1. Sets of Invariant Measures and Cesaro StabilitySergey Kryzhevich, 2017, original scientific article Abstract: We take a space X of dynamical systems that could be: homeomorphisms or continuous maps of a compact metric space K or diffeomorphisms of a smooth manifold or actions of an amenable group. We demonstrate that a typical dynamical system of X is a continuity point for the set of probability invariant measures considered as a function of a map, let Y be the set of all such continuity points. As a corollary we prove that for typical dynamical systems average values of continuous functions calculated along trajectories do not drastically change if the system is perturbed.
Keywords: ergodic theory, invariant measures, shadowing, stability, tolerance stability, topological dynamics
Published in RUNG: 02.10.2017; Views: 5114; Downloads: 0
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2. Weak forms of shadowing in topological dynamicsSergey Kryzhevich, Danila Cherkashin, 2017, original scientific article Abstract: We consider continuous maps of compact metric spaces. It is proved that every pseudotrajectory with sufficiently small errors contains a subsequence of positive density that is point-wise close to a subsequence of an exact trajectory with the same indices. Also, we study homeomorphisms such that any pseudotrajectory can be shadowed by a finite number of exact orbits. In terms of numerical methods this property (we call it multishadowing) implies possibility to calculate minimal points of the dynamical system.
We prove that for the non-wandering case multishadowing is equivalent to density of minimal points. Moreover, it is equivalent to existence of a family of $\varepsilon$-networks ($\varepsilon > 0$) whose iterations are also $\varepsilon$-networks. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.
Keywords: Topological dynamics, minimal points, invariant measure, shadowing, chain recurrence, $\varepsilon$-networks, syndetic sets
Published in RUNG: 27.07.2017; Views: 5528; Downloads: 0
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4. Stability by linear approximation for time scale dynamical systemsSergey Kryzhevich, Alexander Nazarov, 2017, original scientific article Abstract: We study systems on time scales that are generalizations of classical differential or difference equations and appear in numerical methods. In this paper we consider linear systems and their small nonlinear perturbations. In terms of time scales and of eigenvalues of matrices we formulate conditions, sufficient for stability by linear approximation. For non-periodic time scales we use techniques of central upper Lyapunov exponents (a common tool of the theory of linear ODEs) to study stability of solutions. Also, time scale versions of the famous Chetaev’s theorem on conditional instability are proved. In a nutshell, we have developed a completely new technique in order to demonstrate that methods of non-autonomous linear ODE theory may work for time-scale dynamics.
Keywords: Time scale system, Linearization, Lyapunov functions, Millionschikov rotations, Stability
Published in RUNG: 15.03.2017; Views: 5464; Downloads: 162
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