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S. Pilyugin (1999)
Shadowing in dynamical systems
D.V. Anosov (1970)
Proc. 5th Int. Conf. on Nonl. Oscill.
(2010)
Lipschitz shadowing implies structural stability, Nonlinearity
S.Yu. Pilyugin (1999)
Shadowing in dynamical systems, Lecture Notes in Math.
S.Yu. Pilyugin, S.B. Tikhomirov (2010)
Lipschitz shadowing implies structural stabilityNonlinearity, 23
S.Yu. Pilyugin (1994)
Lecture Notes in Math.
R. Bowen (1975)
Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms
S. Pilyugin (2011)
Theory of pseudo-orbit shadowing in dynamical systemsDifferential Equations, 47
(1970)
On a class of invariant sets of smooth dynamical systems
K. Palmer (2000)
Shadowing in Dynamical Systems. Theory and Applications
A. Petrov, S. Pilyugin (2015)
Nonsmooth mappings with Lipschitz shadowingarXiv: Dynamical Systems
S. Pilyugin (1994)
The Space of Dynamical Systems with the C0-Topology
ISSN 0012-2661, Differential Equations, 2016, Vol. 52, No. 13, pp. 1732–1737. Pleiades Publishing, Ltd., 2016. CONTROL THEORY ∗ ∗∗ S. Yu. Pilyugin and A. A. Rodionova St. Peterburg State University, St. Petersburg, 199034 Russia ∗ ∗∗ e-mail: sp@sp1196.spb.edu, a.a.rodionova@gmail.com DOI: 10.1134/S0012266116130048 INTRODUCTION The theory of shadowing of approximate trajectories (pseudotrajectories) has become one of com- prehensively developed fields in the global theory of dynamical systems. th The main results that had been obtained in the theory by the end of the 20 century were covered in detail in the monographs [1, 2]. Many modern results were described in the review [3]. The so-called Lipschitz shadowing plays a special role in the theory. This property implies that there exists an exact trajectory and the distance from the exact to shadowed pseudotrajectory is estimated linearly in terms of a stepwise error of the pseudotrajectory (see Section 1 for the rigorous definition). The first classical results on shadowing in a neighborhood of the hyperbolic set of a diffeo- morphism were obtained by Anosov [4] and Bowen [5]. Analysis of these results shows that the shadowing in their case was Lipschitz. Later it was shown that any structurally stable diffeomor- phism has
Differential Equations – Springer Journals
Published: Mar 4, 2017
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