Scaling of the Durations of Chaotic Transients in Windows of Attracting Periodicity

Joeri Jacobs, Edward Ott, and Brian R. Hunt

Phys. Rev. E 56 (1997), 6508-6515

Online abstract and download information

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 $1/\tau ~ (\Delta\mu)^{1/2}$ 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 $1/\tau ~ (\Delta\mu)^{d-1/2}$, where $d$ is a fractal dimension associated with a typical attractor for chaotic parameter values near the considered periodic windows.

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