Rate-Dependent Bifurcations and Isolating Blocks in Nonautonomous Systems

Abstract

We examine rate-dependent tipping and the behavior of solutions to nonautonomous systems from a topological perspective. We analyze an example of a rate-dependent bifurcation in which steady solutions begin to spiral, yet remain in a bounded region around a moving equilibrium. This example motivates us to develop a theory of isolating blocks for invariant sets in nonautonomous systems. Our examination of these isolating blocks reveals that solutions in rate-dependent systems are structurally stable; the rate-dependent forcing may have some amount of noise while the underlying behavior of solutions to the system remains the same. We also find isolating blocks useful for placing bounds on critical values for rate-dependent tipping in nonautonomous systems. Finally, we introduce rate-dependent tipping for discrete maps.