Brownian Behavior in Coupled Chaotic Oscillators

Alexander N Pisarchik,Francisco Javier Martín-Pasquín

doi:10.3390/math9192503

Alexander N Pisarchik, Francisco Javier Martín-Pasquín

Open Access

https://doi.org/10.3390/math9192503

Copy DOI

Abstract

Since the dynamical behavior of chaotic and stochastic systems is very similar, it is sometimes difficult to determine the nature of the movement. One of the best-studied stochastic processes is Brownian motion, a random walk that accurately describes many phenomena that occur in nature, including quantum mechanics. In this paper, we propose an approach that allows us to analyze chaotic dynamics using the Langevin equation describing dynamics of the phase difference between identical coupled chaotic oscillators. The time evolution of this phase difference can be explained by the biased Brownian motion, which is accepted in quantum mechanics for modeling thermal phenomena. Using a deterministic model based on chaotic Rössler oscillators, we are able to reproduce a similar time evolution for the phase difference. We show how the phenomenon of intermittent phase synchronization can be explained in terms of both stochastic and deterministic models. In addition, the existence of phase multistability in the phase synchronization regime is demonstrated.

Highlights

Brownian motion is the random motion of particles in a fluid medium as a result of collisions with fluid molecules
Brownian motion is among the simplest random processes and is akin to two other simpler and more complex stochastic processes: random walk and Donsker’s theorem [3]
We explore the possibility of simulating the synchronous dynamics of coupled chaotic oscillators with stochastic differential equations, it is not at all obvious that random behavior can be reproduced using deterministic equations

Summary

Introduction

Brownian motion is the random motion of particles in a fluid medium as a result of collisions with fluid molecules This transport phenomenon was named after the Scottish biologist Robert Brown, who in 1827 observed through a microscope how particles are retained in cavities inside a pollen grain in water. With the advent of computers, the scientific community has discovered that chaotic systems behave like stochastic ones when we perform simulations in which we can control the accuracy of parameters and initial conditions [5]. This brings us back to the philosophical question of what behavior in nature is truly random. We demonstrate that the phase difference between two coupled chaotic oscillators behaves like a trajectory derived from the stochastic model, and that parameters obtained with this model can be used to describe the chaotic behavior of the phase difference

Stochastic Model

Modified Kuramoto Model

Langevin Dynamics of Two Coupled Chaotic Oscillators

Mean Angular Velocity and Diffusion Coefficient

Deterministic Model

Chaotic Phase Difference

Brownian-like Phase Diffusion

Matching Both Models

Itô-Langevin Form

Numerical Simulations

Conclusions