Catastrophes with indeterminate outcome

Abstract

A catastrophe in a dissipative dynamical system which causes an attractor to completely lose stability will result in a transient trajectory making a rapid jump in phase space to some other attractor. In systems where more than one other attractor is available, the attractor chosen may depend very sensitively on how the catastrophe is realized. Two examples in forced oscillators of Duffing type illustrate how the probabilities of different outcomes can be estimated using the phase space geometry of invariant manifolds.