Jumps to Resonance: Long Chaotic Transients, Unpredictable Outcome, and the Probability of Restabilization

Abstract

For nonlinear oscillators with the ability to escape from a potential well, jumps to resonance from a fold can result in qualitatively different types of response. They may be safe where the system always restabilizes onto a oscillation within the well; unsafe where the system always escapes out of the well; or indeterminate where the outcome is unpredictable. In the indeterminate case, long chaotic transients may persist before the system decides to which long-term behavior it will settle upon. We determine at which control parameters indeterminate jumps occur. We also examine how the transients scale as well as the probability of restabilization after the bifurcation.