Structures of chaos and q-phase transitions

Abstract

Abstract A statistical-mechanical formalism of chaos based on the geometry of invariant sets in phase space is discussed to show that chaotic dynamical systems can be treated by a formalism analogous to that of thermodynamic systems if one takes a relevant coarse-grained quantity, but their statistical laws are quite different from those of thermodynamic systems. This is a generalization of statistical mechanics for dealing with dissipative and hamiltonian dynamical systems of a few degrees of freedom. Thus the sum of the local expansion rate of nearby orbits along a relevant orbit over a long but finite time has been introduced in order to describe and characterize (1) a drastic change of the structure of a chaotic region at a bifurcation and associated anomalous phenomena, (2) the critical KAM torus, diffusion and repeated sticking of a chaotic orbit to a critical torus. Here a q-phase transition, analogous to the ferromagnetic phase transition, plays an important role.