Recent zbMATH articles in MSC 37Lhttps://www.zbmath.org/atom/cc/37L2021-03-30T15:24:00+00:00WerkzeugComputability 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)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.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)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.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)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.