On the Nonexistence of Continuous Immersions for Discrete-time Systems

TL;DR

This paper extends previous results on the nonexistence of continuous linear immersions from continuous-time to discrete-time nonlinear dynamical systems, highlighting fundamental obstructions in both settings.

Contribution

It generalizes nonexistence results for continuous immersions to discrete-time systems and explores related limitations involving alpha-limit sets.

Findings

Discrete-time systems share similar obstructions to continuous immersions.

Countably many omega-limit sets prevent linear immersion in discrete time.

Examples illustrate the theoretical nonexistence results.

Abstract

Understanding when linear immersions of nonlinear dynamical systems exist is important since such immersions allow us to leverage the rich tools of linear system theory to analyze nonlinear dynamics. Recently, Liu et al. (2023) showed that continuous-time dynamical systems that admit countably many but more than one omega-limit sets cannot be immersed into finite dimensional linear systems with a one-to-one and continuous mapping. In this paper, we extend these results to discrete-time dynamics and show that similar obstructions exist also in discrete time. We further consider a generalization involving alpha-limit sets. Several examples are provided to demonstrate the results.