Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalar ordinary differential equations

TL;DR

This paper investigates the long-term behavior of nonhomogeneous linear dissipative scalar ODEs, revealing conditions for stability and chaos within their invariant sets, thus advancing understanding of their complex dynamics.

Contribution

It provides a detailed analysis of the structure and dynamics of invariant sets in nonhomogeneous linear dissipative ODEs, including stability and chaos conditions.

Findings

Invariant compact sets can be uniformly stable or chaotic.

Presence of Li-Yorke and Auslander-Yorke chaos within attractors.

Conditions on dissipative and linear terms influence dynamics.

Abstract

The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.