.https://www.zbmath.org/1455.370742021-03-30T15:24:00+00:00"Mendanha, Sebastião A."https://www.zbmath.org/authors/?q=ai:mendanha.sebastiao-a"Cardoso, Wesley B."https://www.zbmath.org/authors/?q=ai:cardoso.wesley-b"Avelar, Ardiley T."https://www.zbmath.org/authors/?q=ai:avelar.ardiley-t"Bazeia, Dionisio"https://www.zbmath.org/authors/?q=ai:bazeia.dionisioSummary: In this paper we use a phenomenological field theory model to study the first-order phase transition from the lamellar phase to the inverse hexagonal phase in specific lipid bilayers. The model is described by a real scalar field with potential that supports both symmetric and asymmetric phase conformations. We adapt the coordinate and parameters of the model to describe the phase transition, and we show that the model is capable of correctly inferring the fraction of the inverse hexagonal phase in the phase transition, suggesting an alternative way to be couple to experimental techniques generally required for \(H_{II}\)-phase characterization.A coupled \((2+1)\)-dimensional mKdV system and its nonlocal reductions.https://www.zbmath.org/1455.370552021-03-30T15:24:00+00:00"Zhu, Xiaoming"https://www.zbmath.org/authors/?q=ai:zhu.xiaomingSummary: In this paper, we propose a coupled \((2+1)\)-dimensional modified Korteweg-de Vries (mKdV) system, which admits two kinds of nonlocal reductions. By constructing the Darboux transformation for the considered equation, a variety of exact solutions, such as soliton, soliton-type, kink, kink-type, rational, rogue wave and lump solutions are given explicitly.Non-explosion by Stratonovich noise for ODEs.https://www.zbmath.org/1455.600762021-03-30T15:24:00+00:00"Maurelli, Mario"https://www.zbmath.org/authors/?q=ai:maurelli.marioSummary: We show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension \(d\ge 2\). We also show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE.Analytic deformations of pencils and integrable one-forms having a first integral.https://www.zbmath.org/1455.370442021-03-30T15:24:00+00:00"Scárdua, Bruno"https://www.zbmath.org/authors/?q=ai:scardua.bruno|scardua.bruno-c-azevedoThe paper addresses the problem of persistence of a first integral for a holomorphic integrable one-form in a neighborhood of a singular point.
The author considers integrable analytic deformations of codimension-one holomorphic foliations. Let \(\Omega(x,t)\) be a holomorphic one-form defined in a connected product neighborhood \(U \times D\subset \mathbb{C}^n\times\mathbb{C}\), of the origin of \(\mathbb{C}^{n+1}\), where \(D\subset\mathbb{C}\) is a disc centered at the origin, such that \(\Omega^0:=\Omega(x,0)\) admits a first integral of holomorphic or meromorphic type. For each parameter \(t\in D\), \(\Omega^t\) denotes the restriction \(\Omega^t:=\Omega(x,t)\). The author considers the case where each \(\Omega^t\) is integrable. The family of codimension-one foliations \(\{\mathcal{F}_t:\Omega^t=0~\text{in}~U\}_{t\in(\mathbb{C},0)}\) can be seen as an analytic deformation in \(U\) of the foliation \(\mathcal{F}_0\) which, by hypothesis, admits a holomorphic (or meromorphic) first integral \(f\).
In the first part of the paper, the author studies the case where the foliation \(\mathcal{F}_0\) has a holomorphic first integral \(f\) at the origin in \(\mathbb{C}^n\) with \(n\ge 3\), under the assumption that the germ \(f\) is irreducible and that its typical fiber is simply-connected. In Theorem 1.1, the author first proves that if the codimension of the singular locus of \(\Omega^0\) is greater than or equal to \(2\), then the germ of the foliation \(\mathcal{F}_t\) also has a holomorphic first integral. Then, for the general case, that is when the codimension of the singular locus of \(\Omega^0\) is greater than or equal to \(1\), the author obtains a two-dimensional normal form for the foliation \(\mathcal{F}_t\).
The second part of the paper deals with analytic deformations \(\{\mathcal{F}_t\}_{t\in(\mathbb{C},0)}\) of a local pencil \(\mathcal{F}_0:\frac{f}{g}=\hbox{constant}\) for \(f,g\) belonging to the ring \(\mathcal{O}_n\) of germs of holomorphic functions at the origin of \(\mathbb{C}^n\). In dimension \(n=2\), the author considers \(f=x\) and \(g=y\) and assumes that the axes are invariant for each foliation \(\mathcal{F}_t\).
In higher dimension \(n\ge 3\), the author works under some generic geometric conditions on the germs \(f\) and \(g\). In both cases, the author proves the following (Theorems 1.2 and 1.3):
1) For an analytic deformation there is a multiform formal first integral of type \(\widehat F = \frac{f^{1+\widehat\lambda(t)}}{g^{1+\widehat\mu(t)}} e^{\widehat H(x,y,t)}\) with \(\widehat\lambda(t)\) and \(\widehat\mu(t)\) formal series and some properties on \(\widehat H(x,y,t)\);
2) For an integrable deformation there is a meromorphic first integration of the form \(M = \frac{f}{g} e^{P(t)+H(x,y,t)}\) with \(P(t)\) holomorphic and some additional properties on \(H(x,y,t)\).
Reviewer: Jasmin Raissy (Toulouse)An interpolation lemma.https://www.zbmath.org/1455.111592021-03-30T15:24:00+00:00"Cantat, Serge"https://www.zbmath.org/authors/?q=ai:cantat.sergeSummary: L'inégalité ultramétrique valable dans le monde $p$-adique annihile certains problèmes de convergence que l'on rencontre fréquemment en analyse complexe. Ce texte illustre ce phénomène en présentant un lemme d'interpolation dû à \textit{J. P. Bell} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 367--379 (2006; Zbl 1147.11020)] et \textit{B. Poonen} [Bull. Lond. Math. Soc. 46, No. 3, 525--527 (2014; Zbl 1291.11135)] qui peut être utilisé pour étudier les transformations algébriques, et les groupes qu'elles engendrent.
For the entire collection see [Zbl 1429.00034].Null systems in the non-minimal case.https://www.zbmath.org/1455.370102021-03-30T15:24:00+00:00"Qiu, Jiahao"https://www.zbmath.org/authors/?q=ai:qiu.jiahao"Zhao, Jianjie"https://www.zbmath.org/authors/?q=ai:zhao.jianjieBy topological dynamical system we mean a pair \((X,T)\) consisting of a compact metric space \(X\) and a continuous map \(T : X \to X\). \textit{W. Huang} et al. [Ergodic Theory Dyn. Syst. 23, 1505--1523 (2003; Zbl 1134.37308)] proved that every minimal distal null system is equicontinuous.
In this paper it is shown that the minimality assumption is superfluous, that is, every distal null system is equicontinuous. Actually, it is even proved that every non-equicontinuous distal system has infinite sequence entropy. It is also shown that every null system with closed proximal relation is mean equicontinuous. As a consequence, every null system with dense minimal points is mean equicontinuous. Finally, an example of a distal system with trivial \(\mathrm{Ind}_{\mathrm{fip}}\)-pairs and a non-trivial regionally proximal relation of order \(\infty\) is constructed.
Reviewer: Nilson C. Bernardes Jr. (Rio de Janeiro)Higher prolongations of control affine systems: absolute stability and generalized recurrence.https://www.zbmath.org/1455.370792021-03-30T15:24:00+00:00"Marques, André L."https://www.zbmath.org/authors/?q=ai:marques.andre-l"Tozatti, Hélio V. M."https://www.zbmath.org/authors/?q=ai:tozatti.helio-v-m"Souza, Josiney A."https://www.zbmath.org/authors/?q=ai:souza.josiney-aVariations around Eagleson's theorem on mixing limit theorems for dynamical systems.https://www.zbmath.org/1455.370042021-03-30T15:24:00+00:00"Gouëzel, Sébastien"https://www.zbmath.org/authors/?q=ai:gouezel.sebastienLet \(T\) be an ergodic probability-preserving transformation on a probability space \((X,m)\). Let \(f:X\rightarrow \mathbb{R}\) be a measurable function. The renormalized Birkhoff sums are \(S_n f=\sum_{k=0}^{n-1}f\circ T^k\). In this present paper, taking into account the renormalized Birkhoff sums, the author deals with two variations of \textit{G. K. Eagleson}'s theorem [Theory Probab. Appl. 21, 637--643 (1976; Zbl 0365.60025)].
Firstly, the author gives a general argument to show that it is always equivalent to have an almost sure limit theorem for an invariant probability measure. A version of Eagleson's result that applies to almost sure limit theorems is presented. Let \((X,T)\) be a
measurable map on a standard measurable space. Consider a \(\sigma\)-finite measure \(\mu\) for which \(T\) is non-singular and ergodic. Let \(m_1\) and \(m_2\) be two probability measures that are absolutely continuous with respect to \(\mu\). In this case, the author proves that \(m_1\) and \(m_2\) can be coupled along orbits. Secondly, the author investigates the distributional limit theorems for \(S_n f(x)/B_n\) when one conditions on the positions of both \(x\) and \(T^nx\). He proves that the random variables \(S_n f(x)/B_n\) on the probability spaces \((X, m_n/m_n(X))\) converge in distribution to a real random variable \(Z\), where \(m_n\) is a sequence of measures satisfied some condition.
Reviewer: Hasan Akin (Gaziantep)Noncoercive Lyapunov functions for input-to-state stability of infinite-dimensional systems.https://www.zbmath.org/1455.352742021-03-30T15:24:00+00:00"Jacob, Birgit"https://www.zbmath.org/authors/?q=ai:jacob.birgit"Mironchenko, Andrii"https://www.zbmath.org/authors/?q=ai:mironchenko.andrii"Partington, Jonathan R."https://www.zbmath.org/authors/?q=ai:partington.jonathan-r"Wirth, Fabian"https://www.zbmath.org/authors/?q=ai:wirth.fabian-rogerThe topic of the paper is the input-to-state stability (ISS) of a class of infinite-dimensional dynamical systems, which the authors call forward complete control systems. This class covers a wide range of infinite-dimensional systems.
For this class the authors define several stability and regularity notions and prove a characterization of ISS in terms of these concepts.
As one of the classical tools for studying stability is the notion of a Lyapunov function, the authors give a definition of a noncoercive ISS Lyapunov function and prove that existence of such a function for a forward complete control system implies ISS provided the systems satisfies additional regularity properties. This results is then used to propose a method for constructing ISS Lyapunov functions for linear systems with unbounded input operators in the case when the input space is \(L^{\infty}(\mathbb{R}_+,U)\), where \(U\) is a Banach space. This method is then used to construct ISS Lyapunov function in some specific cases, including the case of a 1D heat equation with Dirichlet boundary conditions.
Reviewer: Ivica Nakić (Zagreb)Equivalent notions of hyperbolicity.https://www.zbmath.org/1455.370302021-03-30T15:24:00+00:00"Barreira, Luis"https://www.zbmath.org/authors/?q=ai:barreira.luis-m"Valls, Claudia"https://www.zbmath.org/authors/?q=ai:valls.claudiaThe main aim of the paper is to give a characterization of the hyperbolicity of a linear nonautonomous dynamics, both for discrete- and continuous-time systems, in terms of a certain cocycle that is often
quite useful in the qualitative theory of equations. The authors obtain the base of the cocycle by closing the translation of the dynamics of the system with respect to the topology of uniform convergence on compact sets. For discrete-time dynamics, this procedure is related to pointwise convergence.
The importance of this approach is that the base of the cocycle may become compact under some additional assumptions. The authors do not provide neither examples nor applications of their general theory.
Reviewer: Anatoly Martynyuk (Kyïv)Freak chimera states in a locally coupled Duffing oscillators chain.https://www.zbmath.org/1455.340312021-03-30T15:24:00+00:00"Clerc, M. G."https://www.zbmath.org/authors/?q=ai:clerc.marcel-g"Coulibaly, S."https://www.zbmath.org/authors/?q=ai:coulibaly.saliya"Ferré, M. A."https://www.zbmath.org/authors/?q=ai:ferre.m-aThis paper considers a chain of periodically forced Duffing oscillators, diffusively coupled to their two nearest neighbours. Parameters are set so that an uncoupled oscillator is in a bistable state. There may exist states for which a group of oscillators are near one of the uncoupled attractors, behaving approximately periodically, while the rest are near the other uncoupled attractor, which has more complex behaviour. Such a state is referred to as a ``chimera''. If one group is near an attractor and behaving in a complex way, and the other group is near the other attractor, also behaving in a complex way, this is referred to as a ``freak chimera''. Transitions from a chimera to a freak chimera are numerically investigated as the amplitude of forcing and strength of coupling are varied.
Reviewer: Carlo Laing (Auckland)Weak amenability for dynamical systems.https://www.zbmath.org/1455.370072021-03-30T15:24:00+00:00"McKee, Andrew"https://www.zbmath.org/authors/?q=ai:mckee.andrewSummary: Using the recently developed notion of a Herz-Schur multiplier of a \(C^*\)-dynamical system we introduce weak amenability of \(C^*\)- and \(W^*\)-dynamical systems. As a special case we recover Haagerup's characterisation of weak amenability of a discrete group. We also consider a generalisation of the Fourier algebra and its multipliers to crossed products.Basic bifurcation scenarios in neighborhoods of boundaries of stability regions of libration points in the three-body problem.https://www.zbmath.org/1455.700072021-03-30T15:24:00+00:00"Yumagulov, M. G."https://www.zbmath.org/authors/?q=ai:yumagulov.marat-gayazovichSummary: In this paper, we construct stability regions (in the linear approximation) of triangular libration points for the planar, restricted, elliptic three-body problem and examine bifurcations that occur when parameters of the system pass through the boundaries of these regions. A new scheme for the construction of stability regions is presented, which leads to approximation formulas describing these boundaries. We prove that on one part of the boundary, the main scenario of bifurcation is the appearance of nonstationary \(4 \pi \)-periodic solutions that are close to a triangular libration point, whereas on the other part, the main scenario is the appearance of quasiperiodic solutions.Sliding bifurcations in the memristive Murali-Lakshmanan-Chua circuit and the memristive driven Chua oscillator.https://www.zbmath.org/1455.370762021-03-30T15:24:00+00:00"Ahamed, A. Ishaq"https://www.zbmath.org/authors/?q=ai:ahamed.a-ishaq"Lakshmanan, M."https://www.zbmath.org/authors/?q=ai:lakshmanan.muthusamyThe structural stability of maps with heteroclinic repellers.https://www.zbmath.org/1455.370202021-03-30T15:24:00+00:00"Chen, Yuanlong"https://www.zbmath.org/authors/?q=ai:chen.yuanlong"Li, Liangliang"https://www.zbmath.org/authors/?q=ai:li.liangliang"Wu, Xiaoying"https://www.zbmath.org/authors/?q=ai:wu.xiaoying"Wang, Feng"https://www.zbmath.org/authors/?q=ai:wang.feng.3|wang.feng.2|wang.feng.1|wang.feng.4Chaos in periodically forced reversible vector fields.https://www.zbmath.org/1455.370222021-03-30T15:24:00+00:00"Labouriau, Isabel S."https://www.zbmath.org/authors/?q=ai:labouriau.isabel-s"Sovrano, Elisa"https://www.zbmath.org/authors/?q=ai:sovrano.elisaIn the present paper,
the chaos appearance
in time-periodic perturbations
of reversible plane vector fields is
discussed.
The following is the main result
of the paper.
Theorem. Let \(X_\lambda(x,y)\) be a fixed type of
normal form for a one-parameter family of
codimension-1 reversible vector fields,
of either saddle type or of cusp type.
Let \(\lambda_1\) and \(\lambda_2\)
be two real distinct values.
Suppose that the dynamical system
\(\dot X=X(x,y)\) switches in a
\(T\)-periodic manner between
\(\dot X=X_{\lambda_1}(x,y)\) for
\(t\in [0,\tau_1)\)
and
\(\dot X=X_{\lambda_2}(x,y)\) for
\(t\in [\tau_1,\tau_1+\tau_2)\)
with \(\tau_1+\tau_2=T\).
Then, for open sets of the
parameters \((\lambda_1,\lambda_2)\)
and for \(\tau_1\) and \(\tau_2\)
in open intervals, there exists infinitely many \(T\)-periodic solutions as well as
chaotic-like dynamics for the problem
\(\dot X=X(x,y)\).
Reviewer: Mihai Turinici (Iaşi)A note on reversibility and Pell equations.https://www.zbmath.org/1455.370392021-03-30T15:24:00+00:00"Bessa, Mário"https://www.zbmath.org/authors/?q=ai:bessa.mario"Carvalho, Maria"https://www.zbmath.org/authors/?q=ai:de-carvalho.maria-pires"Rodrigues, Alexandre A. P."https://www.zbmath.org/authors/?q=ai:rodrigues.alexandre-a-pSummary: This note concerns hyperbolic toral automorphisms which are reversible with respect to a linear area-preserving involution. Due to the low dimension, we will be able to associate the reversibility with a generalized Pell equation from whose set of solutions we will infer further information. Additionally, we will show that reversibility is a rare feature and will characterize the generic setting.Hidden attractors with conditional symmetry.https://www.zbmath.org/1455.370272021-03-30T15:24:00+00:00"Li, Chunbiao"https://www.zbmath.org/authors/?q=ai:li.chunbiao"Sun, Jiayu"https://www.zbmath.org/authors/?q=ai:sun.jiayu"Sprott, Julien Clinton"https://www.zbmath.org/authors/?q=ai:sprott.julien-clinton"Lei, Tengfei"https://www.zbmath.org/authors/?q=ai:lei.tengfeiFixing monotone Boolean networks asynchronously.https://www.zbmath.org/1455.680812021-03-30T15:24:00+00:00"Aracena, Julio"https://www.zbmath.org/authors/?q=ai:aracena.julio"Gadouleau, Maximilien"https://www.zbmath.org/authors/?q=ai:gadouleau.maximilien"Richard, Adrien"https://www.zbmath.org/authors/?q=ai:richard.adrien"Salinas, Lilian"https://www.zbmath.org/authors/?q=ai:salinas.lilianSummary: The asynchronous automaton associated with a Boolean network \(f : \{0, 1\}^n \to \{0, 1\}^n\) is considered in many applications. It is the finite deterministic automaton with set of states \(\{0, 1\}^n\), alphabet \(\{1, \ldots, n\}\), where the action of letter \(i\) on a state \(x\) consists in switching the \(i\) th component if \(f_i(x) \neq x_i\) or doing nothing otherwise. This action is extended to words in the natural way. We then say that a word \(w\) fixes \(f\) if, for all states \(x\), the result of the action of \(w\) on \(x\) is a fixed point of \(f\). In this paper, we ask for the existence of fixing words, and their minimal length. Firstly, our main results concern the minimal length of words that fix \textit{monotone} networks. We prove that there exists a monotone network \(f\) with \(n\) components such that any word fixing \(f\) has length \(\Omega (n^2)\). Conversely, we construct a word of length \(O (n^3)\) that fixes all monotone networks with \(n\) components. Secondly, we refine and extend our results to different classes of networks.The Yamada model for a self-pulsing laser: bifurcation structure for nonidentical decay times of gain and absorber.https://www.zbmath.org/1455.370702021-03-30T15:24:00+00:00"Otupiri, Robert"https://www.zbmath.org/authors/?q=ai:otupiri.robert"Krauskopf, Bernd"https://www.zbmath.org/authors/?q=ai:krauskopf.bernd"Broderick, Neil G. R."https://www.zbmath.org/authors/?q=ai:broderick.neil-g-rConstruction of peakon-antipeakon solutions and ill-posedness for the b-family of equations.https://www.zbmath.org/1455.352252021-03-30T15:24:00+00:00"Novruzov, Emil"https://www.zbmath.org/authors/?q=ai:novruzov.emil-bSummary: For \(s < 3 / 2\), it is known (see [\textit{A. Alexandrou Himonas} et al., J. Nonlinear Sci. 26, No. 5, 1175--1190 (2016; Zbl 1356.37079)]) that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces \(H^s\) when \(b > 1\). The proof of ill-posedness depends naturally on the value of \(b\), and is based on the construction of peakon-antipeakon solutions with interesting properties which allows to make conclusion on ill-posedness. In this context the construction of such type of the solution for \(b < 1\) is very attractive problem which help shed light on the ill-posedness problem in this case. Thus, in the present paper we consider the ill-posedness of the b-family of equations with additional term for insufficiently investigated case \(b < 1\) on the line.Preface.https://www.zbmath.org/1455.680222021-03-30T15:24:00+00:00"Dennunzio, Alberto (ed.)"https://www.zbmath.org/authors/?q=ai:dennunzio.alberto"Formenti, Enrico (ed.)"https://www.zbmath.org/authors/?q=ai:formenti.enricoFrom the text: The 23rd Annual International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2017) was held in Milano (Italy) in the period 7--9 June 2017 at the University of Milano-Bicocca.
The papers presented at the conference were selected following a rigorous peer review process in which all papers had two reviews. After the conference, authors were invited to submit a substantially extended and improved version of their papers. A further review round followed and all papers got two new reviews. Finally, ten of them were selected to appear in this special issue.Hopf bifurcation and stability crossing curve in a planktonic resource-consumer system with double delays.https://www.zbmath.org/1455.370732021-03-30T15:24:00+00:00"Jiang, Zhichao"https://www.zbmath.org/authors/?q=ai:jiang.zhichao"Guo, Yanfen"https://www.zbmath.org/authors/?q=ai:guo.yanfenMulti-scroll chaotic system model and its cryptographic application.https://www.zbmath.org/1455.370782021-03-30T15:24:00+00:00"Liu, Song"https://www.zbmath.org/authors/?q=ai:liu.song"Wei, Yaping"https://www.zbmath.org/authors/?q=ai:wei.yaping"Liu, Jingyi"https://www.zbmath.org/authors/?q=ai:liu.jingyi"Chen, Shiqiang"https://www.zbmath.org/authors/?q=ai:chen.shiqiang"Zhang, Guoping"https://www.zbmath.org/authors/?q=ai:zhang.guopingInductive learning from state transitions over continuous domains.https://www.zbmath.org/1455.681802021-03-30T15:24:00+00:00"Ribeiro, Tony"https://www.zbmath.org/authors/?q=ai:ribeiro.tony"Tourret, Sophie"https://www.zbmath.org/authors/?q=ai:tourret.sophie"Folschette, Maxime"https://www.zbmath.org/authors/?q=ai:folschette.maxime"Magnin, Morgan"https://www.zbmath.org/authors/?q=ai:magnin.morgan"Borzacchiello, Domenico"https://www.zbmath.org/authors/?q=ai:borzacchiello.domenico"Chinesta, Francisco"https://www.zbmath.org/authors/?q=ai:chinesta.francisco"Roux, Olivier"https://www.zbmath.org/authors/?q=ai:roux.olivier-f|roux.olivier-h"Inoue, Katsumi"https://www.zbmath.org/authors/?q=ai:inoue.katsumiSummary: Learning from interpretation transition (LFIT) automatically constructs a model of the dynamics of a system from the observation of its state transitions. So far, the systems that LFIT handles are restricted to discrete variables or suppose a discretization of continuous data. However, when working with real data, the discretization choices are critical for the quality of the model learned by LFIT. In this paper, we focus on a method that learns the dynamics of the system directly from continuous time-series data. For this purpose, we propose a modeling of continuous dynamics by logic programs composed of rules whose conditions and conclusions represent continuums of values.
For the entire collection see [Zbl 1453.68016].Symbolic dynamics for one dimensional maps with nonuniform expansion.https://www.zbmath.org/1455.370122021-03-30T15:24:00+00:00"Lima, Yuri"https://www.zbmath.org/authors/?q=ai:lima.yuriGiven a piecewise \(C^{1+\beta}\) map of the interval, possibly with{critical} points and discontinuities, the author constructs a symbolic model for invariant probability measures with nonuniform expansion that does not approach the critical points and discontinuities exponentially fast almost surely.
More specifically, for each \(\chi>0\) it is constructed a finite-to-one Hölder continuous map from a countable topological Markov shift to the natural extension of the interval map, that codes the lifts of all invariant probability measures with Lyapunov exponent greater than \(\chi\) almost everywhere.
Consider a symbolical dynamical system \(S: \{0,1\}^{\mathbb Z}\to \{0,1\}^{\mathbb Z}\) with shift operator \(S\). If the subshift \(M\subset \{0,1\}^{\mathbb Z}\) is countable, then all recurrent points are periodic, and the theory of such subshift becomes more transparent. The proof of the mentioned result consists on implementing the construction of symbolic dynamics for nonuniformly hyperbolic systems. Applications such as the counting of periodic points and the countability of measures of maximal entropy are discussed.
Reviewer: Alexei Kanel-Belov (Ramat-Gan)Reciprocal transformations of Harry-Dym hierarchy.https://www.zbmath.org/1455.370592021-03-30T15:24:00+00:00"Cai, Liqiang"https://www.zbmath.org/authors/?q=ai:cai.liqiang"Cheng, Jipeng"https://www.zbmath.org/authors/?q=ai:cheng.jipengSummary: In this paper, we work out the reciprocal transformations generated by the adjoint eigenfunctions, and write down the corresponding changes of the Lax operators, eigenfunctions and adjoint eigenfunctions under this kind of new reciprocal transformations at first. Then we discuss the successive actions of the reciprocal transformations generated by the eigenfunctions and the adjoint eigenfunctions. Also we investigate the commutativity of different types of the reciprocal transformations. The above results are very useful to construct the new type solutions.Dimension of Gibbs measures with infinite entropy.https://www.zbmath.org/1455.370252021-03-30T15:24:00+00:00"Pérez Pereira, Felipe"https://www.zbmath.org/authors/?q=ai:perez-pereira.felipeThis article contributes to the dimension theory of measure-preserving maps. It provides an analysis of the Hausdorff and packing dimensions of Gauss-like Markov-Reyni maps preserving a Gibbs measure with infinite entropy and satisfying some decay condition. Interestingly, it derives from the analysis that such measures are not exact dimensional.
Reviewer: Asgar Jamneshan (Istanbul)Complex fractional moments for the characterization of the probabilistic response of non-linear systems subjected to white noises.https://www.zbmath.org/1455.352582021-03-30T15:24:00+00:00"Di Paola, Mario"https://www.zbmath.org/authors/?q=ai:di-paola.mario"Pirrotta, Antonina"https://www.zbmath.org/authors/?q=ai:pirrotta.antonina"Alotta, Gioacchino"https://www.zbmath.org/authors/?q=ai:alotta.gioacchino"Di Matteo, Alberto"https://www.zbmath.org/authors/?q=ai:di-matteo.alberto"Pinnola, Francesco Paolo"https://www.zbmath.org/authors/?q=ai:pinnola.francesco-paoloThe core of the chapter consists in a demonstration of an employment of Complex Fractional Moments (CFM) in an analysis of the PDF of a system response under an excitation of a white noise of several types. Although the chapter has a character of an overview, the rich evaluation of literature resources indicates that the set of papers published by authors of this chapter makes more or less a closed package of information about a new non-conventional approach. Nevertheless, the chapter itself is worthy to be recommended to those, who are interested in FPE analysis of various types. The chapter is transparently written and provides valuable information about the problem and adequate literature.
In the first two sections, a general form of the FPE solution using decomposition with respect to stochastic moments of the non-integer order type is explained clarifying a relation with the Mellin integral transform in continuous or discrete versions. Important details concerning mathematical limitations and pitfalls are discussed before individual applications are presented. Three examples of FPE originating from a nonlinear diffusion dynamic system (Langevin type equation) with different type of a white noise excitation are demonstrated: (i) conventional FPE with additive Gaussian white noise; (ii) FPE with \(\alpha\)-stable white noise excitation; (iii) Generalized FPE (Kolmogorov-Feller) with Poissonian white noise. In the case (i) advantages of the CFM solution procedure are demonstrated in comparison with integer order moments. Problems of non-guaranteed convergence of integer order moments are avoided, hierarchy problems are solved as well, compromises with various types of closure disappeared, etc. Furthermore, convergence seems to be faster and more convenient type. Whereas in the (i) the CFM application offer more convenient and elegant solution, the cas (ii), Lévy \(\alpha\)-stable white noise excitation, explicitly requires to be analyzed by means of the CFM. It follows immediately from the FPE itself, where fractional derivative of the α-order is present. In such a case an application of the CFM is doubtlessly inherent. Similarly the case (iii) is much more related with fractional order moments than those with integer order. Although a procedure with classical moments is also possible, numerical evaluation is very clumsy and convergence problematic. Analytical results are verified by means of stochastic simulation. Comparison of both is excellent. In general, the chapter should be considered as an excellent pioneering work, which gives not only an effective alternative, but also new possibilities in searching of a weak solution of the FPE. The chapter doubtlessly attracts many readers working in area of random vibration of nonlinear dynamic systems as well as people involved in basic theory of random processes.
For the entire collection see [Zbl 1430.74006].
Reviewer: Jiri Náprstek (Praha)On periodic solutions to Lagrangian system with singularities and constraints.https://www.zbmath.org/1455.370512021-03-30T15:24:00+00:00"Zubelevich, O."https://www.zbmath.org/authors/?q=ai:zubelevich.oleg-eduardovichA time-dependent mechanical Lagrangian with gyroscopic forces and ideal constraints is considered. It is shown that, provided certain inequalities are satisfied, the Euler-Lagrange equations admit non-trivial \(\omega\)-periodic (arithmetic quasi-periodic) solutions which are odd as functions of time. The author considers in detail the example of a tube rotating freely in a vertical plane with a small ball rolling in it and of a point moving in a plane under the potential \(V=-\gamma \left( |{r}-{r}_0|^{-d}+|{r}+{r}_0|^{-d} \right)\), where \(\gamma \in \mathbb R^+\) and \({r}_0\neq 0\) is a fixed vector.
Reviewer: Giovanni Rastelli (Vercelli)Application of matrix semi-tensor product in chaotic image encryption.https://www.zbmath.org/1455.940382021-03-30T15:24:00+00:00"Wang, Xingyuan"https://www.zbmath.org/authors/?q=ai:wang.xingyuan"Gao, Suo"https://www.zbmath.org/authors/?q=ai:gao.suoSummary: This paper proposes a new image encryption method based on matrix semi-tensor product theory. Using Hyperchaotic Lorenz system to generate chaotic sequences, and then using this chaotic sequence to generate two fixed scrambling matrices. The plain image is scrambled by chaotic positioning sort scrambled method. Analogous to a chemical reaction, the plain image scrambled image is one of the reactant, then add a matrix which is related to the plain image information and extracted from the chaotic sequence (this matrix can be differ in size from the plain image matrix, different matrices can be intercepted and generated according to different plain image). This matrix is scrambled as another reactant, Apply the method of semi-tensor product to carry on the reaction (diffusion), finally get the product (encryption image). This method breaks the shackles of the traditional matrix operation and makes the reaction matrix have more forms. Compared with other experimental results, the proposed algorithm is more secure and available.On generalized Lattès maps.https://www.zbmath.org/1455.300252021-03-30T15:24:00+00:00"Pakovich, Fedor"https://www.zbmath.org/authors/?q=ai:pakovich.fedorSummary: We introduce a class of rational functions \(A:\mathbb{CP}^1\rightarrow\mathbb{CP}^1\) which can be considered as a natural extension of the class of Lattès maps, and establish basic properties of functions from this class.Dynamics and eigenvalues in dimension zero.https://www.zbmath.org/1455.370152021-03-30T15:24:00+00:00"Hernández-Corbato, Luis"https://www.zbmath.org/authors/?q=ai:hernandez-corbato.luis"Nieves-Rivera, David Jesús"https://www.zbmath.org/authors/?q=ai:nieves-rivera.david-jesus"Ruiz Del Portal, Francisco R."https://www.zbmath.org/authors/?q=ai:ruiz-del-portal.francisco-romero"Sánchez-Gabites, Jaime J."https://www.zbmath.org/authors/?q=ai:sanchez-gabites.jaime-jLet \(X\) be a compact, metric and totally disconnected space and let \(f: X \to X\) be a continuous map.
Then every non-zero eigenvalue of the induced map \(f^*\) on the Čech cohomology group \(\check{H}^0(Z; \mathbb{C})\) is a root of unity.
However, for the induced map \(f_*\) on the Čech homology group \(\check{H}_0(Z; \mathbb{C})\), the authors prove that the existence of an eigenvalue \(\lambda \in \mathbb{C}\) with \(|\lambda| \ne 0,1\) is equivalent to all such \(\lambda\)'s being eigenvalues, and is also equivalent to \((X,f)\) admitting dynamical \(\epsilon\)-partitions for all \(\epsilon>0\).
(The latter condition can also be expressed in terms of adding machines.)
In particular, if the topological entropy of \(f\) is non-zero or \(f\) is positively expansive and \(X\) is infinite, then every \(\lambda \in \mathbb{C}\) with \(|\lambda| \ne 0,1\) is an eigenvalue of \(f_*: \check{H}_0(Z; \mathbb{C}) \to \check{H}_0(Z; \mathbb{C})\).
This stands in contrast with an inequality of \textit{A. Manning} [Lect Notes in Math. 468, 185-190 (1975; Zbl 0307.54042)] that bounds the entropy of \(f\) below by the spectral radius of the induced map \(f_*\) on \(H_1(X;\mathbb{C})\) when \(X\) is a compact manifold.
Reviewer: Wolfgang Steiner (Paris)The approximation of invariant sets in infinite dimensional dynamical systems.https://www.zbmath.org/1455.370652021-03-30T15:24:00+00:00"Gerlach, Raphael"https://www.zbmath.org/authors/?q=ai:gerlach.raphael"Ziessler, Adrian"https://www.zbmath.org/authors/?q=ai:ziessler.adrianThe authors start with a short summary on a novel framework -- developed in their previous works -- for the computation of finite-dimensional invariant sets of infinite-dimensional dynamical systems. After that, by applying their results on embedding techniques for core dynamical system, they extend a classical subdivision scheme as well as a continuation algorithm for the computation of attractors and invariant manifolds of finite-dimensional systems to the infinite-dimensional setting.
They consider dynamical systems of the form \(u_{j+1}=\Phi (u_{j})\), \(j=0, 1,\dots\), where \(\Phi : Y\rightarrow Y\) is Lipschitz continuous in a Banach space \(Y\). Moreover, they assume that \(\Phi\) has an invariant compact set \(\mathcal{A}\), that is \(\Phi(\mathcal{A})=\mathcal{A}\). Further, the core dynamical system is given as \(x_{j+1}=\varphi (x_{j})\), \(j=0, 1, 2,\dots\), with \(\varphi : \mathbb{R}^{k}\rightarrow \mathbb{R}^{k}\).
The authors give Algorithm 1 as the subdivision method for embedded global attractors and Algorithm 2 as the continuation method for embedded unstable manifolds. They use software package GAIO (Global Analysis of Invariant Objects) for the numerical implementation of these algorithms.
For the numerical realization of the core dynamical systems, the authors restrict their attention to delay differential equations of the form
\(\dot{y}(t)=g(y(t), y(t-\tau))\), where \(y(t)\in \mathbb{R}^{n}\), \(\tau >0\) is a constant time delay and \(g:\mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is a smooth map;
and partial differential equations of the form \(\frac{\partial}{\partial t} u(y, t)=F(y, u)\), \(u(y, 0)=u_{0}(y)\), where \(u:\mathbb{R}^{n}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is in some Banach space \(Y\) and \(F\) is a (nonlinear) differential operator.
The authors also explain how to implement the above approach for the analysis of delay differential equations and partial differential equations such as the Kuramoto-Sivashinsky equation, the Ginzburg-Landau equation and reaction-diffusion equations. Further, they illustrate numerically and graphically their results by computing the attractor of the Mackey-Glass equation
\[\dot{u}(t)=\beta\frac{u(t-\tau)}{1+u(t-\tau)^{\eta}}-\gamma u(t)\]
and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation
\[u_{t}+4u_{yyyy}+\mu [u_{yy}+\frac{1}{2}(u_{y})^{2}]=0 \quad 0\leq y\leq 2\pi, \quad u(y, 0)=u_{0}(y),\quad u(y+2\pi, t)=u(y, t).\]
For the entire collection see [Zbl 1445.37003].
Reviewer: Mohammad Sajid (Buraidah)Critically finite random maps of an interval.https://www.zbmath.org/1455.370362021-03-30T15:24:00+00:00"Atnip, Jason"https://www.zbmath.org/authors/?q=ai:atnip.jason"Urbański, Mariusz"https://www.zbmath.org/authors/?q=ai:urbanski.mariuszThe authors consider a class of random dynamical systems called random critically finite maps of the interval \([0,1]\).
For such a map \(T\),
the main goal of this paper is to study the existence, uniqueness, and properties of random conformal measures for \(T\) and random \(T\)-invariant measures absolutely continuous/equivalent with respect to conformal measures.
To do this the authors introduce a subset of \([0,1]\) called the set of \(A\)-admissible parameters, denoted by \(AA(T)\).
They show that for each \(t \in AA(T)\) a \(t\)-conformal random measure \(\nu_t\) exists and given \(t \ge 0\) any two
\(t\)-conformal random measures are equivalent.
The structure of \(AA(T)\) is studied by means of the expected topological pressure for each \(t \in AA(T)\).
Reviewer: Steve Pederson (Atlanta)On sofic groups, Kaplansky's conjectures, and endomorphisms of pro-algebraic groups.https://www.zbmath.org/1455.370132021-03-30T15:24:00+00:00"Phung, Xuan Kien"https://www.zbmath.org/authors/?q=ai:phung.xuan-kienSummary: Let \(G\) be a group. Let \(X\) be a connected algebraic group over an algebraically closed field \(K\). Denote by \(A = X(K)\) the set of \(K\)-points of \(X\). We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over \((G, X, K)\). They are cellular automata \(\tau : A^G \to A^G\) whose local defining map is induced by a homomorphism of algebraic groups \(X^M \to X\) where \(M \subset G\) is a finite memory set of \(\tau \). Our first result is that when \(G\) is sofic, such an algebraic group cellular automaton \(\tau\) is invertible whenever it is injective and \(\text{char}(K) = 0\). When \(G\) is amenable, we show that an algebraic group cellular automaton \(\tau\) is surjective if and only if it satisfies a weak form of pre-injectivity called \((\bullet)\)-pre-injectivity. This yields an analogue of the classical Moore-Myhill Garden of Eden theorem. We also introduce the near ring \(R(K, G)\) which is \(K [ X_g : g \in G]\) as an additive group but the multiplication is induced by the group law of \(G\). The near ring \(R(K, G)\) contains naturally the group ring \(K [G]\) and we extend Kaplansky's conjectures to this new setting. Among other results, we prove that when \(G\) is an orderable group, then all one-sided invertible elements of \(R(K, G)\) are trivial, i.e., of the form \(a X_g + b\) for some \(g \in G, a \in K^\ast \), and \(b \in K\). This in turns allows us to show that when \(G\) is locally residually finite and orderable (e.g. \( \mathbb{Z}^d\) or a free group), all injective algebraic cellular automata \(\tau : \mathbb{C}^G \to \mathbb{C}^G\) are of the form \(\tau(x)(h) = a x( g^{- 1} h) + b\) for all \(x \in \mathbb{C}^G, h \in G\) for some \(g \in G, a \in \mathbb{C}^\ast \), and \(b \in \mathbb{C} \).Braid equivalence in the Hénon family. I.https://www.zbmath.org/1455.370412021-03-30T15:24:00+00:00"de Carvalho, A."https://www.zbmath.org/authors/?q=ai:de-carvalho.andre-f|salles-de-carvalho.andre|nolasco-de-carvalho.alexandre|de-carvalho.alexandre-luis-trovon|de-carvalho.a-a|ponce-de-leon-ferreira-de-carvalho.andre-carlos|de-carvalho.alcides"Hall, T."https://www.zbmath.org/authors/?q=ai:hall.t-j|hall.tony|hall.tom-e|hall.tyson-s|hall.tracy|hall.toby|hall.t-m|hall.thomas-eric"Hazard, P."https://www.zbmath.org/authors/?q=ai:hazard.p-e|hazard.peterThe dynamics of the real quadratic family \(f_{a}(x)\,=\,a-x^{2}\) is well understood (see [\textit{M. Lyubich}, Notices Am. Math. Soc. 47, No. 9, 1042--1052 (2000; Zbl 1040.37032)]). On the other hand the understanding of the two-dimensional Hénon family \(F_{a,\,b}(x,\,y)\,=\,(f_{a}(x)-by,\,x)\) (which for \(b\,=\,0\) degenerates to \(f_{a}(x)\)) is still rather rudimentary.
This article is concerned with periodic orbits in the Hénon familly in the parameter regions close to degeneration, and exploits both similarities and differences between the quadratic family and
Hénon family.
In the paper, the authors describe two mechanisms (Constructions 3.1 and 4.6) which lead to a coalescence on the
level of unimodal permutations (the unimodal permutations are defined in Definition 2.8).
The paper is rather technical and the used terminology cannot be totally reproduced here.
A geometric
braid on \(n\) strands is a diagram with \(n\) arcs (strands) connecting two
ordered sets of \(n\) points lined up vertically, so that only double intersections are
allowed and at each of them it is specified which strand goes above and which goes
below. To each geometric braid is associated a braid type, and
braid types determine geometric braids up to conjugacy.
For a \(p\in \mathbb{N}\) with \(U_{p}\) it is denoted the set of unimodal
permutations on \(p\) symbols.
For any \(u\in U_{p}\) there is unique unimodal braid \(\beta\) which induces \(u\). The set of unimodal braids on \(p\) strands is denoted by \(UB_{p}\).
Theorem A (Theorem 3.3 in the paper). Let \( p \in \mathbb{N}\). Let \(\beta \in UB_{p}\) be cyclic. Assume \(\beta \) is reconnectable at the non-dynamical preimage. Let \(\beta _{-}\) and \(\beta _{+}\) denote the braids from Construction 3.1.
1. If the non-dynamical preimage lies to the left of \(m\), then \(\beta _{+}\,\sim\,\beta _{-}\).
2. If the non-dynamical preimage lies to the right of \(m\), then \(\beta _{+}\,\sim_{r}\,\beta _{-}\).
Theorem B (Theorem 4.8 in the paper). Given \((u_{-}, \,u_{+})\), satisfying Properties (1)--(4) (page 97 in the paper). Let \(\beta _{-}\) and \(\beta _{+}\) denote the pair of braids produced
from Construction 4.6.
1. If \(\beta _{-}\,\sim\,\beta _{+}\), then \(\beta _{-}^{1}\,\sim\,\beta _{+}^{1}\).
2. If \(\beta _{-}\,\sim _{r}\,\beta _{+}\), then \(\beta _{-}^{1}\,\sim _{r}\,\beta _{+}^{1}\).
Moreover, the corresponding unimodal permutations \((u_{-}^{1}\,\sim _{r}\,u_{+}^{1})\) also satisfy the Properties (1)--(4).
The two above results are applied to equivalences realised in the Hénon family.
The paper ends with two questions.
Question A. Let \(a_{-}, \,\,a_{+}\in [-1/4,\,\, 2]\) be such that \(f_{a_{-}}\) and
\(f_{a_{+}}\) have critical orbits \(c_{-}\) and \(c_{+}\)
of types \(u_{-}\) and \(u_{+}\) respectively. Let \(C_{-}\) and \(C_{+}\) denote the corresponding periodic orbits
for \(F_{a_{-},\,0}\) and \(F_{a_{+},\,0}\) respectively. Does there exist a braid equivalence in the family \(F_{a,\,b}\)
connecting \((C_{-},\, F_{a_{-},\,0})\) and \((C_{+},\, F_{a_{+},\,0})\)?
Question B. For each positive integer \(i\), let \(a^{i}_{-} ,\,\,a^{i}_{+}\in [-1/4,\,\, 2]\) be parameters such that \(f_{a^{i}_{-}}\) and
\(f_{a^{i}_{+}}\) have critical orbits \(c^{i}_{-}\) and \(c^{i}_{+}\)
of types \(u^{i}_{-}\) and \(u^{i}_{+}\) respectively.
Let \(C^{i}_{-}\) and \(C^{i}_{+}\) denote the corresponding periodic orbits
for \(F_{a^{i}_{-},\,0}\) and \(F_{a^{i}_{+},\,0}\) respectively. Does there exist,
for each \(i\), a braid equivalence in the family \(F_{a,\,b}\)
connecting \((C^{i}_{-},\, F_{a^{i}{-},\,0})\) and \((C^{i}_{+},\, F_{a^{i}{+},\,0})\)?
Are the paths \(\gamma ^{i}\) in the \((a,\, b)\)-plane which realise these braid equivalences pairwise disjoint?
For the entire collection see [Zbl 1432.37002].
Reviewer: Dimitrios Varsos (Athína)New Weierstrass elliptic wave solutions of the Davey-Stewartson equation with power law nonlinearity.https://www.zbmath.org/1455.352192021-03-30T15:24:00+00:00"El Achab, Abdelfattah"https://www.zbmath.org/authors/?q=ai:el-achab.abdelfattahIn this paper the author considers the (2+1)-dimensional
Davey-Stewartson (DS) equations and obtains some previously known
and new solutions through the Weierstrass elliptic function
method. The paper is organized as follows. The first section is an
introduction to the subject. Section 2 deals with the Weierstrass
elliptic function method. In section 3, the author gives some
particular travelling wave solutions of the (2+1)-dimensional (DS)
equations and restates the main points in section 4. The paper is
supported by an appendix containing some properties of Weierstrass
elliptic functions. The paper is well documented.
Reviewer: Ahmed Lesfari (El Jadida)Lump, rogue wave, multi-waves and homoclinic breather solutions for \((2+1)\)-modified Veronese web equation.https://www.zbmath.org/1455.370602021-03-30T15:24:00+00:00"Rizvi, S. T. R."https://www.zbmath.org/authors/?q=ai:rizvi.syed-tahir-raza"Seadawy, Aly R."https://www.zbmath.org/authors/?q=ai:seadawy.aly-r"Ahmed, S."https://www.zbmath.org/authors/?q=ai:ahmed.syed-ali|ahmed.s-k|ahmed.sarafima|ahmed.shareef|ahmed.s-shahanawaz|ahmed.shazad-shawki|ahmed.sidrah|ahmed.said-mohamed|ahmed.sanaa-h|ahmed.sajid|ahmed.syed-iqbal|ahmed.safwat-abdel-radi-saleh|ahmed.shams-a|ahmed.shabir|ahmed.shabbir|ahmed.syed-faraz|ahmed.syed-sabbir|ahmed.sultan|ahmed.shahnawaz|ahmed.samah-m|ahmed.s-basheer|ahmed.shehzad|ahmed.saneeha|ahmed.sayad-ali|ahmed.segun|ahmed.sahin|ahmed.syed-ishtiaque|ahmed.saifuddin|ahmed.salman|ahmed.saim|ahmed.shafique|ahmed.sk-shamim|ahmed.syed-imtiaz|ahmed.s-reaz|ahmed.saleh|ahmed.saad-zagloul-rida|ahmed.syed-a-s|ahmed.suman|ahmed.sherif|ahmed.selim|ahmed.sabbir|ahmed.sajeer|ahmed.shaikh-s|ahmed.shakeel|ahmed.s-nomaan|ahmed.shohel|ahmed.shafiq-u|ahmed.sayed-attia|ahmed.syed-ejaz|ahmed.shamsuddin|ahmed.suhaib|ahmed.sameh-elsayed|ahmed.sameed|ahmed.saeed|ahmed.shakil|ahmed.shaimaa-a-a|ahmed.said-gamil|ahmed.syed-afrozuddin|ahmed.sarfraz|ahmed.shehab|ahmed.shaikh-fokor-uddin-ali|ahmed.safayet|ahmed.salim"Younis, M."https://www.zbmath.org/authors/?q=ai:younis.muhammad"Ali, K."https://www.zbmath.org/authors/?q=ai:ali.kashifThe Liouville equation as a Hamiltonian system.https://www.zbmath.org/1455.370542021-03-30T15:24:00+00:00"Kozlov, V. V."https://www.zbmath.org/authors/?q=ai:kozlov.vasilii-vasilevich|kozlov.vladimir-vasilievich|kozlov.v-v|kozlov.victor-v|kozlov.valerij-vasilievichThe author considers smooth dynamical systems on closed manifolds with invariant measure. The evolution of the density of a nonstationary invariant measure is described by the well-known Liouville equation and for ergodic dynamical systems, the latter is expressed in Hamiltonian form. An infinite collection of quadratic invariants that are pairwise in involution with respect to the Poisson bracket generated by the Hamiltonian structure is indicated. Some remarks and open questions are mentioned at the end of the paper.
Reviewer: Ahmed Lesfari (El Jadida)Global dynamics of Filippov-type plant disease models with an interaction ratio threshold.https://www.zbmath.org/1455.340522021-03-30T15:24:00+00:00"Li, Wenxiu"https://www.zbmath.org/authors/?q=ai:li.wenxiu"Huang, Lihong"https://www.zbmath.org/authors/?q=ai:huang.lihong"Wang, Jiafu"https://www.zbmath.org/authors/?q=ai:wang.jiafuSummary: A Filippov-type plant disease model is developed by introducing a interaction ratio threshold, the number of susceptible plants infected by per diseased plant, which determines whether control measures including replanting or roguing are carried out. The main purpose of this paper is to give a completely qualitative analysis of the model. By employing Poincaré maps, our analysis reveals rich dynamics including a global attractor bounded by a touching closed orbit, which is convergent in finite time from its outside, a global attractor bounded by two touching closed orbits and a pseudo-saddle, and a globally asymptotically stable pseudo-node. Moreover, we give biological implications of our results in implementing control strategies for plant diseases.Ergodic properties of some negatively curved manifolds with infinite measure.https://www.zbmath.org/1455.370352021-03-30T15:24:00+00:00"Vidotto, Pierre"https://www.zbmath.org/authors/?q=ai:vidotto.pierreSummary: Let $M=X/ \Gamma$ be a geometrically finite negatively curved manifold with fundamental group $\Gamma$ acting on $X$ by isometries. The purpose
of this paper is to study the mixing property of the geodesic flow on $T^1M$, the asymptotic equivalent as $R\longrightarrow+\infty$ of the
number of closed geodesics on $M$ of length less than $R$ and of the orbital
counting function $\sharp\{\gamma\in \Gamma \ |\ d(o,\gamma.o)\leqslant R\}$.
These properties are well known when the Bowen-Margulis measure on $T^1M$ is finite. We consider here divergent Schottky groups whose Bowen-Margulis measure is infinite and ergodic, and we precise these ergodic properties using a suitable symbolic coding.Asymptotic autonomous attractors for a stochastic lattice model with random viscosity.https://www.zbmath.org/1455.370632021-03-30T15:24:00+00:00"Yang, Shuang"https://www.zbmath.org/authors/?q=ai:yang.shuang"Li, Yangrong"https://www.zbmath.org/authors/?q=ai:li.yangrongSummary: We study forward asymptotic autonomy of a pullback random attractor for a non-autonomous random lattice system and establish the criteria in terms of convergence, recurrence, forward-pullback absorption and asymptotic smallness of the discrete random dynamical system. By applying the abstract result to both non-autonomous and autonomous stochastic lattice equations with random viscosity, we show the existence of both pullback and global random attractors such that the time-component of the pullback attractor semi-converges to the global attractor as the time-parameter tends to infinity.Expanding measures for homeomorphisms with eventually shadowing property.https://www.zbmath.org/1455.370082021-03-30T15:24:00+00:00"Dong, Meihua"https://www.zbmath.org/authors/?q=ai:dong.meihua"Lee, Keonhee"https://www.zbmath.org/authors/?q=ai:lee.keonhee"Nguyen, Ngocthach"https://www.zbmath.org/authors/?q=ai:nguyen.ngoc-thachGiven a homeomorphism \(f:X\to X\), where \(X\)~is compact metric, define
for \(x\in X\) and \(\delta>0\) the set \(\Gamma_\delta^f(x)\) to be the set
of~\(y\) for which \(d(f^n(x),f^n(y))\le\delta\) for all~\(n\in\mathbb{Z}\).
One can generally define \(f\) to be expansive if \(\Gamma_\delta^f(x)\) is
small for some \(\delta\) and all~\(x\).
Simply expansive means \(\Gamma_\delta^f(x)=\{x\}\) for some \(\delta\)
and all~\(x\).
If \(\mu\)~is a Borel (probability) measure then \(\mu\)~is expansive for~\(f\)
if \(\mu(\Gamma_\delta^f(x))=0\) for some \(\delta\) and all~\(x\).
Demanding \(\mu(\Gamma_\delta^f(x))=\mu(\{x\})\) (if \(\mu\)~is not necessarily
non-atomic) yields strong expansiveness.
The authors call \(f\) \(\mu\)-expanding if
\(\mu(\Gamma_\delta^f(x)\setminus O_f(x))=0\) for some \(\delta\) and all~\(x\),
where \(O_f(x)\)~is the (full) orbit of~\(x\).
And \(f\)~is (invariant) measure-expanding if every (invariant) measure is
expanding for~\(f\).
The main result of the paper is a spectral decomposition theorem for
homeomorphisms that are invariant measure-expanding and eventually
shadowing on the chain-recurrent sets.
The paper ends with some applications to diffeomorphisms of compact smooth
manifolds.
Reviewer: K. P. Hart (Delft)On inhomogeneous nonholonomic Bilimovich system.https://www.zbmath.org/1455.370532021-03-30T15:24:00+00:00"Borisov, A. V."https://www.zbmath.org/authors/?q=ai:borisov.alexey-v"Tsiganov, A. V."https://www.zbmath.org/authors/?q=ai:tsiganov.andrey-vladimirovich"Mikishanina, E. A."https://www.zbmath.org/authors/?q=ai:mikishanina.evgeniya-arifzhanovnaSummary: We consider a simple motivating example of a non-Hamiltonian dynamical system with time-dependent constraints obtained by imposing rheonomic non-integrable Bilimovich's constraint on a freely rotating rigid body. Dynamics of this low-dimensional nonlinear nonautonomous dynamic system involves different kinds of stable and unstable attractors, quasi and strange attractors, compact and noncompact invariant attractive curves, etc. To study this cautionary example we apply the Poincaré map method to disambiguate and discover multiscale temporal dynamics, specifically the coarse-grained dynamics resulting from fast-scale nonlinear control via nonholonomic Bilimovich's constraint.Reduction of divisors for classical superintegrable \(\mathrm{GL}(3)\) magnetic chain.https://www.zbmath.org/1455.140702021-03-30T15:24:00+00:00"Tsiganov, A. V."https://www.zbmath.org/authors/?q=ai:tsiganov.andrey-vladimirovichSummary: Separated variables for a classical \(\mathrm{GL}(3)\) magnetic chain are coordinates of a generic positive divisor \(D\) of degree \(n\) on a genus \(g\) non-hyperelliptic algebraic curve. Because \(n > g\), this divisor \(D\) has unique representative \(\rho(D)\) in the Jacobian, which can be constructed by using \(\dim|D| = n - g\) steps of Abel's algorithm. We study the properties of the corresponding chain of divisors and prove that the classical \(\mathrm{GL}(3)\) magnetic chain is a superintegrable system with \(\dim|D| = 2\) superintegrable Hamiltonians.
{\copyright 2020 American Institute of Physics}Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case.https://www.zbmath.org/1455.340602021-03-30T15:24:00+00:00"Mégret, Lucile"https://www.zbmath.org/authors/?q=ai:megret.lucile"Demongeot, Jacques"https://www.zbmath.org/authors/?q=ai:demongeot.jacquesSummary: We generalize the results on the existence of an over-stable solution of singularly perturbed differential equations to the equations of the form
\[
\varepsilon\ddot{x}-F(x,t,\dot{x},k(t), \varepsilon) = 0.
\]
In this equation, the time dependence prevents from returning to the well known case of an equation of the form \(\varepsilon dy/dx = F(x,y,a,\varepsilon)\) where \(a\) is a parameter. This can have important physiological applications. Indeed, the coupling between the cardiac and the respiratory activity can be modeled with two coupled van der Pol equations. But this coupling vanishes during the sleep or the anesthesia. Thus, in a perspective of an application to optimal awake, we are led to consider a non autonomous differential equation.Hodge-GUE correspondence and the discrete KdV equation.https://www.zbmath.org/1455.370562021-03-30T15:24:00+00:00"Dubrovin, Boris"https://www.zbmath.org/authors/?q=ai:dubrovin.boris-a"Liu, Si-Qi"https://www.zbmath.org/authors/?q=ai:liu.siqi"Yang, Di"https://www.zbmath.org/authors/?q=ai:yang.di"Zhang, Youjin"https://www.zbmath.org/authors/?q=ai:zhang.youjinThe authors prove the conjectural relationship between the certain special cubic Hodge integrals of the Gopkumar-Marino-Vafa type and GUE correlators. They show that the partition function of these Hodge integrals is a tau function of the discrete KdV hierarchy. Here, they mainly use the Virasoro constraints to show this equivalence. It may be more interesting to consider the equivalence of the bilinear equations.
Reviewer: Jipeng Cheng (Xuzhou)Dynamics of the Chaplygin ball with variable parameters.https://www.zbmath.org/1455.370522021-03-30T15:24:00+00:00"Borisov, Alexei V."https://www.zbmath.org/authors/?q=ai:borisov.alexey-v"Mikishanina, Evgeniya A."https://www.zbmath.org/authors/?q=ai:mikishanina.evgeniya-arifzhanovnaSummary: This work is devoted to the study of the dynamics of the Chaplygin ball with variable moments of inertia, which occur due to the motion of pairs of internal material points, and internal rotors. The components of the inertia tensor and the gyrostatic momentum are periodic functions. In general, the problem is nonintegrable. In a special case, the relationship of the problem under consideration with the Liouville problem with changing parameters is shown. The case of the Chaplygin ball moving from rest is considered separately. Poincaré maps are constructed, strange attractors are found, and the stages of the origin of strange attractors are shown. Also, the trajectories of contact points are constructed to confirm the chaotic dynamics of the ball. A chart of dynamical regimes is constructed in a separate case for analyzing the nature of strange attractors.Analysis of special cases in the study of bifurcations of periodic solutions of the Ikeda equation.https://www.zbmath.org/1455.370462021-03-30T15:24:00+00:00"Kubyshkin, Evgenii P."https://www.zbmath.org/authors/?q=ai:kubyshkin.evgenii-p"Moriakova, Alena R."https://www.zbmath.org/authors/?q=ai:moriakova.alena-rSummary: This paper deals with bifurcations from the equilibrium states of periodic solutions of the Ikeda equation, which is well known in nonlinear optics as an equation with a delayed argument, in two special cases that have not been considered previously. Written in a characteristic time scale, the equation contains a small parameter with a derivative, which makes it singular. Both cases share a single mechanism of the loss of stability of equilibrium states under changes of the parameters of the equation associated with the passage of a countable number of roots of the characteristic equation through the imaginary axis of the complex plane, which are in this case in certain resonant relations. It is shown that the behavior of solutions of the equation with initial conditions from fixed neighborhoods of the studied equilibrium states in the phase space of the equation is described by countable systems of nonlinear ordinary differential equations that have a minimal structure and are called the normal form of the equation in the vicinity of the studied equilibrium state. An algorithm for constructing such systems of equations is developed. These systems of equations allow us to single out one ``fast'' variable and a countable number of ``slow'' variables, which makes it possible to apply the averaging method to the systems of equations obtained. Equilibrium states of the averaged system of equations of ``slow'' variables in the original equation correspond to periodic solutions of the same nature of sustainability. In the special cases under consideration, the possibility of simultaneous bifurcation from equilibrium states of a large number of stable periodic solutions (multistability bifurcation) and evolution of these periodic solutions to chaotic attractors with changing bifurcation parameters is shown. One of the special cases is associated with the formation of paired equilibrium states (a stable and an unstable one). An analysis of bifurcations in this case provides an explanation of the formation of the ``boiling points of trajectories'', when a periodic solution arises ``out of nothing'' at some point in the phase space under changes of the parameters of the equation and quickly becomes chaotic.Control of an inverted wheeled pendulum on a soft surface.https://www.zbmath.org/1455.370772021-03-30T15:24:00+00:00"Kiselev, Oleg M."https://www.zbmath.org/authors/?q=ai:kiselev.oleg-mSummary: The dynamics of an inverted wheeled pendulum controlled by a proportional plus integral plus derivative action controller in various cases is investigated. The properties of trajectories are studied for a pendulum stabilized on a horizontal line, an inclined straight line and on a soft horizontal line. Oscillation regions on phase portraits of dynamical systems are shown. In particular, an analysis is made of the stabilization of the pendulum on a soft surface, modeled by a differential inclusion. It is shown that there exist trajectories tending to a semistable equilibrium position in the adopted mathematical model. However, in numerical simulations, as well as in the case of real robotic devices, such trajectories turn into a limit cycle due to round-off errors and perturbations not taken into account in the model.Local bifurcation structure of a bouncing ball system with a piecewise polynomial function for table displacement.https://www.zbmath.org/1455.370642021-03-30T15:24:00+00:00"Okishio, Yudai"https://www.zbmath.org/authors/?q=ai:okishio.yudai"Ito, Hiroaki"https://www.zbmath.org/authors/?q=ai:ito.hiroaki"Kitahata, Hiroyuki"https://www.zbmath.org/authors/?q=ai:kitahata.hiroyukiThis work considers a ball bouncing on a heavy flat table subject to a predefined periodic vertical motion, taking into account the gravity force and neglecting air resistance. In this setting, the authors compare the system behavior assuming the classical sinusoidal table excitation with the case where the excitation is given by periodic piecewise polynomial functions. In the latter case, the authors present analytical approximations of period-doubling bifurcation points. The periodic solutions are characterized in this work by the pair \((n,k)\), representing the number \(n\) of ball bounces per \(k\) table oscillations. The analytical predictions are compared with bifurcations diagrams obtained numerically by varying the amplitude of excitation, considering sinusoidal and piecewise polynomial periodic functions. In the sinusoidal case the authors detect the presence of a period-doubling cascade leading to chaos, following Feigenbaum's universality. For the piecewise polynomial excitation, the bifurcation diagram differs considerably in that the period-doubling route to chaos is replaced by the disappearance of a period-two solution via a border-collision bifurcation, after which chaotic solutions appear.
Reviewer: Joseph Páez Chávez (Guayaquil)On linear invariant manifolds in the generalized problem of motion of a top in a magnetic field.https://www.zbmath.org/1455.700122021-03-30T15:24:00+00:00"Irtegov, Valentin"https://www.zbmath.org/authors/?q=ai:irtegov.valentin-dmitrievich"Titorenko, Tatiana"https://www.zbmath.org/authors/?q=ai:titorenko.tatiana-nSummary: Differential equations describing the rotation of a rigid body with a fixed point under the influence of forces generated by the Barnett-London effect are analyzed. They are a multiparametric system of equations. A technique for finding their linear invariant manifolds is proposed. With this technique, we find the linear invariant manifolds of codimension 1 and use them in the qualitative analysis of the equations. Computer algebra tools are applied to obtain the invariant manifolds and to analyze the equations. These tools proved to be essential.
For the entire collection see [Zbl 1428.68016].Explicit examples in ergodic optimization.https://www.zbmath.org/1455.370662021-03-30T15:24:00+00:00"Ferreira, Hermes H."https://www.zbmath.org/authors/?q=ai:ferreira.hermes-h"Lopes, Artur O."https://www.zbmath.org/authors/?q=ai:lopes.artur-oscar"Oliveira, Elismar R."https://www.zbmath.org/authors/?q=ai:oliveira.elismar-rSummary: Denote by \(T\) the transformation \(T(x)=2 \,x\) (mod 1). Given a potential \(A:S^1\rightarrow\mathbb{R}\) we are interested in exhibiting in several examples the explicit expression for the calibrated subaction \(V:S^1\rightarrow\mathbb{R}\) for \(A\). The action of the 1/2 iterative procedure \(\mathcal{G}\), acting on continuous functions \, was analyzed in a companion paper. Given an initial condition \(f_0\), the sequence, \(\mathcal{G}^n(f_0)\) will converge to a subaction. The sharp numerical evidence obtained from this iteration allow us to guess explicit expressions for the subaction in several worked examples: among them for \(A(x)=\sin^2(2\pi x)\) and \(A(x)=\sin(2\pi x)\). Here, among other things, we present piecewise analytical expressions for several calibrated subactions. The iterative procedure can also be applied to the estimation of the joint spectral radius of matrices. We also analyze the iteration of \(\mathcal{G}\) when the subaction is not unique. Moreover, we briefly present the version of the 1/2 iterative procedure for the estimation of the main eigenfunction of the Ruelle operator.Existence of homoclinic orbits and heteroclinic cycle in a class of three-dimensional piecewise linear systems with three switching manifolds.https://www.zbmath.org/1455.370232021-03-30T15:24:00+00:00"Zhu, Bin"https://www.zbmath.org/authors/?q=ai:zhu.bin.6"Wei, Zhouchao"https://www.zbmath.org/authors/?q=ai:wei.zhouchao"Escalante-González, R. J."https://www.zbmath.org/authors/?q=ai:escalante-gonzalez.r-j"Kuznetsov, Nikolay V."https://www.zbmath.org/authors/?q=ai:kuznetsov.nikolay-vSummary: In this article, we construct a kind of three-dimensional piecewise linear (PWL) system with three switching manifolds and obtain four theorems with regard to the existence of a homoclinic orbit and a heteroclinic cycle in this class of PWL system. The first theorem studies the existence of a heteroclinic cycle connecting two saddle-foci. The existence of a homoclinic orbit connecting one saddle-focus is investigated in the second theorem, and the third theorem examines the existence of a homoclinic orbit connecting another saddle-focus. The last one proves the coexistence of the heteroclinic cycle and two homoclinic orbits for the same parameters. Numerical simulations are given as examples and the results are consistent with the predictions of theorems.
{\copyright 2020 American Institute of Physics}On the Medvedev-Scanlon conjecture for minimal threefolds of nonnegative Kodaira dimension.https://www.zbmath.org/1455.140272021-03-30T15:24:00+00:00"Bell, Jason P."https://www.zbmath.org/authors/?q=ai:bell.jason-p"Ghioca, Dragos"https://www.zbmath.org/authors/?q=ai:ghioca.dragos"Reichstein, Zinovy"https://www.zbmath.org/authors/?q=ai:reichstein.zinovy-b"Satriano, Matthew"https://www.zbmath.org/authors/?q=ai:satriano.matthewSummary: Motivated by work of Zhang from the early `90s, \textit{A. Medvedev} and \textit{T. Scanlon} [Ann. Math. (2) 179, No. 1, 81--177 (2014; Zbl 1347.37145)] formulated the following conjecture. Let \(F\) be an algebraically closed field of characteristic 0 and let \(X\) be a quasiprojective variety defined over \(F\) endowed with a dominant rational self-map \(\phi \).
Then there exists a point \(x \in X(F)\) with Zariski dense orbit under \(\phi\) if and only if \(\phi\) preserves no nontrivial rational fibration, i.e., there exists no nonconstant rational functions \(f\in F(X)\) such that \(\phi^\ast(f)=f\).
The Medvedev-Scanlon conjecture holds when \(F\) is uncountable. The case where \(F\) is countable (e.g., \(F = \overline{\mathbb Q})\) is much more difficult; here the Medvedev-Scanlon conjecture has only been proved in a small number of special cases. In this paper we show that the Medvedev-Scanlon conjecture holds for all varieties of positive Kodaira dimension, and explore the case of Kodaira dimension 0. Our results are most definitive in dimension 3.The Zak transform on Gelfand-Shilov and modulation spaces with applications to operator theory.https://www.zbmath.org/1455.420322021-03-30T15:24:00+00:00"Toft, Joachim"https://www.zbmath.org/authors/?q=ai:toft.joachimSummary: We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.Topological conjugacy for the Morse minimal system: an example.https://www.zbmath.org/1455.370162021-03-30T15:24:00+00:00"Dykstra, Andrew"https://www.zbmath.org/authors/?q=ai:dykstra.andrew"Lemasurier, Michelle"https://www.zbmath.org/authors/?q=ai:lemasurier.michelleIn this short and elegant paper, the authors provide an example of a subshift that is topologically conjugate to a Morse minimal system but is not generated by a constant-length substitution. This confirms that the property of being generated by a constant-length substitution is not a topological conjugacy invariant. The proof uses results developed in [\textit{E. M. Coven} et al., Indag. Math., New Ser. 28, No. 1, 91--107 (2017; Zbl 1356.37021)].
Reviewer: Alexei Kanel-Belov (Ramat-Gan)The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with two equal masses.https://www.zbmath.org/1455.370692021-03-30T15:24:00+00:00"Kuang, Wentian"https://www.zbmath.org/authors/?q=ai:kuang.wentian"Ouyang, Tiancheng"https://www.zbmath.org/authors/?q=ai:ouyang.tiancheng"Xie, Zhifu"https://www.zbmath.org/authors/?q=ai:xie.zhifu"Yan, Duokui"https://www.zbmath.org/authors/?q=ai:yan.duokuiThe paper deals with periodic orbits in the three body problem. The authors study the collisionless Broucke-Hénon orbit and the Schubart orbit with collision using variational methods. The variational approach is outlined very clear from the Lagrangian action. Scenarios with and without collisions are studied.
Reviewer: Wolfgang G. Hollik (Karlsruhe)Kantorovich-Rubinstein-Wasserstein distance between overlapping attractor and repeller.https://www.zbmath.org/1455.370402021-03-30T15:24:00+00:00"Chigarev, Vladimir"https://www.zbmath.org/authors/?q=ai:chigarev.vladimir"Kazakov, Alexey"https://www.zbmath.org/authors/?q=ai:kazakov.alexey-o"Pikovsky, Arkady"https://www.zbmath.org/authors/?q=ai:pikovsky.arkady-sIn this paper, the Kantorovich-Rubinstein-Wasserstein distance is used to characterize the difference between overlapping attractors and repellers.
``Recently, a mixed dynamics attracted much attention, where attractors (limiting sets forward in time) and repellers (limiting sets at the dynamics backward in time) overlap but do not coincide. Here, we introduce a simple way to generate such an overlapping by adding dissipation to the conservative systems in a special controlled way.''
In Section 2, some systems with chaotic behavior are considered. Three of them are maps defined on a two-dimensional torus possessing the uniform invariant measure, namely the Anosov cat map, the Chirikov standard map, and the skew shift. By employing a Möbius circle map, a dissipation to these maps is added. For the perturbed maps, attractors and repellers are investigated. The last model is the incompressible three-dimensional flow of the ABC-type on a three-dimensional torus.
The Kantorovich-Rubinstein-Wasserstein distance is presented in deep detail.
A numerical study of attractors and repellers is presented.
Reviewer: Cristian Lăzureanu (Timisoara)Limit cycle bifurcations from an order-3 nilpotent center of cubic Hamiltonian systems perturbed by cubic polynomials.https://www.zbmath.org/1455.340412021-03-30T15:24:00+00:00"Zhang, Li"https://www.zbmath.org/authors/?q=ai:zhang.li.1|zhang.li.10|zhang.li|zhang.li.4|zhang.li.6|zhang.li.11|zhang.li.3|zhang.li.9|zhang.li.2|zhang.li.8|zhang.li.5|zhang.li.7|zhang.li.12"Wang, Chenchen"https://www.zbmath.org/authors/?q=ai:wang.chenchen"Hu, Zhaoping"https://www.zbmath.org/authors/?q=ai:hu.zhaopingAuthors' abstract: From [\textit{M. A. Han} et al., ``Polynomial Hamiltonian systems with a nilpotent critical point'', Adv. Space Res. 46, No. 4, 521--525 (2009; \url{doi:10.1016/j.asr.2008.08.025})] we know that the highest order of the nilpotent center of cubic Hamiltonian system is \(3\). In this paper, perturbing the Hamiltonian system which has a nilpotent center of order \(3\) at the origin by cubic polynomials, we study the number of limit cycles of the corresponding cubic near-Hamiltonian systems near the origin. We prove that we can find seven and at most seven limit cycles near the origin by the first-order Melnikov function.
Reviewer: Majid Gazor (Isfahan)Embedding fractals in Banach, Hilbert or Euclidean spaces.https://www.zbmath.org/1455.280042021-03-30T15:24:00+00:00"Banakh, Taras"https://www.zbmath.org/authors/?q=ai:banakh.taras-o"Nowak, Magdalena"https://www.zbmath.org/authors/?q=ai:nowak.magdalena"Strobin, Filip"https://www.zbmath.org/authors/?q=ai:strobin.filipThe authors associate to a given metric fractal with the doubling property, another metric fractal that is equi-Hölder equivalent to an Euclidean fractal. This leads to their main result, saying that each finite-dimensional compact metrizable space K containing an open uncountable zero-dimensional space Z is homeomorphic to an Euclidean fractal.
Reviewer: George Stoica (Saint John)Global dynamics for mathematical model of \textit{Echinococcus multilocularis} in rodents and red foxes.https://www.zbmath.org/1455.370722021-03-30T15:24:00+00:00"Hassan, Adamu Shitu"https://www.zbmath.org/authors/?q=ai:hassan.adamu-shitu"Munganga, Justin M. W."https://www.zbmath.org/authors/?q=ai:munganga.justin-manango-wSummary: A deterministic model for transmission of Echinococcus multilocularis \textit{(EM)}, a parasitic disease responsible for human alveolar echinococcosis, is formulated and analyzed rigorously. The model consists of two hosts, with three compartments each, and concentration of the parasites from environment as sources of infection. The model takes into account a predator-prey relationship between the major hosts and obtained a threshold value for their existence. Systematic derivation of basic reproduction number, \( \mathcal{R}_0\), is provided. Thorough qualitative analysis of the model reveals that it has a local and global asymptotic stable disease-free equilibrium when \(\mathcal{R}_0 < 1\); thus, (\textit{EM}) will die out in the populations. However, when \(\mathcal{R}_0\) exceeds unity, the model exhibits a unique endemic equilibrium, which is globally asymptotic stable; hence, disease will persist. The elasticity indices and partial rank correlation coefficients of the basic reproduction number and cumulative new cases of the two hosts with respect to parameter values are computed. Sensitivity analyses identified key parameters that are the most sensitive and can be used for control strategies in reducing \(\mathcal{R}_0\) below unity. Numerical simulations are used to verify theoretical results and quantify prevalence of the disease in host populations.The limiting distribution of a non-stationary integer valued GARCH\((1,1)\) process.https://www.zbmath.org/1455.621802021-03-30T15:24:00+00:00"Michel, Jon"https://www.zbmath.org/authors/?q=ai:michel.jonAuthor's abstract: ``We consider the integer-valued GARCH\((1,1)\) process defined by the two equation system \(Y_n\overset{d}{\sim}\text{Poisson}(\lambda_n)\) and \(\lambda_{n+1}= \omega+\alpha Y_n+\beta\lambda_n\). When \(\alpha+\beta< 1\), this process has a stationary solution and the properties are well understood. In this note, we find the limiting distribution of \(\lambda_n\) and \(Y_n\) for the case of \(\alpha+\beta= 1\).''
The paper is structured in 3 chapters: 1. Introduction; 2. Limit distribution (with Theorem 2.1); 3. Conclusion; References (20 references); Appendix A (Mathematical appendix, Proof of Theorem 2.1).
The INGARCH (integer-valued GARCH) process was first introduced by \textit{T. H. Rydberg} and \textit{N. Shephard} (1999, unpublished paper, see \url{http://fmwww.bc.edu/RePEc/es2000/0740.pdf}), further research see [Zbl 1150.62046; Zbl 1205.62130; Zbl 1253.62058; Zbl 1277.60089; Zbl 1290.62092; Zbl 1367.62267; Zbl 1207.62165; Zbl 1331.62003].
Reviewer: Ludwig Paditz (Dresden)Quasi-doubling of self-similar measures with overlaps.https://www.zbmath.org/1455.280062021-03-30T15:24:00+00:00"Hare, Kathryn E."https://www.zbmath.org/authors/?q=ai:hare.kathryn-e"Hare, Kevin G."https://www.zbmath.org/authors/?q=ai:hare.kevin-g"Troscheit, Sascha"https://www.zbmath.org/authors/?q=ai:troscheit.saschaSummary: The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which we call the Assouad dimension (of the measure) and is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and only if it is doubling.
Motivated by recent progress on both the Assouad dimension of measures that satisfy a strong separation condition and the quasi-Assouad dimension of metric spaces, we introduce the notion of the quasi-Assouad dimension of a measure. As with sets, the quasi-Assouad dimension of a measure is dominated by its Assouad dimension. It dominates both the quasi-Assouad dimension of its support and the supremal local dimension of the measure, with strict inequalities possible in all cases.
Our main focus is on self-similar measures in \(\mathbb R\) whose support is an interval and which may have `overlaps'. For measures that satisfy a weaker condition than the weak separation condition we prove that finite quasi-Assouad dimension is equivalent to quasi-doubling of the measure, a strictly less restrictive property than doubling. Further, we exhibit a large class of such measures for which the quasi-Assouad dimension coincides with the maximum of the local dimension at the endpoints of the support. This class includes all regular, equicontractive self-similar measures satisfying the weak separation condition, such as convolutions of uniform Cantor measures with integer ratio of dissection. Other properties of this dimension are also established and many examples are given.On interpreting Patterson-Sullivan measures of geometrically finite groups as Hausdorff and packing measures.https://www.zbmath.org/1455.370342021-03-30T15:24:00+00:00"Simmons, David"https://www.zbmath.org/authors/?q=ai:simmons.davidSummary: We provide a new proof of a theorem whose proof was sketched by \textit{D. Sullivan} [Acta Math. 149, 215--237 (1982; Zbl 0517.58028)], namely that if the Poincaré exponent of a geometrically finite Kleinian group \(G\) is strictly between its minimal and maximal cusp ranks, then the Patterson-Sullivan measure of \(G\) is not proportional to the Hausdorff or packing measure of any gauge function. This disproves a conjecture of \textit{B. O. Stratmann} [in: Non-Euclidean geometries. János Bolyai memorial volume. Papers from the international conference on hyperbolic geometry, Budapest, Hungary, July 6--12, 2002. New York, NY: Springer. 227--247 (2006; Zbl 1104.37020)].Counting periodic points in parallel graph dynamical systems.https://www.zbmath.org/1455.370382021-03-30T15:24:00+00:00"Aledo, Juan A."https://www.zbmath.org/authors/?q=ai:aledo.juan-angel"Barzanouni, Ali"https://www.zbmath.org/authors/?q=ai:barzanouni.ali"Malekbala, Ghazaleh"https://www.zbmath.org/authors/?q=ai:malekbala.ghazaleh"Sharifan, Leila"https://www.zbmath.org/authors/?q=ai:sharifan.leila"Valverde, Jose C."https://www.zbmath.org/authors/?q=ai:valverde.jose-cSummary: Let \(F: \{0,1\}^n\longrightarrow \{0,1\}^n\) be a parallel dynamical system over an undirected graph with a Boolean maxterm or minterm function as a global evolution operator. It is well known that every periodic point has at most two periods. Actually, periodic points of different periods cannot coexist, and a fixed point theorem is also known. In addition, an upper bound for the number of periodic points of \(F\) has been given. In this paper, we complete the study, solving the minimum number of periodic points' problem for this kind of dynamical systems which has been usually considered from the point of view of complexity. In order to do this, we use methods based on the notions of minimal dominating sets and maximal independent sets in graphs, respectively. More specifically, we find a lower bound for the number of fixed points and a lower bound for the number of 2-periodic points of \(F\). In addition, we provide a formula that allows us to calculate the exact number of fixed points. Furthermore, we provide some conditions under which these lower bounds are attained, thus generalizing the fixed-point theorem and the 2-period theorem for these systems.A new mean ergodic theorem for tori and recurrences.https://www.zbmath.org/1455.370052021-03-30T15:24:00+00:00"Nguyen, Khoa D."https://www.zbmath.org/authors/?q=ai:nguyen.khoa-d|nguyen.khoa-dangSummary: Let \(X\) be a finite-dimensional connected compact abelian group equipped with the normalized Haar measure \(\mu\). We obtain the following mean ergodic theorem over `thin' phase sets. Fix \(k\geq 1\) and, for every \(n\geq 1\), let \(A_n\) be a subset of \(\mathbb{Z}^k\cap [-n,n]^k\). Assume that \((A_n)_{n\geq 1}\) has \(\omega (1/n)\) density in the sense that \(\lim_{n\rightarrow\infty}(|A_n|/n^{k-1})=\infty\). Let \(T_1,\dots,T_k\) be ergodic automorphisms of \(X\). We have
\[
\frac{1}{|A_n|}\mathop{\sum}_{(n_1,\dots ,n_k)\in A_n}f_1(T_1^{n_1}(x))\cdots f_k(T_k^{n_k}(x))\stackrel{L_\mu^2}{\longrightarrow }\int f_1\,d\mu\cdots \int f_k\,d\mu,
\]
for any \(f_1,\dots ,f_k\in L_\mu^{\infty}\). When the \(T_i\) are ergodic epimorphisms, the same conclusion holds under the further assumption that \(A_n\) is a subset of \([0,n]^k\) for every \(n\). The density assumption on the \(A_i\) is necessary. Immediate applications include certain Poincaré style recurrence results.Equidistribution of Farey sequences on horospheres in covers of \(\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})\) and applications.https://www.zbmath.org/1455.370032021-03-30T15:24:00+00:00"Heersink, Byron"https://www.zbmath.org/authors/?q=ai:heersink.byronAuthor's abstract: We establish the limiting distribution of
certain subsets of Farey sequences, i.e., sequences of primitive
rational points, on expanding horospheres in covers
\({\Delta\setminus SL(n+1,\mathbb{R})}\) of
\({SL(n+1,\mathbb{Z})\setminus SL(n+1,\mathbb{R})}\), where \(\Delta\)
is a finite-index subgroup of \({SL(n+1,\mathbb{Z})}\). These subsets
can be obtained by projecting to the hyperplane
\({\{(x_1,...,x_{n+1})\in\mathbb{R}^{n+1}:x_{n+1}=1\}}\) sets of the
form \({\mathbf{A}=\cup_{j=1}^Ja_j\Delta}\), where for all \(j\), \(a_j\)
is a primitive lattice point in \(\mathbb{Z}^{n+1}\). Our method
involves applying the equidistribution of expanding horospheres in
quotients of \({SL(n+1,\mathbb{R})}\) developed by \textit{J. Marklof} and \textit{A. Strömbergsson} [Ann. Math. (2) 172, No. 3, 1949--2033 (2010; Zbl 1211.82011)], and more precisely understanding how the
full Farey sequence distributes in \({\Delta\setminus
SL(n+1,\mathbb{R})}\) when embedded on expanding horospheres as done
in previous work by Marklof. For each of the Farey sequence subsets,
we extend the statistical results by \textit{J. Marklof} [in: Limit theorems in probability, statistics and number theory. In honor of Friedrich Götze on the occasion of his 60th birthday. Selected papers based on the presentations at the workshop, Bielefeld, Germany, August 4--6, 2011. Berlin: Springer. 49--57 (2013; Zbl 1291.37044)]
regarding the full multidimensional Farey sequences, and solutions
by \textit{J. S. Athreya} and \textit{A. Ghosh} [Enseign. Math. (2) 64, No. 1--2, 1--21 (2018; Zbl 1435.37014)] to Diophantine approximation
problems of \textit{P. Erdős} et al. [Colloq. Math. 6, 119--126 (1958; Zbl 0087.04305)] and \textit{H. Kesten} [Trans. Am. Math. Soc. 103, 189--217 (1962; Zbl 0105.03805)]. We also prove that
\textit{J. Marklof}'s result [Invent. Math. 181, No. 1, 179--207 (2010; Zbl
1200.11022)] on the asymptotic distribution of Frobenius numbers
holds for sets of primitive lattice points of the form \(\mathbf{A}\).
Reviewer: Ivan Podvigin (Novosibirsk)Distributional chaos in multifractal analysis, recurrence and transitivity.https://www.zbmath.org/1455.370092021-03-30T15:24:00+00:00"Chen, An"https://www.zbmath.org/authors/?q=ai:chen.an"Tian, Xueting"https://www.zbmath.org/authors/?q=ai:tian.xuetingSummary: There is much research on the dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, the Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure), etc. However, this is not the case from the viewpoint of chaos. There are many results on the relationship of positive topological entropy and various chaos. However, positive topological entropy does not imply a strong version of chaos, called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. In this paper, we will show that, for dynamical systems with specification properties, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, we prove that several recurrent level sets of points with different recurrent frequency have uncountable DC1-scrambled subsets. The major argument in proving the above results is that there exists uncountable DC1-scrambled subsets in saturated sets.Topological entropy of Markov set-valued functions.https://www.zbmath.org/1455.370142021-03-30T15:24:00+00:00"Alvin, Lori"https://www.zbmath.org/authors/?q=ai:alvin.lori"Kelly, James P."https://www.zbmath.org/authors/?q=ai:kelly.james-pierreThe authors deal with the topological entropy of a family of set-valued functions and study Markov set-valued functions defined in [the authors, Topology Appl. 241, 102--114 (2018; Zbl 1401.54016)]. They investigate the topological entropy of such maps. Taking into account symbolic techniques, the authors determine upper and lower bounds of the entropy for a Markov set-valued function. The authors calculate an upper bound for the entropy of a Markov set-valued function in the case where each map in the associated Markov system is monotone and then get a method for calculating a lower bound for the entropy. An equivalent definition of topological entropy that uses open covers is given.
Reviewer: Hasan Akin (Gaziantep)On centro-affine curves and Bäcklund transformations of the KdV equation.https://www.zbmath.org/1455.370582021-03-30T15:24:00+00:00"Tabachnikov, Serge"https://www.zbmath.org/authors/?q=ai:tabachnikov.serge-lAuthor's abstract: We continue the study of the Korteweg-de Vries equation in terms of cento-affine curves, initiated by \textit{U. Pinkall} [Result. Math. 27, No. 3--4, 328--332 (1995; Zbl 0835.35128)]. A centro-affine curve is a closed parametric curve in the affine plane such that the determinant made by the position and the velocity vectors is identically one. The space of centro-affine curves is acted upon by the special linear group, and the quotient is identified with the space of Hill's equations with periodic solutions. It is known that the space of centro-affine curves carries two pre-symplectic structures, and the KdV flow is identified with is a bi-Hamiltonian dynamical system therein. We introduce a one-parameter family of transformations on centro-affine curves, prove that they preserve both presymplectic structures, commute with the KdV flow, and share the integrals with it. Furthermore, the transformation commute with each other (Bianchi permutability). We also describe integrals of the KdV equation as arising from the monodromy of Riccati equations associated with centro-affine curves. We are motivated by our work (joint with \textit{M. Arnold} et al. [Cross-ratio dynamics on ideal polygons, Preprint \url{https://arxiv.org/pdf/1812.05337.pdf}]), concerning the cross-ratio dynamics on ideal polygons in the hyperbolic plane and hyperbolic space, whose continuous version is studied in the present paper.
Reviewer: Ivan C. Sterling (St. Mary's City)From variational to bracket formulations in nonequilibrium thermodynamics of simple systems.https://www.zbmath.org/1455.800042021-03-30T15:24:00+00:00"Gay-Balmaz, François"https://www.zbmath.org/authors/?q=ai:gay-balmaz.francois"Yoshimura, Hiroaki"https://www.zbmath.org/authors/?q=ai:yoshimura.hiroakiAmong the approaches that generalize the fundamental principles of thermodynamics to the case of irreversible processes, one may mention the variational and bracket formulations. The first approach is a systematic one: it allows to tackle many different situations from a common point of view. The second one is a somewhat case-to-case approach: different types of brackets (Poisson bracket is the most known) may be used in different situations.
In this paper the authors unities both approaches.
The generalization of the Lagrange-d'Alembert principle itself begins from the mechanical system with friction; for this system, the notions of entropy and temperature are defined. Then for systems with internal mass transfer, the thermodynamic displacements are defined besides the well-known thermodynamic forces and fluxes. It allows to derive the evolution equations for such systems.
Later three main bracket formalisms, namely single generator, double generator and GENERIC formalism, are derived from the the ,obtained principles and equations. Explicit expressions for all these types of brackets in terms of entropy, Lagrangian and Hamiltonian of the system under discussion are obtained.
The authors also discuss case when configuration space is a Lie group and the system has some symmetry connected with this group. For that case reduced expressions for the brackets are derived. The behaviour of coadjoint orbits in this case is discussed, too.
Reviewer: Aleksey Syromyasov (Saransk)The short pulse equation: Bäcklund transformations and applications.https://www.zbmath.org/1455.350582021-03-30T15:24:00+00:00"Mao, Hui"https://www.zbmath.org/authors/?q=ai:mao.hui"Liu, Q. P."https://www.zbmath.org/authors/?q=ai:liu.qingpingSummary: A Bäcklund transformation (BT), which involves both independent and dependent variables, is established and studied for the short pulse (SP) equation. Based it, the nonlinear superposition formulae for 2-, 3-, and 4-BT are presented. The general result for the composition of \(N\)-BTs is achieved and given in terms of determinants. As applications, various solutions including loop solitons, breather solutions, and their interaction solutions are worked out.A guide to Lie systems with compatible geometric structures.https://www.zbmath.org/1455.370012021-03-30T15:24:00+00:00"de Lucas, Javier"https://www.zbmath.org/authors/?q=ai:de-lucas.javier"Sardón Muñoz, Cristina"https://www.zbmath.org/authors/?q=ai:sardon-munoz.cristinaAs recalled at the beginning of the book, ``A Lie system is a non-autonomous system of first-order ordinary differential equations whose general solution can be written as a function, the so-called superposition rule, of a generic family of particular solutions and a set of constants related to initial conditions.'' The interesting fact is that the defining property of Lie systems is determined by an autonomous (i.e., time-independent) structure via the Lie-Scheffers theorem: the system of ordinary differential equations is a Lie system if and only if it is linear combination, with coefficients depending on time only, of a finite number of autonomous vector fields spanning a finite-dimensional Lie algebra. The analysis of a time-dependent system is therefore completely determined by a time-independent algebraic structure.
The aim of the authors of this book is to show that the existence of additional structures, typical of Hamiltonian mechanics, sheds new light on the known results and allows the determination of new properties of the systems. It is remarkable that some Hamiltonian structure can be found in several relevant Lie systems, starting from the first- and second-order Riccati system itself and including among others the Smorodinsky-Winternitz system and the Kummer-Schwarz equation. The authors collect here the results of most, if not all, of the Lie systems from the beginning until the most recent ones, particularly studied by the schools of Winternitz, Carinena and the authors themselves.
The introduction includes a short but complete historical presentation of Lie systems.
The first two chapters of the book introduce the Hamiltonian geometric structures necessary in the following as well as the basics of Lie systems theory, making the exposition completely self-contained, while Chapters 3--7 develop the main topic of the book.
First, a geometric version of the Lie-Scheffers Theorem is given, and the superposition rule is characterized by a zero-curvature connection on a suitable bundle. The superposition rules are properly defined for second- and third-order systems. Lie-Hamilton systems are then defined as those Lie systems such that the generators of their finite-dimensional Lie algebra are all Hamiltonian with respect to some Poisson structure. It is then possible to associate with these systems a Poisson coalgebra and the superposition rule can be determined from the Casimirs of this coalgebra (Theorem 4.37). Several examples are given and a classification of Lie-Hamilton systems in the plane is derived with many instances of application in physics.
As a generalization for the geometric structure, Dirac-Lie systems are defined, in order to go from symplectic to presymplectic geometries, so that the generators of the Lie algebra of a Lie system are required to be Hamiltonian on some Dirac manifold. For example, third-order Kummer-Schwarz equations can be written as Lie systems, but not as Lie-Hamilton systems; the introduction of this more general structure allows a pure algebraic determination of the known superposition rule for these equations.
A further analogous generalization is provided by Jacobi-Lie systems, where the generators of the Lie algebra of a Lie system must be Hamiltonian on a Jacobi manifold. Jacobi-Lie systems are classified in dimension two and the example of coupled Riccati equations is discussed.
The \(k\)-symplectic-Lie systems complete the generalization of geometric structures, and the generators of the Lie algebra of a Lie systems are required to be \(k\)-Hamiltonian, i.e., Hamiltonian with respect to the \(k\)-presymplectic forms of the structure. \(k\)-symplectic Lie systems can be considered Dirac-Lie in different ways. Schwarzian equations are given as an example.
Chapter 8 is devoted to the classification of finite-dimensional Lie algebras of conformal Killing vector fields on two-dimensional pseudo-Riemannian manifolds and known results are reformulated in the perspective of application to Lie systems. Indeed, the Poisson tensor of some Lie-Hamilton system, such as Milne-Pinney equations, can be determined algebraically, instead of being integrated, by the approach described here.
In the last chapter it is shown that Lie symmetries techniques can be applied to partial differential equations such as partial Riccati equations.
The book is clearly written and rich of examples, all of them presented with many details. Thanks to a large bibliography, the whole research work done on Lie systems so far is offered to the reader.
Reviewer: Giovanni Rastelli (Vercelli)Thermodynamic formalism and integral means spectrum of logarithmic tracts for transcendental entire functions.https://www.zbmath.org/1455.300192021-03-30T15:24:00+00:00"Mayer, Volker"https://www.zbmath.org/authors/?q=ai:mayer.volker"Urbański, Mariusz"https://www.zbmath.org/authors/?q=ai:urbanski.mariuszIn this very interesting pioneering paper the authors develop an entirely new approach to the thermodynamic formalism for entire functions with bounded singular sets. This approach covers all entire functions for which the thermodynamic formalism has been so far established and goes far beyond. They show that there is a strong relation between the transfer operator, on whose properties the thermodynamic formalisms relay, and the integral means spectrum for logarithmic tracts. They derive from this that the negative spectrum property implies good behavior of this operator. The negative spectrum property turns out to be a very general condition which holds as soon as the tracts have some sufficiently nice geometry, for example, for quasidisks, John, or Hölder tracts. In these cases the authors get a good control of the corresponding transfer operators, leading to the full thermodynamic formalism along with its applications such as exponential decay of correlations, a central limit theorem, and a Bowen's formula for the Hausdorff dimension of radial Julia sets.
In particular, they show that the thermodynamic formalism holds for every hyperbolic function from the Eremenko-Lyubich analytic family of the Speicer class \(\mathcal{S}\) (the class \(S\) consists of entire functions with finite singular set), provided this family contains at least one function with Hölder tracts. The latter is, for example, the case if the Eremenko-Lyubich analytic family \(\mathcal{M}_g \) is associated with the Poincaré function of a polynomial \(g\in \mathcal{S}\) having a connected Julia set.
Reviewer: Olga M. Katkova (Boston)Spectral analysis of Morse-Smale flows. I: Construction of the anisotropic spaces.https://www.zbmath.org/1455.370242021-03-30T15:24:00+00:00"Dang, Nguyen Viet"https://www.zbmath.org/authors/?q=ai:dang.nguyen-viet"Rivière, Gabriel"https://www.zbmath.org/authors/?q=ai:riviere.gabrielGiven a smooth \(C^{\infty}\) flow \(\varphi^t:M\rightarrow M\) on a closed oriented manifold \(M^n\), one can define what is called the \textit{correlation function}:
\[ C_{\psi_1,\psi_2}(t):=\int_{M} (\varphi ^{-t})^{\ast} (\psi_1) \wedge (\psi_2),\]
where \(\psi_1,\psi_2\) are differential \(k\) and \(n-k\) forms on \(M\), respectively. Its Laplace transform is given by
\[ \widehat{C}_{\psi_1,\psi_2}(z):=\int_{0}^{\infty} e^{-tz} C_{\psi_1,\psi_2}(t) dt\]
which is well-defined for \(\text{Re}(z)>c>0\) (where \(c\) depends on the flow). If this holomorphic function admits a meromorphic extension to the entire
complex plane, the poles and their residues (\textit{correlation spectrum}) provide some information on the long time dynamics of the flow.
The authors [J. Inst. Math. Jussieu
19, No. 5, 1409--1465 (2020, Zbl 07238540); Am. J. Math. 142, No. 2, 547--593 (2020; Zbl 1444.37026); ``Topology of Pollicott-Ruelle resonant states'', Preprint, \url{arXiv:1703.08037}]
used spectral analysis to explore Morse-Smale dynamical systems, providing a generalization of their work done in
[Ann. Sci. Éc. Norm. Supér. (4) 52, No. 6, 1403--1458 (2019, Zbl 1448.37029)]
for the particular case of Morse-Smale gradient flows. The novelty in this context is that Morse-Smale flows admit not only hyperbolic fixed points (as in the gradient case) but also hyperbolic periodic orbits; moreover a transversality property between stable and unstable manifolds is satisfied.
The current paper is the first in this sequence. The main theorem proved herein guarantees that
the Laplace transformed correlator \(\widehat{C}_{\psi_1,\psi_2}(z)\) associated to a Morse-Smale flow \(\varphi\) has a meromorphic extension to \(\mathbb{C}\) whose poles are of finite order and are contained in a discrete subset of \(\mathbb{C}\). It is interesting to note that this result holds globally on the manifold in the sense that it is not required any restriction on the supports of \(\psi_1\) and \(\psi_2\).
This is done by developing a global functional groundwork adapted to a Morse-Smale flow.
The strategy is to apply a microlocal analysis for such flows following the ideas presented in [\textit{F. Faure} and \textit{J. Sjöstrand}, Commun. Math. Phys. 308, No. 2, 325--364 (2011, Zbl 1260.37016)]. The authors study the properties of the symplectic lift of such flow to the cotangent space; they show how to construct an escape function adapted to the Hamiltonian dynamics; then
they present anisotropic Sobolev spaces of currents adapted to Morse-Smale vector fields which allows them to show the existence of a discrete dynamical spectrum on these spaces.
Reviewer: Dahisy Lima (Santo André)Time correlation functions of equilibrium and nonequilibrium Langevin dynamics: derivations and numerics using random numbers.https://www.zbmath.org/1455.650192021-03-30T15:24:00+00:00"Shang, Xiaocheng"https://www.zbmath.org/authors/?q=ai:shang.xiaocheng"Kröger, Martin"https://www.zbmath.org/authors/?q=ai:kroger.martinStatistical solutions to the barotropic Navier-Stokes system.https://www.zbmath.org/1455.351722021-03-30T15:24:00+00:00"Fanelli, Francesco"https://www.zbmath.org/authors/?q=ai:fanelli.francesco"Feireisl, Eduard"https://www.zbmath.org/authors/?q=ai:feireisl.eduardIn this article, the authors apply the concept of statistical solutions to the analysis of the barotropic Navier-Stokes system with inhomogeneous boundary conditions. The motion of a compressible viscous fluid contained in a bounded domain \(\Omega\subset\mathbb{R}^d\), \(d=2,3\) is described by the mass density \(\rho(x,t)\) and the velocity \(u(x,t)\), \(x\in\Omega\), \(t>0\). \(\rho\) and \(u\) satisfy to the equations
\[
\begin{array}{l}
\frac{\partial \rho}{\partial t}+\mbox{div}\,(\rho u)=0,
\vphantom{\begin{array}{c}I\\I\end{array}}\\
\frac{\partial}{\partial t}(\rho u)+\mbox{div}\,(\rho u\otimes u)-\mbox{div}\,\mathbb{S}(u)+\nabla p(\rho)=\rho g,
\vphantom{\begin{array}{c}I\\I\end{array}}\\
u=u_B,\quad \rho=\rho_B \quad \mbox{on}\ \partial\Omega,
\vphantom{\begin{array}{c}I\\I\end{array}}\\
\rho(x,0)=\rho_0,\quad (\rho u)(x,0)=m_0.
\end{array}
\]
Here \(g,u_B,\rho_B,\rho_0,m_0\) are given functions, the pressure \(p(\rho)\) is a certain power function, \(\mathbb{S}(u)\) is the viscous stress tensor
\[
\mathbb{S}(u)= \mu\left(\nabla u +(\nabla u)^T-\frac{2}{d}\,\mbox{div}\,u\,\mathbb{I}\right)+\lambda\,\mbox{div}\,u\,\mathbb{I},\quad \mu>0,\ \lambda>0.
\]
A statistical solution of the problem is defined as a time depended family of Markov operators on the set of probability measures related to the data of the problem. The paper proves the existence of a statistical solution to the problem and discusses its relation to the weak solution.
Reviewer: Il'ya Sh. Mogilevskii (Tver')Taylor coefficients of the conformal map for the exterior of the reciprocal of the multibrot set.https://www.zbmath.org/1455.370432021-03-30T15:24:00+00:00"Shimauchi, Hirokazu"https://www.zbmath.org/authors/?q=ai:shimauchi.hirokazuSummary: In this paper we investigate normalized conformal mappings of the exterior of the reciprocal of the Multibrot set and analyze the growth of the denominator of the coefficients. Our inequality improves Ewing and Schober's result which was presented in [\textit{J. H. Ewing} and \textit{G. Schober}, J. Math. Anal. Appl. 170, No. 1, 104--114 (1992; Zbl 0766.30012)]. We use the coefficient formula of the author [in: Topics in finite or infinite dimensional complex analysis. Proceedings of the 19th international conference on finite or infinite dimensional complex analysis and applications (ICFIDCAA), Hiroshima, Japan, December 11--15, 2011. Sendai: Tohoku University Press. 237--248 (2013; Zbl 1333.37055)]. The straightforward adaptation of the proof in this paper slightly improves the main theorem of the author [Osaka J. Math. 52, No. 3, 737--746 (2015; Zbl 1352.37142)].Magneto-optical/ferromagnetic-material computation: Bäcklund transformations, bilinear forms and \(N\) solitons for a generalized \((3+1)\)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system.https://www.zbmath.org/1455.352482021-03-30T15:24:00+00:00"Gao, Xin-Yi"https://www.zbmath.org/authors/?q=ai:gao.xin-yi"Guo, Yong-Jiang"https://www.zbmath.org/authors/?q=ai:guo.yongjiang"Shan, Wen-Rui"https://www.zbmath.org/authors/?q=ai:shan.wenrui"Yuan, Yu-Qiang"https://www.zbmath.org/authors/?q=ai:yuan.yu-qiang"Zhang, Chen-Rong"https://www.zbmath.org/authors/?q=ai:zhang.chen-rong"Chen, Su-Su"https://www.zbmath.org/authors/?q=ai:chen.su-suThe authors study a generalized \((3+1)\)-dimensional modified Kadomtsev-Petviashvili system for electromagnetic waves in a ferromagnetic material. In particular, the authors derive variable coefficient dependent Auto-Bäcklund transformations for this system; from this, soliton examples are obtained. The authors also derive associated bilinear forms with the Hirota method and obtain two branches of \(N\)-solitonic solutions for the system.
Reviewer: Eric Stachura (Marietta)On the choice of the best members of the Kim family and the improvement of its convergence.https://www.zbmath.org/1455.370752021-03-30T15:24:00+00:00"Chicharro López, Francisco Israel"https://www.zbmath.org/authors/?q=ai:lopez.francisco-israel-chicharro"Cordero, Alicia"https://www.zbmath.org/authors/?q=ai:cordero.alicia"Garrido, Neus"https://www.zbmath.org/authors/?q=ai:garrido.neus"Torregrosa, Juan Ramon"https://www.zbmath.org/authors/?q=ai:torregrosa.juan-ramonSummary: The best members of the Kim family, in terms of stability, are obtained by using complex dynamics. From this elements, parametric iterative methods with memory are designed. A dynamical analysis of the methods with memory is presented in order to obtain information about the stability of them. Numerical experiments are shown for confirming the theoretical results.On stable and finite Morse index solutions of the fractional Toda system.https://www.zbmath.org/1455.352842021-03-30T15:24:00+00:00"Fazly, Mostafa"https://www.zbmath.org/authors/?q=ai:fazly.mostafa"Yang, Wen"https://www.zbmath.org/authors/?q=ai:yang.wenSummary: We develop a monotonicity formula for solutions of the fractional Toda system
\[
( - {\Delta} )^s f_\alpha = e^{- ( f_{\alpha + 1} - f_\alpha )} - e^{- ( f_\alpha - f_{\alpha - 1} )} \;\; \text{in} \;\; \mathbb{R}^n,
\]
when \(0 < s < 1, \alpha = 1, \cdots, Q, f_0 = - \infty, f_{Q + 1} = \infty\) and \(Q \geq 2\) is the number of equations in this system. We then apply this formula, technical integral estimates, classification of stable homogeneous solutions, and blow-down analysis arguments to establish Liouville type theorems for finite Morse index (and stable) solutions of the above system when \(n > 2 s\) and
\[
\frac{ {\Gamma} ( \frac{ n}{ 2} ) {\Gamma} ( 1 + s )}{ {\Gamma} ( \frac{ n - 2 s}{ 2} )} \frac{ Q ( Q - 1 )}{ 2} > \frac{ {\Gamma}^2 ( \frac{ n + 2 s}{ 4} )}{ {\Gamma}^2 ( \frac{ n - 2 s}{ 4} )} .
\]
Here, \( \Gamma\) is the Gamma function. When \(Q = 2\), the above equation is the classical (fractional) Gelfand-Liouville equation.Traveling chimera states in continuous media.https://www.zbmath.org/1455.740102021-03-30T15:24:00+00:00"Alvarez-Socorro, Alejandro J."https://www.zbmath.org/authors/?q=ai:alvarez-socorro.alejandro-j"Clerc, M. G."https://www.zbmath.org/authors/?q=ai:clerc.marcel-g"Verschueren, N."https://www.zbmath.org/authors/?q=ai:verschueren.nicolasSummary: Coupled oscillators exhibit intriguing dynamical states characterized by the coexistence of coherent and incoherent domains known as chimera states. Similar behaviors have been observed in coupled systems and continuous media. Here we investigate the transition from motionless to traveling chimera states in continuous media. Based on a prototype model for pattern formation, we observe coexistence between motionless and traveling chimera states. The spatial disparity of chimera states allows us to reveal the motion mechanism. The propagation of chimera states is described by their median and centroidal point. The mobility of these states depends on the size of the incoherent domain. The bifurcation diagram of traveling chimeras is elucidated.\(\lambda\)-Hypersurfaces on shrinking gradient Ricci solitons.https://www.zbmath.org/1455.370572021-03-30T15:24:00+00:00"Barbosa, Ezequiel"https://www.zbmath.org/authors/?q=ai:barbosa.ezequiel-r"Santana, Farley"https://www.zbmath.org/authors/?q=ai:santana.farley"Upadhyay, Abhitosh"https://www.zbmath.org/authors/?q=ai:upadhyay.abhitoshAuthors' abstract: We consider an orientable smooth non-totally geodesic \(\lambda\)-hypersurface \(\Sigma\) properly immersed on the cylinder shrinking soliton \(\left(\mathbb{S}_{\sqrt{2(k - 1)}}^k \times \mathbb{R}^l, \overline{g}, f\right)\), where \(\overline{g}\) is the product metric and the weight function \(f\) is defined by \(f(p, x) = \frac{|x|^2}{4}\). If \(\Sigma\) has weighted weak index less than or equal to \(l - 1\), we show that \(\Sigma\) splits off a linear space. Also we classify the \(f\)-stable orientable complete \(\lambda \)-hypersurfaces properly immersed on \((\mathbb{S}_{\sqrt{2 (k - 1)}}^k \times \mathbb{R}, \overline{g}, f)\), with \(f(p, t) = \frac{t^2}{4}\), obtaining as result \(\mathbb{S}_{\sqrt{2 (k - 1)}}^k \times \{t\}\) or \(\mathbb{S}_{\sqrt{2(k - 1)}}^{k - 1} \times \mathbb{R}\).
Reviewer: Ivan C. Sterling (St. Mary's City)Spectral gap property for random dynamics on the real line and multifractal analysis of generalised Takagi functions.https://www.zbmath.org/1455.370472021-03-30T15:24:00+00:00"Jaerisch, Johannes"https://www.zbmath.org/authors/?q=ai:jaerisch.johannes"Sumi, Hiroki"https://www.zbmath.org/authors/?q=ai:sumi.hirokiThis paper studies the connection between chaos and order by considering the random iteration of finitely many expanding \(C^{1+\epsilon}\) diffeomorphisms on the real line.
The authors prove the spectral gap property of the associated transition operator acting on spaces of Hölder continuous functions.
They apply their results to generalized Takagi functions \(\mathcal{T}\) on the real line including a multifractal analysis of the pointwise Hölder exponents in this setting.
Furthermore, the global Hölder continuity and the non-differentiability of the elements of \(\mathcal{T}\) are investigated.
Reviewer: Steve Pederson (Atlanta)Multidimensional diffeomorphisms with stable periodic points.https://www.zbmath.org/1455.370212021-03-30T15:24:00+00:00"Vasil'eva, E. V."https://www.zbmath.org/authors/?q=ai:vasileva.ekaterina-vladimirovna|vasileva.ekaterina-viktorovnaSummary: Diffeomorphisms of a multidimensional space into itself with a hyperbolic fixed point are discussed in this paper. It is assumed that at the intersection of stable and unstable manifolds, there are points that are different from the hyperbolic point. Such points are called homoclinic and are divided into transversal and non-transversal, depending on the behavior of stable and unstable manifolds. It follows from articles by \textit{S. E. Newhouse} [Topology 13, 9--18 (1974; Zbl 0275.58016)], \textit{S. V. Gonchenko} et al. [Russ. Acad. Sci., Dokl., Math. 47, No. 3, 1 (1993; Zbl 0864.58043); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 330, No. 2, 144--147 (1993)], \textit{B. F. Ivanov} [Differ. Uravn. 15, 1411--1414 (1979; Zbl 0423.34066)] and others, that with a certain method of tangency of a stable manifold with an unstable one, the neighborhood of a non-transversal homoclinic point contains an infinite number of stable periodic points, but at least one of the characteristic exponents at these points tends to zero with increasing period. The present study is a continuation of previous studies by the author. In previously published papers, restrictions were imposed on the eigenvalues of the Jacobi matrix of the original diffeomorphism at a hyperbolic point. More precisely, it was assumed that either all eigenvalues are real and the Jacobi matrix is diagonal, or the matrix has only one real eigenvalue less than one in modulus, while all other eigenvalues are various complex integers greater than one in modulus. Within this framework, conditions are obtained for the presence of an infinite set of stable periodic points with characteristic exponents separated from zero in an arbitrary neighborhood of a non-transversal homoclinic point. It is assumed in this paper that the Jacobi matrix of a diffeomorphism has an arbitrary set of eigenvalues at a hyperbolic point. In this case, the conditions are obtained for the existence of an infinite set of stable periodic points whose characteristic exponents are separated from zero in the neighborhood of the non-transversal homoclinic point. The conditions are imposed, first of all, on the method of tangency of a stable manifold with an unstable one; however, the proof of the theorem essentially uses the properties of the eigenvalues of the Jacobi matrix at a hyperbolic point.Tetrahedral chains and a curious semigroup.https://www.zbmath.org/1455.200282021-03-30T15:24:00+00:00"Stewart, Ian"https://www.zbmath.org/authors/?q=ai:stewart.ian-nSummary: In 1957 Steinhaus asked for a proof that a chain of identical regular tetrahedra joined face to face cannot be closed. \textit{S. Świerczkowski} [Colloq. Math. 7, 9--10 (1959; Zbl 0092.38701)] gave a proof in 1959. Several other proofs are known, based on showing that the four reflections in planes though the origin parallel to the faces of the tetrahedron generate a group \(\mathcal{R}\) isomorphic to the free product \(\mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z}_2* \mathbb{Z}_2\). We relate the reflections to elements of a semigroup of \(3\times 3\) matrices over the finite field \(\mathbb{Z}_3\), whose structure provides a simple and transparent new proof that \(\mathcal{R}\) is a free product. We deduce the non-existence of a closed tetrahedral chain, prove that \(\mathcal{R}\) is dense in the orthogonal group \(\mathbb{O}(3)\), and show that every \(\mathcal{R}\)-orbit on the 2-sphere is equidistributed.Existence of \(\omega \)-periodic solutions for a delayed chemostat with periodic inputs.https://www.zbmath.org/1455.340812021-03-30T15:24:00+00:00"Amster, Pablo"https://www.zbmath.org/authors/?q=ai:amster.pablo"Robledo, Gonzalo"https://www.zbmath.org/authors/?q=ai:robledo.gonzalo"Sepúlveda, Daniel"https://www.zbmath.org/authors/?q=ai:sepulveda.danielSummary: This paper proposes an \(\omega \)-periodic version of the Ellermeyer model of delayed chemostat. We obtain a sufficient condition ensuring the existence of a positive \(\omega \)-periodic solution. Our proof is based on the application of the generalized continuation theorem. In addition, as a consequence of the implicit function theorem, we obtain a uniqueness result for sufficiently small delays.A note on the growth of solutions of second-order complex linear differential equations.https://www.zbmath.org/1455.340912021-03-30T15:24:00+00:00"Qiao, Jianyong"https://www.zbmath.org/authors/?q=ai:qiao.jianyong"Zhang, Qi"https://www.zbmath.org/authors/?q=ai:zhang.qi|zhang.qi-shuhuason|zhang.qi.4|zhang.qi.2|zhang.qi.1"Long, Jianren"https://www.zbmath.org/authors/?q=ai:long.jianren"Li, Yezhou"https://www.zbmath.org/authors/?q=ai:li.yezhouLet \(g\) be an entire function. Then the order of growth of \(g\) is defined by
\[
\rho (g)=\limsup_{r\rightarrow +\infty} \frac{\log ^{+}T(r,g)}{\log r}=\limsup_{r\rightarrow +\infty}\frac{\log \log^{+}M(r,g)}{\log r},
\]
where \(T(r,g)\) is the Nevanlinna characteristic function and \(M(r,g)=\max_{|z|=r}|g(z)|\).
Let \(\alpha \), \(\beta \) be two constants with \(0\leq \alpha<\beta \leq 2\pi\). Set
\begin{gather*}
\Omega (\alpha,\beta)=\left\{ z\in\mathbb{C}:\alpha <\arg z<\beta \right\} , \\
\overline{\Omega}(\alpha,\beta)=\left\{ z\in\mathbb{C}:\alpha \leq \arg z\leq \beta \right\} , \\
\Omega \left( \alpha ,\beta ,r\right) =\left\{ z\in\mathbb{C}:\alpha <\arg z<\beta \right\} \cap \left\{ z\in\mathbb{C}:\left\vert z\right\vert <r\right\}.
\end{gather*}
Let \(g\) be an analytic function in \(\overline{\Omega}\left( \alpha ,\beta\right) \). Then the growth order \(\rho_{\alpha ,\beta}(g)\) of \(g\) in \(\Omega (\alpha,\beta)\) is defined as
\[
\rho_{\alpha ,\beta}(g)=\limsup_{r\rightarrow +\infty} \frac{\log ^{+}\log ^{+}M(r,\Omega (\alpha,\beta),g)}{\log r},
\]
where \(M(r,\Omega (\alpha,\beta),g)=\sup_{\alpha \leq \arg z\leq \beta}\left\vert g\left( re^{i\theta}\right) \right\vert\). Moreover the radial order \(\rho_{\theta}(g)\) of \(g\) is denoted by
\[
\rho_{\theta}(g)=\lim_{\varepsilon \rightarrow 0^{+}}\limsup_{r\rightarrow +\infty} \frac{\log ^{+}\log^{+}M(r,\Omega \left( \alpha -\varepsilon ,\beta +\varepsilon \right) ,g)}{\log r}.
\]
In this paper under review, the authors study the growth of solutions of the second-order linear differential equation
\[
f^{\prime\prime}+A(z)f^{\prime}+B(z)f=0,
\tag{1}
\]
where \(A(z)\) and \(B(z)\) are entire functions.
Firstly, the authors assumes that the coefficient \(A(z)\) of \((1)\) is a solution of another equation. Secondly, they consider the lower bound estimate of the linear measure of the angular domain \(I(f)=\left\{ \theta \in \left[ 0,2\pi \right] :\rho_{\theta}(f)=\infty \right\} \) of every non-trivial solution \(f\) of \((1)\).
The results obtained improve and extend those of \textit{J. R. Long} [``On the radial distribution of Julia set of solutions of \(f^{\prime\prime}+A(z)f^{\prime}+B(z)f=0\)'', J. Comput. Anal. Appl. 24, No. 4, 675--691 (2018); \textit{Z.-G. Huang} and \textit{J. Wang}, J. Math. Anal. Appl. 431, No. 2, 988--999 (2015; Zbl 1320.30057)] and [\textit{Z. Zhou} et al., J. Guizhou Norm. Univ., Nat. Sci. 31, No. 2, 50--53, 111 (2013; Zbl 1289.30202)].
Some examples are given to illustrate the results obtained.
Reviewer: Benharrat Belaidi (Mostaganem)On the notion of fuzzy shadowing property.https://www.zbmath.org/1455.370262021-03-30T15:24:00+00:00"Nia, Mehdi Fatehi"https://www.zbmath.org/authors/?q=ai:nia.mehdi-fatehiSummary: This paper is concerned with the study of fuzzy dynamical systems. Let \((X,M,*)\) be a fuzzy metric space in the sense of George and Veeramani. A fuzzy discrete dynamical system is given by any fuzzy continuous self-map defined on \(X\). We introduce the various fuzzy shadowing and fuzzy topological transitivity on a fuzzy discrete dynamical systems. Some relations between this notions have been proved.Research of four-dimensional dynamic systems describing processes of three'level assimilation.https://www.zbmath.org/1455.000262021-03-30T15:24:00+00:00"Chilachava, Temur"https://www.zbmath.org/authors/?q=ai:chilachava.temur"Pinelas, Sandra"https://www.zbmath.org/authors/?q=ai:pinelas.sandra"Pochkhua, George"https://www.zbmath.org/authors/?q=ai:pochkhua.georgeThe authors perform a new nonlinear mathematical model for process of three-level assimilation based on four-dimensional dynamical system under proper assumptions.
For the entire collection see [Zbl 1445.34003].
Reviewer: Vasile Postolică (Piatra Neamt)