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It is often of interest to know which systems will approach a periodictrajectory when given a periodic input. Results are available for certainclasses of systems, such as contracting systems, showing that they alwaysentrain to periodic inputs. In contrast to this, we demonstrate that thereexist systems which are globally exponentially stable yet do not entrain to aperiodic input. This could be seen as surprising, as it is known that globallyexponentially stable systems are in fact contracting with respect to someRiemannian metric.

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