Geometric phases in discrete dynamical systems

TL;DR

This paper extends the concept of geometric phase to discrete dynamical systems, analyzing its relationship with system rotation, chaos onset, and diffusive behavior in various rotator models.

Contribution

It introduces a generalized notion of geometric phase for discrete systems and explores its connections to rotation number, chaos, and Lyapunov exponents.

Findings

Analytical relationship between geometric phase and rotation number in the rotated sine circle map.

Connection between geometric phase and chaos onset in the rotated standard map.

Relation of geometric phase to diffusive behavior and Lyapunov exponent in chaotic regimes.

Abstract

In order to study the behaviour of discrete dynamical systems under adiabatic cyclic variations of their parameters, we consider discrete versions of adiabatically-rotated rotators. Paralleling the studies in continuous systems, we generalize the concept of geometric phase to discrete dynamics and investigate its presence in these rotators. For the rotated sine circle map, we demonstrate an analytical relationship between the geometric phase and the rotation number of the system. For the discrete version of the rotated rotator considered by Berry, the rotated standard map, we further explore this connection as well as the role of the geometric phase at the onset of chaos. Further into the chaotic regime, we show that the geometric phase is also related to the diffusive behaviour of the dynamical variables and the Lyapunov exponent.