Dynamics of non-autonomous systems with nested invariant cone structure and its applications

TL;DR

This paper investigates how nested invariant cone structures influence the dynamics of non-autonomous, almost periodic systems, showing their persistence under perturbations and reducing complex dynamics to finite-dimensional systems.

Contribution

It is the first to analyze global dynamics of non-autonomous systems with invariant nested cones, extending previous results to more general settings including almost-periodic and periodic perturbations.

Findings

Nested invariant cone structures persist under C1 perturbations.

Omega-limit set dynamics can be reduced to finite-dimensional systems.

In special cases, omega-limit sets resemble one-dimensional systems with minimal sets.

Abstract

The current paper is devoted to the investigation of the influence of nested invariant cone structure on the dynamics, in the context of non-autonomous (time almost periodic)cases. We first prove that the nested invariant cone structure can persistent under C1 perturbations; and the dynamics of the omega-limit set of any precompact orbit can be reduced to the dynamics of a compact invariant of a suitable finite dimensional system(see Theorem 2.1). In some special cases, the dynamics of any omega-limit set generated by the skew product semiflow of such a system is similar to a one-dimensional system, that is, the omega-limit set contains at most two minimal sets, and any minimal set is an almost automorphic extension of its base flow(a universal phenomenon in multi-frequency driven systems, introduced by S. Bochner), these results are also correct for such systems under C1 small perturbations(see Theorems 2.2, 2.3). To our best knowledge, it is the first paper to touch the global dynamics of abstract non-autonomous systems with invariant nested cones; the setting is general, since it contains an autonomous system plus an almost-periodic perturbation term, and a periodic system with another periodic perturbation term (these two periods are irrationally dependent)as special cases. The results can be viewed as a generalization of important works of W. Shen and Y. Yi(1995 J. Differential Equations 122 114-136) for scalar parabolic equations with separated boundary conditions, Y. Wang(2007 Nonlinearity 20 831-843) for tridiagonal competitive cooperative systems(see Section 4).