Scaling of the durations of chaotic transients in windows of attracting periodicity

As a bifurcation parameter $\mu$ is varied it is common for chaotic systems to display windows of width $\Delta \mu$ in which there is stable periodic behavior. In this paper we examine the dependence of the transient time $\tau$ of a periodic window (i.e., the typical time an initial condition wanders around chaotically before settling into periodic behavior) on the size of the periodic window $\Delta \mu$. We argue and numerically verify that for one-dimensional maps with a quadratic extremum $1/\tau \sim (\Delta \mu {)}^{1/2}$ and we find an asymptotic universal form for the parameter dependence of $\tau$ within individual high-period windows. For two-dimensional maps, we conjecture that for small windows the scaling changes to $1/\tau \sim (\Delta \mu {)}^{d-1/2}$, where $d$ is a fractal dimension associated with a typical attractor for chaotic parameter values near the considered periodic windows.