Arnol'd cat map lattices

TL;DR

This paper constructs and analyzes lattice field theories based on Arnol'd cat maps, exploring their chaotic dynamics, symplectic structure, and ergodic properties in phase and configuration spaces.

Contribution

It introduces a novel class of Arnol'd cat map lattice models with symplectic evolution and investigates their chaotic and ergodic behavior using standard chaos benchmarks.

Findings

Chaotic properties depend on interaction strength and range

Systems exhibit ergodicity and mixing over long periods

Spectrum of periods varies with system parameters

Abstract

We construct Arnol'd cat map lattice field theories in phase space and configuration space. In phase space we impose that the evolution operator of the linearly coupled maps be an element of the symplectic group, in direct generalization of the case of one map. To this end we exploit the correspondence between the cat map and the Fibonacci sequence. The chaotic properties of these systems can be, also, understood from the equations of motion in configuration space, where they describe inverted harmonic oscillators, with the runaway behavior of the potential competing with the toroidal compactification of the phase space. We highlight the spatio-temporal chaotic properties of these systems using standard benchmarks for probing deterministic chaos of dynamical systems, namely the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. The spectrum of the periods exhibits a strong dependence on the strength and the range of the interaction.