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## Регулярные аттракторы автономных и неавтономных динамических систем

Доклады Академии наук. 2014. Т. 455. № 5. С. 512-517.

Зелик С. В., Chepyzhov V. V.

Translator: О. Сипачева

Editor of translation: В. А. Ильин

We study regular global attractors of the dynamical systems corresponding to dissipative evolution equations and their nonautonomous perturbations. We prove that the nonautonomous dynamical system (process) resulting from a small nonautonomous perturbation of an autonomous dynamical system (semigroup) having a regular attractor has a regular nonautonomous attractor as well. Moreover, the symmetric Hausdorff deviation of the perturbed attractors from the unperturbed ones is bounded above by O(ε^κ), where ε is a perturbation parameter and 0 < κ < 1. We apply the obtained results to weakly dissipative wave equations in a bounded domain in the three-dimensional space perturbed by timedependent external forces.

Вишик М. И., Зелик С. В., Chepyzhov V. V., Математический сборник 2013 Т. 204 № 1 С. 3-46

We study regular global attractors of dissipative dynamical semigroups with discrete and continuous time and we
investigate attractors for non-autonomous perturbations of such semigroups. The main theorem states that regularity of global
attractors preserves under small non-autonomous perturbations. Besides, non-autonomous regular global attractors remain
exponential and robust. We apply these general results to model non-autonomous reaction-diffu\-sion systems in ...

Added: February 17, 2013

Romanov A., Izvestiya. Mathematics 2011 Vol. 75 No. 6 P. 1165-1183

For a linear contraction U in a Banach space X we discuss conditions for the convergence of ergodic operator nets corresponding to the adjoint operator U* in the W*O-topology of the space End X*. The accumulation points of all possible nets of this kind form a compact convex set L = Ker G in End ...

Added: October 6, 2012

NY : Springer, 2012

The volume is dedicated to Stephen Smale on the occasion of his 80th birthday. Besides his startling 1960 result of the proof of the Poincaré conjecture for all dimensions greater than or equal to five, Smale’s ground breaking contributions in various fields in Mathematics have marked the second part of the 20th century and beyond. ...

Added: December 19, 2012

Demina M.V., Kudryashov N. A., Regular and Chaotic Dynamics 2016 Vol. 21 No. 3 P. 351-366

Polynomial dynamical systems describing interacting particles in the plane are
studied. A method replacing integration of a polynomial multi-particle dynamical system
by finding polynomial solutions of partial differential equations is introduced. The method
enables one to integrate a wide class of polynomial multi-particle dynamical systems. The
general solutions of certain dynamical systems related to linear second-order partial differential
equations are ...

Added: October 5, 2018

Stanislav Minkov, Ivan Shilin, Attractors of direct products / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2011.04824.

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows φ on S^2 for which the attractor of the Cartesian square of φ is smaller than the Cartesian square of the attractor of φ. In the example for the minimal attractors, the flow φ also has an SRB-measure such that its square does ...

Added: November 12, 2020

Kleptsyn V., Alvarez S., Malicet D. et al., Groups with infinitely many ends acting analytically on the circle / Cornell University. Series math "arxiv.org". 2015.

Added: June 22, 2016

Mikheev A. V., Теория. Практика. Инновации 2017 № 9 (21)

In this paper we consider the calculation of a dynamical system described by a second-order differential equation in which a fundamental system of solutions consisting of functions of exponential type is replaced by bounded functions of the Verhulst model. The time dependence of the forces acting on the dynamical system is analyzed, and the obtained ...

Added: September 6, 2017

Romaskevich O. L., L'Enseignement Mathématique 2014

We consider 3 -periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection. ...

Added: December 25, 2014

Aranson S. K., Belitsky G. R., Zhuzhoma E. V., American Mathematical Society, 1996

The book is an introduction to the qualitative theory of dynamical systems on manifolds of low dimension (on the circle and on surfaces). Along with classical results, it reflects the most significant achevements in this area obtained in recent times. The reader of this book need to be familiar only with basic courses in differential ...

Added: October 2, 2014

Blokh A., Oversteegen L., Ptacek R. et al., The parameter space of cubic laminations with a fixed critical leaf / Cornell University. Series math "arxiv.org". 2015.

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by ...

Added: February 11, 2015

Romanov A., Известия РАН. Серия математическая 2011 Т. 75 № 6 С. 79-98

Для линейного сжатия U в пространстве Банаха X обсуждаются условия сходимости соответствующих его сопряженному оператору U* эргодических операторных сетей в W*O-топологии пространства End X*. Точки накопления всевозможных таких сетей образуют выпуклое компактное множество L в End X*, представляющее собой ядро полугруппы операторов G - замкнутой выпуклой оболочки степеней U*. Показано, что все эргодические сети слабо* ...

Added: October 10, 2012

Protasov V., Systems and Control Letters 2016 Vol. 90 P. 54-60

We prove the existence of positive linear switching systems (continuous time), whose trajectories grow to infinity, but slower than a given increasing function. This implies that, unlike the situation with linear ODE, the maximal growth of trajectories of linear systems may be arbitrarily slow. For systems generated by a finite set of matrices, this phenomenon ...

Added: February 22, 2016

Stankevich N., Shchegoleva N. A., Sataev I. R. et al., Journal of Computational and Nonlinear Dynamics 2020 Vol. 15 No. 11 P. 111001

Using an example a system of two coupled generators of quasiperiodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involve saddle tori occurring at their ...

Added: September 4, 2020

Shilin I., Доклады Академии наук 2016 Т. 469 № 3 С. 287-290

В работе показано, что неустойчивость аттракторов Милнора по Ляпунову является локально топологически типичным динамическим явлением, которое наблюдается в присутствии устойчивых гомоклинических касаний для 2-сжимающих периодических седел. ...

Added: October 14, 2018

Lerman L., Gubina E., Discrete and Continuous Dynamical Systems - Series S 2020 Vol. 13 No. 4 P. 1341-1367

We study a class of scalar differential equations on the circle S1. This class is characterized mainly by the property that any solution of such an equation possesses an exponential dichotomy both on the semi-axes R+ and R+. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations ...

Added: September 16, 2020

Filimonov D., Клепцын В. А., Nonlinearity 2014 Vol. 27 No. 6 P. 1205-1223

We study possible one-end finitely presented subgroups of <img />, acting without finite orbits. Our main result, theorem 1, establishes that any such action possesses the so-called property (<img />), that allows one to make distortion-controlled expansion and is thus sufficient to conclude that the action is Lebesgue-ergodic. We also propose a path towards full ...

Added: October 23, 2014

Blokh A., Oversteegen L., Ptacek R. et al., Smart criticality / Cornell University. Series math "arxiv.org". 2014.

A crucial fact established by Thurston in his 1985 preprint is that distinct \emph{minors} of quadratic laminations do not cross inside the unit disk; this led to his construction of a combinatorial model of the Mandelbrot set. Thurston's argument is based upon the fact that \emph{majors} of a quadratic lamination never enter the region between ...

Added: February 11, 2015

Pardalos P. M., Rassias T. undefined., Springer, 2014

The contributions in this volume have been written by eminent scientists from the international mathematical community and present significant advances in several theories, methods and problems of Mathematical Analysis, Discrete Mathematics, Geometry and their Applications. The chapters focus on both old and recent developments in Functional Analysis, Harmonic Analysis, Complex Analysis, Operator Theory, Combinatorics, Functional ...

Added: May 30, 2014

Stankevich N., Kazakov A., Gonchenko S., Chaos 2020 Vol. 30 Article 123129

The generalized four-dimensional Rössler system is studied. Main bifurcation scenarios leading to a hyperchaos are described phenomenologically and their implementation in the model is demonstrated. In particular, we show that the formation of hyperchaotic invariant sets is related mainly to cascades (finite or infinite) of nondegenerate bifurcations of two types: period-doubling bifurcations of saddle cycles with a ...

Added: January 18, 2021

Smilga I., Harmonic functions on the Sierpinski triangle / Cornell University. Series arXiv "math". 2012. No. 1205.4442.

In this paper, we give a few results on the local behavior of harmonic functions on the Sierpinski triangle - more precisely, of their restriction to a side of the triangle. First we present a general formula that gives the Hölder exponent of such a function in a given point. From this formula, we deduce ...

Added: September 26, 2018

V.L. Chernyshev, Tolchennikov A. A., Russian Journal of Mathematical Physics 2017 Vol. 24 No. 3 P. 290-298

In the problem of determining the asymptotics for the number of points moving along a metric tree, a polynomial approximation that uses Barnes’ multiple Bernoulli polynomials is found. The connection between the second term of the asymptotic expansion and the graph structure is discussed. ...

Added: October 3, 2017

Springer, 2009

Vladimir Arnold is one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This first volume of his Collected Works focuses on representations of functions, celestial mechanics, and ...

Added: February 20, 2013

Lerman L. M., Non-autonomous vector fields on $S^3$: simple dynamics and wild separatrices embedding / Cornell University. Series math "arxiv.org". 2021. No. 00526.

We construct new substantive examples of non-autonomous vector fields on 3-dimensional sphere having a simple dynamics but non-trivial topology. The construction is based on two ideas: the theory of diffeomorpisms with wild separatrix embedding (Pixton, Bonatti-Grines, etc.) and the construction of a non-autonomous suspension over a diffeomorpism (Lerman-Vainshtein). As a result, we get periodic, almost periodic or even nonrecurrent ...

Added: December 2, 2021

Pardalos P. M., Rassias T. undefined., Springer, 2014

This volume consists of chapters written by eminent scientists and engineers from the international community and presents significant advances in several theories, and applications of an interdisciplinary research. These contributions focus on both old and recent developments of Global Optimization Theory, Convex Analysis, Calculus of Variations, and Discrete Mathematics and Geometry, as well as several ...

Added: May 30, 2014