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

Abstract

In general we are interested in dynamical systems coupled to complex hysteresis. Therefore as a first step we did some investigation on the dynamics of a periodically driven damped harmonic oscillator coupled to independent Ising spins with a local quenched disorder at zero temperature in the past. Although such a system does not produce hysteresis, we showed how to characterize the dynamics of such a piecewise-smooth system, specially in case of a large number of spins [P. Zech, A. Otto, and G. Radons, Phys. Rev. E101,042217 (2020)]. In this paper we want to extend our model to spins dimers, thus spins with pairwise interaction. We will show in which cases two interacting spins can show elementary hysteresis and we will give a connection to the Preisach Model (PM), when superpose a infinite number of spin-pairs in the thermodynamic limit. We will see, that this will lead us to a dynamical system with an additional hysteretic force in form of a play operator. By using methods from general chaos theory, piecewise-smooth system theory and statistics we will investigate the chaotic behavior of the dynamical system for a few spins and also in case of larger number of spins by calculating bifurcation diagrams, fractal dimensions and self-averaging properties. In doing so we show, how the dynamical properties of the piecewise-smooth system for a large number of spins differs from the system in its thermodynamic limit.

Analysis of a mass-spring-relay system with periodic forcing

“…where the switching time t I I is the first positive root of Eq. (22). The quantities α I I (φ I I ) and β I I are defined as…”

“…Let us now investigate the conditions under which the switching time function is continuous. First we recast the switching time equations (20) and (22) in the following general form…”

“…Lacarbonara and Vestroni [21] investigated the responses and codimension-one bifurcations in Masing-type and Bouc-Wen hysteretic oscillators. A periodically driven damped harmonic oscillator coupled to a random field Ising model showing complex hysteresis is studied in [22].…”

Analysis of a Mass-Spring-Relay System With Periodic Forcing

“…Lacarbonara and Vestroni [19] investigated the responses and codimension-one bifurcations in Masing-type and Bouc-Wen hysteretic oscillators. A periodically driven damped harmonic oscillator coupled to a random field Ising model showing complex hysteresis is studied in [20].…”

Suppression of Quasiperiodicity in Circle Maps with Quenched Disorder

“…Replacing the latter by a spatially random potential leads in the same limit to the systems studied in this paper. While spatial randomness is a fundamental concept in solid state physics [58][59][60], it is largely unexplored for dynamical systems although it is known that it can change the dynamics drastically (for instance, see [61][62][63][64][65]). Results for the special case of expanding circle maps and general expanding dynamical systems, which is not considered here, can be found in [66,67] and [68], respectively.…”