An observation regarding systems which converge to steady states for all constant inputs, yet become chaotic with periodic inputs

TL;DR

This paper presents a construction of systems that stabilize to steady states under constant inputs but exhibit chaotic behavior under periodic inputs, highlighting complex dynamics in forced differential equations.

Contribution

It introduces a general construction and concrete examples of systems with these contrasting behaviors, expanding understanding of forced differential system dynamics.

Findings

Systems converge to steady states for constant inputs

Systems can become chaotic with periodic inputs

Examples include systems with universal convergence to a single state

Abstract

This note provides a general construction, and gives a concrete example of, forced ordinary differential equation systems that have these two properties: (a) for each constant input u, all solutions converge to a steady state but (b) for some periodic inputs, the system has arbitrary (for example, "chaotic") behavior. An alternative example has the property that all solutions converge to the same state (independently of initial conditions as well as input, so long as it is constant).