Thermodynamics of attractor enlargement.

Abstract

Sudden enlargement of a small-size chaotic attractor occurs when it collides with a coexisting nonattracting chaotic set. Before the collision the two dynamically independent invariant sets are characterized by different thermodynamical potentials, e.g., by free energies. The infinitesimally weak dynamical coupling appearing at crisis generates a third component, and the resultant free energy of the enlarged attractor is obtained as the minimum of the three partial free energies. Not far beyond the crisis the free energy of the enlarged attractor is still very close to the minimum of those belonging to the remnant of the old attractor and the other nonattracting chaotic set. We demonstrate this general phenomenon by one-dimensional maps. By extending the concept of Frobenius-Perron operators we invent the constrained generalized Frobenius-Perron operator providing us with a method to compute the free energies of invariant chaotic sets which are either coexisting side by side each other independently or being embedded in a larger set.