The dynamics of a driven harmonic oscillator coupled to independent Ising spins in random fields

TL;DR

This paper investigates the complex dynamical behavior of a hybrid system combining a driven harmonic oscillator with a Random Field Ising Model, revealing phenomena like chaos, multistability, and bifurcations through advanced dynamical systems analysis.

Contribution

It introduces a detailed analysis of the hybrid oscillator-Spin system using bifurcation theory and chaos analysis, highlighting the transition from finite to thermodynamic limits.

Findings

Identification of chaos and multistability in the system

Bifurcation diagrams and Lyapunov exponents computed

Transition to smooth nonlinear behavior in the thermodynamic limit

Abstract

We aim at an understanding of the dynamical properties of a periodically driven damped harmonic oscillator coupled to a Random Field Ising Model (RFIM) at zero temperature, which is capable to show complex hysteresis. The system is a combination of a continuous (harmonic oscillator) and a discrete (RFIM) subsystem, which classifies it as a hybrid system. In this paper we focus on the hybrid nature of the system and consider only independent spins in quenched random local fields, which can already lead to complex dynamics such as chaos and multistability. We study the dynamic behavior of this system by using the theory of piecewise-smooth dynamical systems and discontinuity mappings. Specifically, we present bifurcation diagrams, Lyapunov exponents as well as results for the shape and the dimensions of the attractors and the self-averaging behavior of the attractor dimensions and the magnetization. Furthermore we investigate the dynamical behavior of the system for an increasing number of spins and the transition to the thermodynamic limit, where the system behaves like a driven harmonic oscillator with an additional nonlinear smooth external force.