A remark on omega limit sets for non-expansive dynamics

TL;DR

This paper investigates omega-limit sets in non-expansive dynamical systems, revealing that the nature of these sets depends on the chosen norm, with Euclidean norms allowing more complex limit sets like tori, unlike polyhedral norms.

Contribution

It demonstrates that for non-expansive systems with strictly convex norms, omega-limit sets are essentially those of linear systems, clarifying the influence of norm geometry on long-term behavior.

Findings

Omega-limit sets depend on the norm used.

Polyhedral norms restrict omega-limit sets to equilibria.

Euclidean norms can admit complex limit sets like tori.

Abstract

In this paper, we study systems of time-invariant ordinary differential equations whose flows are non-expansive with respect to a norm, meaning that the distance between solutions may not increase. Since non-expansiveness (and contractivity) are norm-dependent notions, the topology of $\omega$-limit sets of solutions may depend on the norm. For example, and at least for systems defined by real-analytic vector fields, the only possible $\omega$-limit sets of systems that are non-expansive with respect to polyhedral norms (such as $\ell^p$ norms with $p =1$ or $p=\infty$) are equilibria. In contrast, for non-expansive systems with respect to Euclidean ($\ell^2$) norm, other limit sets may arise (such as multi-dimensional tori): for example linear harmonic oscillators are non-expansive (and even isometric) flows, yet have periodic orbits as $\omega$-limit sets. This paper shows that the Euclidean linear case is what can be expected in general: for flows that are contractive with respect to any strictly convex norm (such as $\ell^p$ for any $p\not=1,\infty$), and if there is at least one bounded solution, then the $\omega$-limit set of every trajectory is also an omega limit set of a linear time-invariant system.