Sergei Yu. Pilyugin

Limit shadowing property

In this paper pseudoorbits of discrete dynamical systems are considered such that the one-step errors of the orbits tend to zero with increasing indices. First it is shown that close to hyperbolic sets such orbits are shadowed by true trajectories of the system with shadowing errors also tending to zero. Then the rates of convergence are studied via considering pseudoorbits such that the error sequences belong to certain (weighted) lp-spaces and showing that the corresponding shadowing errors are there, too. Under certain conditions on the weights we establish weighted shadowing near nonhyperbolic sets.

Lipschitz shadowing implies structural stability

We show that the Lipschitz shadowing property of a diffeomorphism is equivalent to structural stability. As a corollary, we show that an expansive diffeomorphism having the Lipschitz shadowing property is Anosov.

Spaces of dynamical systems

Dynamical systems are abundant in theoretical physics and engineering. Their understanding, with sufficient mathematical rigor, is vital to solving many problems. This work conveys the modern theory of dynamical systems in a didactically developed fashion. In addition to topological dynamics, structural stability and chaotic dynamics, also generic properties and pseudotrajectories are covered, as well as nonlinearity. The author is an experienced book writer and his work is based on years of teaching.

Vector fields with the oriented shadowing property

Abstract We give a description of the C 1 -interior ( Int 1 ( OrientSh ) ) of the set of smooth vector fields on a smooth closed manifold that have the oriented shadowing property. A special class B of vector fields that are not structurally stable is introduced. It is shown that the set Int 1 ( OrientSh ∖ B ) coincides with the set of structurally stable vector fields. An example of a field of the class B belonging to Int 1 ( OrientSh ) is given. Bibliography: 18 titles.

Generalizations of the notion of hyperbolicity

We describe several generalizations of the classical notion of hyperbolicity for a sequence of linear mappings. It is shown that the following three statements are equivalent: (i) the corresponding linear non-homogeneous system has a bounded solution for any bounded nonhomogeneity, (ii) the sequence has a (C, λ)-structure, (iii) the sequence is piecewise hyperbolic with long enough intervals of hyperbolicity.

Pseudotrajectories generated by a discretization of a parabolic equation

A semiimplicit discretization of a parabolic equation is considered. The resulting diffepmorphism is shown to be generically Morse-Smale. Uniform bounds for the dimension of its attractor are given and numerical trajectories—including round-off errors—are shown to approximate the attractor.

Lyapunov functions, shadowing, and topological stability

We use Lyapunov type functions to give new conditions under which a homeomorphism of a compact metric space has the shadowing property. These conditions are applied to establish the topological stability of some homeomorphisms with nonhyperbolic behavior.

Main Definitions and Basic Results

In this preliminary chapter, we define pseudotrajectories and various shadowing properties for dynamical systems with discrete and continuous time (Sects. 1.1 and 1.2), study the notion of chain transitivity (Sect. 1.1), describe hyperbolicity, Ω-stability, and structural stability (Sect. 1.3), and prove a lemma on finite Lipschitz shadowing in a neighborhood of a hyperbolic set (Sect. 1.4).

C 1 Interiors of Sets of Systems with Various Shadowing Properties

In this chapter, we study the structure of C1 interiors of some basic sets of dynamical systems having various shadowing properties. We give either complete proofs or schemes of proof of the following main results: The C1 interior of the set of diffeomorphisms having the standard shadowing property is a subset of the set of structurally stable diffeomorphisms (Theorem 3.1.1); this result and Theorem 1.4.1 (a) imply that the C1 interior of the set of diffeomorphisms having the standard shadowing property coincides with the set of structurally stable diffeomorphisms; the set \(\mbox{ Int}^{1}(\mbox{ OrientSP}_{F}\setminus \mathcal{B})\) is a subset of the set of structurally stable vector fields (Theorem 3.3.1); similarly to the case of diffeomorphisms, this result and Theorem 1.4.1 (b) imply that the set \(\mbox{ Int}^{1}(\mbox{ OrientSP}_{F}\setminus \mathcal{B})\) coincides with the set of structurally stable vector fields; the set Int1(OrientSP F ) contains vector fields that are not structurally stable (Theorem 3.4.1).

Chain Transitive Sets and Shadowing

In this chapter, we study relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets. We prove the following two main results: Let Λ be a closed invariant set of f ∈ Diff1(M). Then f | Λ is chain transitive and C1-stably shadowing in a neighborhood of Λ if and only if Λ is a hyperbolic basic set (Theorem 4.2.1); there is a residual set \(\mathcal{R} \subset \text{Diff}^{1}(M)\) such that if \(f \in \mathcal{ R}\) and Λ is a locally maximal chain transitive set of f, then Λ is hyperbolic if and only if f | Λ is shadowing (Theorem 4.3.1).