Role of unstable periodic orbits in phase transitions of coupled map lattices

TL;DR

This paper explores how unstable periodic orbits influence phase transitions in coupled map lattices, revealing that anomalous orbit distributions underpin these transitions and offering a new perspective on their classification.

Contribution

It extends thermodynamic formalism to coupled map lattices, linking unstable periodic orbits to phase transitions and introducing a way to define transition order in chaotic systems.

Findings

Phase transitions are associated with anomalous distributions of unstable periodic orbits.

A q-phase transition characterizes the spatio-temporal configuration space.

Control parameters can be treated as macroscopic temperature in some cases.

Abstract

The thermodynamic formalism for dynamical systems with many degrees of freedom is extended to deal with time averages and fluctuations of some macroscopic quantity along typical orbits, and applied to coupled map lattices exhibiting phase transitions. Thereby, it turns out that a seed of phase transition is embedded as an anomalous distribution of unstable periodic orbits, which appears as a so-called q-phase transition in the spatio-temporal configuration space. This intimate relation between phase transitions and q-phase transitions leads to one natural way of defining transitions and their order in extended chaotic systems. Furthermore, a basis is obtained on which we can treat locally introduced control parameters as macroscopic ``temperature'' in some cases involved with phase transitions.