CHARACTERIZATION OF THE NATURAL MEASURE BY UNSTABLE PERIODIC ORBITS IN CHAOTIC ATTRACTORS

The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of the unstable periodic orbits embedded in that set. This result has been proven to be valid for hyperbolic chaotic systems. We test the goodness of such a periodic-orbit characterization of the natural measure for nonhyperbolic chaotic systems by comparing the natural measure of a typical chaotic trajectory with that computed from unstable periodic orbits. Our results suggest that the unstable periodic-orbit formulation of the natural measure is typically valid for nonhyperbolic chaotic systems. [S0031-9007(97)03650-8] PACS numbers: 05.45. + b In studying chaotic systems, one is often interested in long term statistics such as averages, Lyapunov exponents, dimensions, and other invariants of the probability density or the measure. Both these statistical quantities are physically meaningful only when the measure being considered is the one generated by a typical trajectory in phase space. This measure is called the natural measure [1] and it is invariant under the evolution of the dynamics. Therefore, it is of paramount physical importance to be able to understand and to be able to characterize the natural measure [2] in terms of fundamental dynamical quantities. And there is nothing more fundamental than to express the natural measure in terms of the periodic orbits embedded in a chaotic attractor. A key contribution along these lines was made in Ref. [3] in which the authors obtained an expression for the invariant natural measure in terms of the magnitude of the eigenvalues of the unstable periodic orbits embedded in the chaotic attractor. They proved [3] the correctness of their expression but only for the special case of a hyperbolic dynamics [4]. The validity of their results for physical, which are typically nonhyperbolic, situations remained, however, only a conjecture. The purpose of this Letter is to provide strong evidence for the applicability of the results of Grebogi et al. [3] to nonhyperbolic chaotic systems and, hence, validating their conjecture. The long-time probability density or the natural measure generated by typical trajectories of chaotic dynamical systems is generally highly singular. A trajectory originated from a random initial condition in the basin of attraction of a chaotic attractor visits different parts of the attractor with drastically different probabilities. Call regions with high probabilities “hot” spots and regions with low probabilities “cold” spots. Such hot and cold spots in the attractor can be interwoven on arbitrarily fine scales. In this sense, chaotic attractors are said to possess a multifractal structure, which is a property of the natural measure. To obtain the natural measure, one covers the chaotic attractor with a grid of cubes and examines the frequency with which a typical trajectory visits these cubes in the limit that both the length of the trajectory goes to infinity and the size of the grid goes to zero [5]. Except for an initial condition set of Lebesgue measure zero in the basin of attraction of the chaotic attractor, these frequencies in the cubes are the same for different choices of the initial condition x0, and they are called the natural measure. Specifically, let fsx0, T , eid be the amount of time that a trajectory from a random initial condition x0 in the basin of attraction spends in the ith covering cube Ci of edge length ei in a time T . The natural measure of the attractor in the cube Ci is