Boundary crises and supertrack orbits in the Gauss map

Abstract

Supertrack orbits are used to investigate boundary crises in an one-dimensional, two-parameter ($\nu ,\beta$), nonlinear Gauss map. After the crises, the time evolution of the orbit is shown to be pseudo-chaotic. We investigate the chaotic transient, that is, the time an orbit spends in a region where the chaotic attractor existed prior to the crisis, and confirm it decays exponentially with time. The relaxation time is given by a power-law $\tau \propto \mu ^{\gamma }$ with $\mu =|\beta -\beta _c|$ corresponding to the distance measured in the parameter where the crises are observed. $\beta _c$ is the parameter that characterizes the occurrence of a boundary crisis and the numerical value of the power measured was $\gamma =1/2$.