Critical Exponent of Chaotic Transients in Nonlinear Dynamical Systems

Abstract

The average lifetime of a chaotic transient versus a system parameter is studied for the case wherein a chaotic attractor is converted into a chaotic transient upon collision with its basin boundary (a crisis). Typically the average lifetime $T$ depends upon the system parameter $p$ via $T\sim {|p-p_c|}^{-\gamma }$, where $p_c$ denotes the value of $p$ at the crisis and we call $\gamma$ the critical exponent of the chaotic transient. A theory determining $\gamma$ for two-dimensional maps is developed and compared with numerical experiments. The theory also applies to critical behavior at interior crises.