Crisis in Time-Dependent Dynamical Systems

Abstract

Many dynamical systems operate in a fluctuating environment. However, even in low-dimensional setups, transitions and bifurcations have not yet been fully understood. In this Letter we focus on crises, a sudden flooding of the phase space due to the crossing of the boundary of the basin of attraction. We find that crises occur also in nonautonomous systems although the underlying mechanism is more complex. We show that in the vicinity of the transition, the escape probability scales as exp[-α(lnδ)^{2}], where δ is the distance from the critical point, while α is a model-dependent parameter. This prediction is tested and verified in a few different systems, including the Kuramoto model with inertia, where the crisis controls the loss of stability of a chimera state.