How Fast is Too Fast? Rate-induced Bifurcations in Multiple Time-scale Systems

Abstract

This thesis studies the phenomena of rate-induced bifurcations. Externally forced systems may have a critical rate, above which they undergo some sort of destabilisation, and move away suddenly to a new state. Mathematically, the phenomenon is a non-autonomous instability.

We present a framework in which rate-induced bifurcations can be studied. This is based on geometric singular perturbation theory which is derived from Fenichel’s Theorem. In particular we make use of folded singularities and canard trajectories, which are modern concepts from geometric singular perturbation theory.

We concentrate on systems with multiple time-scales where the mechanism for a rate-induced bifurcation is not obvious. So much so, that once a multiple time-scale system has undergone a rate-induced bifurcation, the instability threshold which separates initial states that destabilise from those that adiabatically follow a changing stable state is described as non-obvious. We study in detail the complicated non-obvious instability threshold that arises near a folded saddle-node (type I) singularity. In particular, we show how the complicated threshold structure depends on two parameters – the ratio of time-scales and the the folded singularity bifurcation parameter. In contrast, we also show single time-scale systems where the rate-induced bifurcation is caused by a large perturbation in the boundary of the basin of attraction for the stable state.