Scaling of periodic orbits in two-dimensional chaotic systems.

Abstract

Invariant sets arising in chaotic dynamics can be organized around their underlying skeleton of unstable periodic orbits. In this paper a scaling function for the eigenvalues of the unstable periodic orbits of strange sets embedded in two dimensions is introduced. For the piecewise-linear Lozi mapping, the scaling function is obtained analytically, as a convergent series in b, the inverse dissipation strength. In general, the scaling function is shown to converge only for uniformly hyperbolic systems, while the inadequacy of this description for other systems is demonstrated and discussed. The converged periodic-orbit scaling function is applied to yield accelerated convergence of the multifractal spectrum of a strange set.