Global exponential stability (and contraction of an unforced system) does not imply entrainment to periodic inputs

TL;DR

This paper shows that global exponential stability does not guarantee a system will synchronize with periodic inputs, challenging assumptions based on contraction theory.

Contribution

It demonstrates the existence of globally exponentially stable systems that do not entrain to periodic inputs, contrary to common beliefs in contraction analysis.

Findings

Existence of systems that are stable but do not synchronize with periodic inputs

Contradicts the assumption that exponential stability implies entrainment

Highlights limitations of contraction-based stability analysis

Abstract

It is often of interest to know which systems will approach a periodic trajectory when given a periodic input. Results are available for certain classes of systems, such as contracting systems, showing that they always entrain to periodic inputs. In contrast to this, we demonstrate that there exist systems which are globally exponentially stable yet do not entrain to a periodic input. This could be seen as surprising, as it is known that globally exponentially stable systems are in fact contracting with respect to some Riemannian metric.