Recent zbMATH articles in MSC 70Khttps://www.zbmath.org/atom/cc/70K2021-05-28T16:06:00+00:00WerkzeugNew vertically planed pendulum motion.https://www.zbmath.org/1459.700172021-05-28T16:06:00+00:00"Ismail, A. I."https://www.zbmath.org/authors/?q=ai:ismail.ahmad-izani-bin-md|ismail.ahmad-izani-md|ismail.ahamd-izani-bin-md|ismail.ahmad-izani-mohamedSummary: This article is concerned about the planed rigid body pendulum motion suspended with a spring which is suspended to move on a vertical plane moving uniformly about a horizontal \(X\)-axis. This model depends on a system containing three generalized coordinates. The three nonlinear differential equations of motion of the second order are obtained to the elastic string length and the oscillation angles \(\varphi_1\) and \(\varphi_2\) which represent the freedom degrees for the pendulum motions. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity \(\omega \). The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange's equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the approximated fourth-order Runge-Kutta method through programming packages. These solutions are represented graphically to describe and discuss the behavior of the body at any instant for different values of the different physical parameters of the body. The obtained results have been discussed and compared with some previously published works. Some concluding remarks have been presented at the end of this work. The value of this study comes from its wide applications in both civil and military life. The main findings and objectives of the current study are obtaining periodic solutions for the problem and satisfying their accuracy and stabilities through the numerical procedure.A modified incremental harmonic balance method for 2-DOF airfoil aeroelastic systems with nonsmooth structural nonlinearities.https://www.zbmath.org/1459.700082021-05-28T16:06:00+00:00"Ni, Ying-Ge"https://www.zbmath.org/authors/?q=ai:ni.yingge"Zhang, Wei"https://www.zbmath.org/authors/?q=ai:zhang.wei.19"Lv, Yi"https://www.zbmath.org/authors/?q=ai:lv.yiSummary: A modified incremental harmonic balance method is presented to analyze the aeroelastic responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity. The current method, which combines the traditional incremental harmonic balance method and a fast Fourier transform, can be used to obtain the higher-order approximate solution for the aeroelastic responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity using significantly fewer linearized algebraic equations than the traditional method, and the dominant frequency components of the response can be obtained by a fast Fourier transform of the numerical solution. Thus, periodic solutions can be obtained, and the calculation process can be simplified. Furthermore, the nonsmooth nonlinearity was expanded into a Fourier series. The procedures of the modified incremental harmonic balance method were demonstrated using systems with hysteresis and free play nonlinearities. The modified incremental harmonic balance method was validated by comparing with the numerical solutions. The effect of the number of harmonics on the solution precision as well as the effect of the free-play and stiffness ratio on the response amplitude is discussed.Global effect of non-conservative perturbations on homoclinic orbits.https://www.zbmath.org/1459.370502021-05-28T16:06:00+00:00"Gidea, Marian"https://www.zbmath.org/authors/?q=ai:gidea.marian"de la Llave, Rafael"https://www.zbmath.org/authors/?q=ai:de-la-llave.rafael"Musser, Maxwell"https://www.zbmath.org/authors/?q=ai:musser.maxwellSummary: We study the effect of time-dependent, non-conservative perturbations on the dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. We assume that the unperturbed system is Hamiltonian, and the normally hyperbolic invariant manifold is parametrized via action-angle coordinates. The homoclinic excursions can be described via the scattering map, which gives the future asymptotic of an orbit as a function of its past asymptotic. We provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map expressed in action-angle coordinates. We illustrate these formulas for perturbations of both uncoupled and coupled rotator-pendulum systems.On mechanisms of destruction of adiabatic invariance in slow-fast Hamiltonian systems.https://www.zbmath.org/1459.700392021-05-28T16:06:00+00:00"Neishtadt, Anatoly"https://www.zbmath.org/authors/?q=ai:neishtadt.anatolii-iStudy on chaotic peculiarities of magnetic-mechanical coupled system of giant magnetostrictive actuator.https://www.zbmath.org/1459.700452021-05-28T16:06:00+00:00"Yan, Hongbo"https://www.zbmath.org/authors/?q=ai:yan.hongbo"Liu, Enzuo"https://www.zbmath.org/authors/?q=ai:liu.enzuo"Zhao, Pengbo"https://www.zbmath.org/authors/?q=ai:zhao.pengbo"Liu, Pei"https://www.zbmath.org/authors/?q=ai:liu.pei"Cao, Rui"https://www.zbmath.org/authors/?q=ai:cao.ruiSummary: We studied the chaotic peculiarities of magnetic-mechanical coupled system of GMA. Based on the working principle of GMA and according to Newton's second law of motion, first piezomagnetic equation, disk spring design theory, and structural dynamics principle of GMA, the present study established a GMA magnetic-mechanical coupled system model. By carrying out data modeling of this coupled system model, the bifurcation chart of the system with the variation of damping factor, excitation force, and exciting frequency parameters as well as the homologous offset oscillogram, phase plane trace chart, and Poincaré diagram was obtained, and the chaotic peculiarities of the system were analyzed. The influence of parametric errors on the coupled system was studied. The analytical results showed that the oscillation equation of the GMA magnetic-mechanical coupled system had nonlinearity and the movement morphology was complicated and diversified. By adjusting the damping factor, exciting frequency, and excitation force parameters of the system, the system could work under the stable interval, which provided theoretical support for the stability design of GMA.Local bifurcation analysis of a rotating blade.https://www.zbmath.org/1459.700442021-05-28T16:06:00+00:00"Zhang, Xiaohua"https://www.zbmath.org/authors/?q=ai:zhang.xiaohua"Chen, Fangqi"https://www.zbmath.org/authors/?q=ai:chen.fangqi"Zhang, Baoqiang"https://www.zbmath.org/authors/?q=ai:zhang.baoqiang"Jing, Taiyan"https://www.zbmath.org/authors/?q=ai:jing.taiyanSummary: In this paper, the stability and local bifurcation for the rotating blade under high-temperature supersonic gas flow are investigated using analytical and numerical methods. Based on obtained four-dimensional averaged equation for the case of \(1:1\) internal resonance and primary resonance, two types of critical points for the bifurcation response equations are considered. The points are characterized by a double zero and two negative eigenvalues and two pairs of purely imaginary eigenvalues, respectively. For each type, the steady state solutions and the stability region is obtained with the aid of center manifold theory and normal form theory. We find the Hopf bifurcation solution which indicates the blade will flutter. In summary, the numerical solutions, whose initial conditions are chosen in the stability region, agree with the analytic results.Parabolic invariant tori in quasi-periodically forced skew-product maps.https://www.zbmath.org/1459.370512021-05-28T16:06:00+00:00"Guan, Xinyu"https://www.zbmath.org/authors/?q=ai:guan.xinyu"Si, Jianguo"https://www.zbmath.org/authors/?q=ai:si.jianguo"Si, Wen"https://www.zbmath.org/authors/?q=ai:si.wenSummary: We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps \(\varphi : \mathbb{R}^n \times \mathbb{T}^d \to \mathbb{R}^n \times \mathbb{T}^d\):
\[
\varphi \begin{pmatrix} z \\ \theta \end{pmatrix} = \begin{pmatrix} z + \phi (z) + h (z, \theta) + \epsilon f (z, \theta) \\ \theta + \omega \end{pmatrix},
\]
where \(\phi : \mathbb{R}^n \to \mathbb{R}^n\) is a homogeneous function of degree \(l\) with \(l \geq 2\) and \(h = \mathcal{O}(|z|^{l + 1})\). We obtain the following results: (a) For \(n = 1, l\) being odd and \(\varepsilon\) sufficiently small, parabolic invariant tori exist if \(\omega\) satisfies the Brjuno-Rüssmann's non-resonant condition. (b) For \(n = 1\), and \(\varepsilon\) sufficiently small, parabolic invariant tori also exist if one of the following conditions holds: (i) First order average is non-zero, first order non-average part is small enough and the forcing frequency \(\omega\) does not need any arithmetic condition. (ii) First order average is non-zero and \(\omega\) satisfies the Brjuno-type weak non-resonant condition; (iii) \(l = 2\), first order average is zero, both first and second order non-average parts are small enough and \(\omega\) satisfying Brjuno-type weak non-resonant condition; (iv) \(l > 2\), first order average is zero, the second order average is non-zero, both first and second order non-average parts are small enough and \(\omega\) satisfies the Brjuno-type weak non-resonant condition. (c) In the case \(n > 1\), if first order average belongs to the interior of the range of \(\varphi, Spec(D \phi) \cap \text{i} \mathbb{R} = \emptyset\), first order non-average part is small enough and the forcing frequency \(\omega\) does not need any arithmetic condition, then the quasi-periodically forced skew-product maps above admit parabolic invariant tori for \(\varepsilon\) sufficiently small. The main methods of this paper are KAM theory and fixed point theorem, which are finally shown that it can be directly applied to the existence problem of quasi-periodic response solutions of degenerate harmonic oscillators.Higher order normal modes.https://www.zbmath.org/1459.370412021-05-28T16:06:00+00:00"Gaeta, Giuseppe"https://www.zbmath.org/authors/?q=ai:gaeta.giuseppe"Walcher, Sebastian"https://www.zbmath.org/authors/?q=ai:walcher.sebastianThe authors study Hamiltonian systems with Hamiltonian given by the sum of a classical kinetic energy term (a quadratic function of the momentum) and a potential energy term which is a homogeneous function of the position. Solutions to the equation of motion are found. They are in the form of normal modes, namely the position vector remains on a fixed line passing through the origin. The example of a potential energy which is homogenous of order four is studied in detail.
Reviewer: Mohammad Khorrami (Tehran)Multifarious chaotic attractors and its control in rigid body attitude dynamical system.https://www.zbmath.org/1459.700152021-05-28T16:06:00+00:00"Wang, Yang"https://www.zbmath.org/authors/?q=ai:wang.yang.1|wang.yang"Wang, Zhen"https://www.zbmath.org/authors/?q=ai:wang.zhen.2|wang.zhen.6|wang.zhen.1|wang.zhen|wang.zhen.7|wang.zhen.3|wang.zhen.5"Kong, Dezhi"https://www.zbmath.org/authors/?q=ai:kong.dezhi"Kong, Lingyun"https://www.zbmath.org/authors/?q=ai:kong.lingyun"Qiao, Yukun"https://www.zbmath.org/authors/?q=ai:qiao.yukunSummary: The Euler dynamical equation which describes the attitude motion of a rigid body will exhibit very complex dynamic behaviors under the action of different external torques. Many special types of new chaotic attractors are presented, including hidden attractors, double-body-double-core chaotic attractors, and single-body-three-core-tree-wing chaotic attractors. The position of equilibrium points in several typical cases of the Euler dynamic equation is solved, and the stability of linearized equation at each equilibrium point and its influence on the formation of the chaotic attractor are analyzed. An improved nonlinear relay control law based on Euler angle feedback is developed to stabilize a new chaotic spacecraft attitude motion to an appointed equilibrium point or a periodic orbit.Existence of periodic solutions for Hamiltonian systems with super-linear and sign-changing nonlinearities.https://www.zbmath.org/1459.370542021-05-28T16:06:00+00:00"Jia, Liqian"https://www.zbmath.org/authors/?q=ai:jia.liqian"Chen, Guanwei"https://www.zbmath.org/authors/?q=ai:chen.guanweiSummary: In this paper, we consider the existence of periodic solutions for the super quadratic second order Hamiltonian system, and primitive functions of nonlinearities are allowed to be sign-changing. By using some weaker conditions, our result extends and improves some existed results in the literature.Applications of the RST algorithm to nonlinear systems in real-time hybrid simulation.https://www.zbmath.org/1459.860142021-05-28T16:06:00+00:00"Tang, Yu"https://www.zbmath.org/authors/?q=ai:tang.yu"Qin, Hui"https://www.zbmath.org/authors/?q=ai:qin.huiSummary: Real-time substructure testing (RST) algorithm is a newly developed integration method for real-time hybrid simulation (RTHS) which has structure-dependent and explicit formulations for both displacement and velocity. The most favourable characteristics of the RST algorithm is unconditionally stable for linear and no iterations are needed. In order to fully evaluate the performance of the RST method in solving dynamic problems for nonlinear systems, stability, numerical dispersion, energy dissipation, and overshooting properties are discussed. Stability analysis shows that the RST method is only conditionally stable when applied to nonlinear systems. The upper stability limit increases for stiffness-softening systems with an increasing value of the instantaneous degree of nonlinearity while decreases for stiffness-hardening systems when the instantaneous degree of nonlinearity becomes larger. Meanwhile, the initial damping ratio of the system has a negative impact on the upper stability limit especially for instantaneous stiffness softening systems, and a larger value of the damping ratio will significantly decrease the upper stability limit of the RST method. It is shown in the accuracy analysis that the RST method has relatively smaller period errors and numerical damping ratios for nonlinear systems when compared with other two well-developed algorithms. Three simplified engineering cases are presented to investigate the dynamic performance of the RST method, and the numerical results indicate that this method has a more desirable accuracy than other methods in solving dynamic problems for both linear and nonliner systems.High-order study of the canard explosion in an aircraft ground dynamics model.https://www.zbmath.org/1459.341332021-05-28T16:06:00+00:00"Qin, Bo-Wei"https://www.zbmath.org/authors/?q=ai:qin.bo-wei"Chung, Kwok-Wai"https://www.zbmath.org/authors/?q=ai:chung.kwok-wai"Algaba, Antonio"https://www.zbmath.org/authors/?q=ai:algaba.antonio"Rodríguez-Luis, Alejandro J."https://www.zbmath.org/authors/?q=ai:rodriguez-luis.alejandro-jSummary: A planar system has been proposed in the paper [\textit{J. Rankin} et al., ``Canard cycles in aircraft ground dynamics'', Nonlinear Dyn. 66, No. 4, 681--688 (2011; \url{doi:10.1007/s11071-010-9940-y})] to understand the canard explosion detected in a 6D aircraft ground dynamics model. A specific feature of this minimal 2D system is a critical manifold with a single fold and an asymptote. In this paper, we provide a high-order analytical prediction (in fact, up to any wanted order) of the canard explosion in this system. Using a nonlinear time transformation method, we are able to approximate not only the critical parameter value, but also the critical manifold in the phase space. The comparison of our theoretical results with the corresponding numerical continuations shows a very good agreement.Delay-coordinate maps and the spectra of Koopman operators.https://www.zbmath.org/1459.370232021-05-28T16:06:00+00:00"Das, Suddhasattwa"https://www.zbmath.org/authors/?q=ai:das.suddhasattwa"Giannakis, Dimitrios"https://www.zbmath.org/authors/?q=ai:giannakis.dimitriosSummary: The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the point spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions in systems with pure point or mixed spectra. We illustrate our results with applications to product dynamical systems with mixed spectra.Construction of a periodic solution to the equations of motion of generalized Atwood's machine using computer algebra.https://www.zbmath.org/1459.700092021-05-28T16:06:00+00:00"Prokopenya, A. N."https://www.zbmath.org/authors/?q=ai:prokopenya.alexander-nSummary: The problem of finding a periodic motion of generalized Atwood's machine in which the pulley of a finite radius is replaced by two small pulleys and one of the objects may oscillate in the vertical plane is discussed. Using symbolic computations, the equations of motion are derived and their periodic solutions in the form of power series in a small parameter in the case of small oscillations are obtained. It is shown that if the difference of the objects' masses is small, then the system has a dynamic equilibrium state in which the oscillating object behaves like a pendulum the length of which performs small oscillations. In this case, the frequency resonance \(2:1\) is observed; i.e., the pendulum length oscillation frequency is twice as large as the oscillation frequency of the angular variable. The comparison of the analytical results with the numerical solutions to the equations of motion confirms the validity of the analytical computations. All computations are performed using the computer algebra system Wolfram Mathematica.Harmonic resonances of graphene-reinforced nonlinear cylindrical shells: effects of spinning motion and thermal environment.https://www.zbmath.org/1459.741182021-05-28T16:06:00+00:00"Dong, Youheng"https://www.zbmath.org/authors/?q=ai:dong.youheng"Li, Xiangyu"https://www.zbmath.org/authors/?q=ai:li.xiangyu"Gao, Kang"https://www.zbmath.org/authors/?q=ai:gao.kang"Li, Yinghui"https://www.zbmath.org/authors/?q=ai:li.yinghui"Yang, Jie"https://www.zbmath.org/authors/?q=ai:yang.jie.1Summary: This work investigates nonlinear harmonic resonance behaviors of graded graphene-reinforced composite spinning thin cylindrical shells subjected to a thermal load and an external excitation. The volume fraction of graphene platelets varies continuously in the shell's thickness direction, which generates position-dependent useful material properties. Natural frequencies of shell traveling waves are derived by considering influences of the initial hoop tension, centrifugal and Coriolis forces, thermal expansion deformation, and thermal conductivity. A new Airy stress function is introduced. Harmonic resonance behaviors and their stable solutions for the spinning cylindrical shell are analyzed based on an equation of motion which is established by adopting Donnell's nonlinear shell theory. The necessary and sufficient conditions for the existence of the subharmonic resonance of the spinning composite cylindrical shell are given. Besides the shell's intrinsic structural damping, the Coriolis effect due to the spinning motion has a contribution to the damping terms of the system as well. Comparisons between the present analytical results and those in other papers are made to validate the existing solutions. Influences of main factors on vibration characteristics, primary resonance, and subharmonic resonance behaviors of the novel composite cylindrical shell are discussed. Furthermore, the mechanism of how the spinning motion affects the amplitude-frequency curves of harmonic resonances of the cylindrical shell is analyzed.Persistence of degenerate lower dimensional invariant tori in reversible systems with Bruno non-degeneracy conditions.https://www.zbmath.org/1459.370532021-05-28T16:06:00+00:00"Yang, Xiaomei"https://www.zbmath.org/authors/?q=ai:yang.xiaomei"Xu, Junxiang"https://www.zbmath.org/authors/?q=ai:xu.junxiangSummary: In this paper we consider a class of degenerate reversible systems with Bruno non-degeneracy conditions, and prove the persistence of a lower dimensional invariant torus, whose frequency vector is only a small dilation of the prescribed one.The transition to chaos. Conservative classical and quantum systems. 3rd edition.https://www.zbmath.org/1459.700012021-05-28T16:06:00+00:00"Reichl, Linda"https://www.zbmath.org/authors/?q=ai:reichl.linda-ePublisher's description: The classical and quantum dynamics of conservative systems governs the behavior of much of the world around us -- from the dynamics of galaxies to the vibration and electronic behavior of molecules and the dynamics of systems formed from or driven by laser radiation. Most conservative dynamical systems contain some degree of chaotic behavior, ranging from a self-similar mixture of regular and chaotic motion, to fully developed chaos. This chaotic behavior has a profound effect on the dynamics.
This book combines mathematical rigor with examples that illuminate the dynamical theory of chaotic systems. The emphasis of the 3rd Edition is on topics of modern interest, including scattering systems formed from molecules and nanoscale quantum devices, quantum control and destabilization of systems driven by laser radiation, and thermalization of condensed matter systems. The book is written on a level accessible to graduate students and to the general research community.
See the reviews of the first and second editions in [Zbl 0776.70003; Zbl 1061.70001].A new sufficient condition in order that the real Jacobian conjecture in \(\mathbb{R}^2\) holds.https://www.zbmath.org/1459.140232021-05-28T16:06:00+00:00"Giné, Jaume"https://www.zbmath.org/authors/?q=ai:gine.jaume"Llibre, Jaume"https://www.zbmath.org/authors/?q=ai:llibre.jaumeThe authors prove that \(F: \mathbb{R}^2\rightarrow \mathbb{R}^2\) is injective if \(\operatorname{det}JF(x,y)\) is nowhere zero, \(F(0,0)=(0,0)\) and \(y'(x)=-(ff_x+gg_x)/(ff_y+gg_y)\) does not have any Puiseux solution at infinity except \(y=\pm ix\) and they conjectured that the inverse direction is also true. In the proof, they need to prove that the origin is a global center of a vector field in order to prove the conclusion.
Reviewer: Yan Dan (Changsha)Forced vibration in cutting process considering the nonlinear curvature and inertia of a rotating composite cutter bar.https://www.zbmath.org/1459.740772021-05-28T16:06:00+00:00"Ren, Yongsheng"https://www.zbmath.org/authors/?q=ai:ren.yongsheng"Yao, Donghui"https://www.zbmath.org/authors/?q=ai:yao.donghuiSummary: Forced vibration of the cutting system with a three-dimensional composite cutter bar is investigated. The composite cutter bar is simplified as a rotating cantilever shaft which is subjected to a cutting force including regenerative delay effects and harmonic exciting items. The nonlinear curvature and inertia of the cutter bar are taken into account based on inextensible assumption. The effects of the moment of inertia, gyroscopic effect, and internal and external damping are also considered, but shear deformation is neglected. Equation of motion is derived based on Hamilton's extended principle and discretized by the Galerkin method. The analytical solutions of the steady-state response of the cutting system are constructed by using the method of multiple scales. The response of the cutting system is studied for primary and superharmonic resonances. The effects of length-to-diameter ratio, damping ratio, cutting force coefficients, ply angle, rotating speed, and internal and external damping are investigated. The results show that nonlinear curvature and inertia imposed a significant effect on the dynamic behavior of the cutting process. The equivalent nonlinearity of the cutting system shows hard spring characteristics. Multiple solutions and jumping phenomenon of typical Duffing system are found in forced response curves.Modelling Atwood's machine with three degrees of freedom.https://www.zbmath.org/1459.700432021-05-28T16:06:00+00:00"Prokopenya, Alexander N."https://www.zbmath.org/authors/?q=ai:prokopenya.alexander-nSummary: A generalized model of the Atwood machine when two bodies can swing in a plane is considered. Combining symbolic and numerical calculations, we have obtained equations of motion of the system and analyzed their solutions. We have shown that oscillations can completely modify motion of the system while the simple Atwood machine demonstrates only the uniformly accelerated motion of the bodies. In particular, a quasi-periodic motion of the system can take place even in case of equal masses of the bodies. We have also obtained a differential equation determining an averaged translational motion of the system and have shown that its solution corresponds completely to the numerical solution of the exact differential equations of motion. The validity of the results obtained is demonstrated by means of the simulation of motion of swinging Atwood's machine with the computer algebra system Wolfram Mathematica.