On the Structure of a Neighborhood of Stable Compact Invariant Sets

Abstract

The analysis of the phase portrait of trajectories in a neighborhood of stable invariant sets of dynamical systems is one of the main problems in qualitative theory of stability of motion. This problem has been under study starting from Lyapunov’s work [1]; in particular, for abstract dynamical systems, it was considered in [2–8], where a wide bibliography on the topic can be found. However, only the case of asymptotic stability has been sufficiently well analyzed (see [1–3, 9]). The so-far existing practice of scientific research shows that, in the case of nonasymptotic stability, the qualitative picture of trajectories near compact invariant sets is very complicated and immensely variegated, which explains why the problem of qualitatively distinguishing all possible situations has not been satisfactorily solved yet. In the present paper, we suggest an approach to the study of the structure of a neighborhood of compact invariant sets for locally compact dynamical systems and use it to give some classification of types of orbitally nonasymptotically stable invariant sets. The analysis carried out below permits one to single out five basic classes of first-level sets and 121 subclasses of second-level sets. Related illustrative examples of dynamical systems are given. Let the triple (X, , π) denote the dynamical system defined on a locally compact metric space (X, d) with phase mapping π : X × → X (π : (x, t) → xt) [2, 3]. For each point x ∈ X, the mapping x : t → xt is referred to as the motion passing through x, and the sets