Phase transitions in thermodynamics of a local lyapunov exponent for fully-developed chaotic systems

Abstract

Fluctuations in the divergence of nearby orbits are studied at a crisis point of chaos. A statistical-thermodynamic method for the description of the fluctuations is developed by using symbolic dynamics, which can explicitly write a relation between a fluctuation and reference orbit. The thermodynamics (the free energy and entropy) is exactly analyzed on a nonhyperbolic attractor of maps conjugate to the map:u→u/a for 0</u<a andu→(1−u)/(1−a) fora⩽u⩽1. Te free energy has discontinuities in its slope. The entropy is directly calculated from the partition function. Then, it becomes clear that the collision of a chaotic attractor with a particular fixed point yields a singular local structure in the distribution of fluctuations. The existence of first-order phase transitions depends on the asymmetry of a map. It is shown that each of the coexisting states at the phase transition points is realized with the same probability in the thermodynamic limit.