Influence of Spatial Inhomogeneity on the Formation of Chaotic Modes at the Self-Organization Process

I A Lyashenko,Z M Liashenko

doi:10.15407/ujpe65.2.130

Abstract

The Lorentz system of equations, in which gradient terms are taken into account, has been solved numerically. Three fundamentally different modes of evolution are considered. In the first mode, the spatial distribution of the order parameter permanently changes in time, and domains of two types with positive and negative order parameter values are formed. In the second mode, the order parameter distribution is close to the stationary one. Finally, in the third mode, the order parameter is identical over the whole space. The dependences of the average area of domains, their number, and their total area on the time are calculated in the first two cases. In the third case, the contribution of gradient terms completely vanishes, and a classical Lorenz attractor is realized.

Highlights

Self-organization is a process of ordered structure formation at which the structure-forming parameter is not changed, i.e., when a specific external influence is absent [1]
In the third case with DuS = DvS = 10−1, the system quickly transits into a mode in which the order parameter acquires the same value over the whole simulation region, so that the contribution of gradient terms completely vanishes according to Eq (11)
We numerically studied the Lorentz system of equations with gradient terms describing the spatial distribution of basic parameters, being taken into account in each equation

Summary

Introduction

Self-organization is a process of ordered structure formation at which the structure-forming parameter is not changed, i.e., when a specific external influence is absent [1]. Influence of Spatial Inhomogeneity on the Formation of Chaotic Modes lowing a self-organization process to take place for a long time consists in an approximate equality between the input and output energy fluxes (in the case of the Earth, the latter should not be monotonically heated up or cooled down). From the aforesaid, it follows that there are stationary nonequilibrium processes in self-organized systems. We will consider the case where the spatial distribution of the order parameter permanently changes in the course of system’s evolution

Classical Lorenz Attractor

Account for a Spatial Inhomogeneity and the Numerical Calculation Procedure

Simulation Results and Their Discussion

Analysis of the Fractal Properties of Order Parameter Distributions

Conclusions