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

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).