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

As a bifurcation parameter μ is varied it is common for chaotic systems to display windows of width Δμ in which there is stable periodic behavior. In this paper we examine the dependence of the transient time τ 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 Δμ. We argue and numerically verify that for one-dimensional maps with a quadratic extremum 1/τ~(Δμ)1/2 and we find an asymptotic universal form for the parameter dependence of τ within individual high-period windows. For two-dimensional maps, we conjecture that for small windows the scaling changes to 1/τ~(Δμ)^d-1/2, where d is a fractal dimension associated with a typical attractor for chaotic parameter values near the considered periodic windows.