GENERALIZED ENTROPIES OF CHAOTIC MAPS AND FLOWS - A UNIFIED APPROACH

A thermodynamic study of nonlinear dynamical systems, based on the orb its' return times to the elements of a generating partition. is propos ed. Its grand canonical nature makes it suitable for application to bo th maps and flows, including autonomous ones. When specialized to the evaluation of the generalized entropies K-q, this technique reproduces a well-known formula for the metric entropy K-l and clarifies the rel ationship between a flow and the associated Poincare maps, beyond the straightforward case of periodically forced nonautonomous systems. Num erical estimates of the topological and metric entropy are presented f or the Lorenz and Rossler systems. The analysis has been carried out e xclusively by embedding scalar time series, ignoring any further knowl edge about the systems, in order to illustrate its usefulness for expe rimental signals as well. Approximations to the generating partitions have been constructed by locating the unstable periodic orbits of the systems up to order 9. The results agree with independent estimates ob tained from suitable averages of the local expansion rates along the u nstable manifolds. (C) 1997 American Institute of Physics.