Discrete dynamics of complex systems

Hermann Haken

doi:10.1155/s1026022697000022

Abstract

This article extends the slaving principle of synergetics to processes with discrete time steps. Starting point is a set of nonlinear difference equations which contain multiplicative noise and which refer to multidimensional state vectors. The system depends on a control parameter. When its value is changed beyond a critical value, an instability of the solution occurs. The stability analysis allows us to divide the system into stable and unstable modes. The original equations can be transformed to a set of difference equations for the unstable and stable modes. The extension of the slaving principle to the time‐discrete case then states that all the stable modes can be explicitly expressed by the unstable modes or so‐called order‐parameters.

Highlights

When we look at the various scientific disciplines, quite generally speaking we can observe the following main trends: 1. Instead of describing and studying states of a system, science is more and more interested in their temporal evolution, or, in other words, in their dynamics.2
In most cases the dynamics of systems has been studied in the frame of evolution equations that in one way or another were modeled in analogy to basic laws of physics, which describe the temporal evolution of the state of a system, for instance of its electro-magnetic fields
Space, time and variables are usually treated as continuous quantities

Summary

HERMANN HAKEN

This article extends the slaving principle of synergetics to processes with discrete time steps. Starting point is a set of nonlinear difference equations which contain multiplicative noise and which refer to multidimensional state vectors. The system depends on a control parameter. When its value is changed beyond a critical value, an instability of the solution occurs. The stability analysis allows us to divide the system into stable and unstable modes. The original equations can be transformed to a set of difference equations for the unstable and stable modes. The extension of the slaving principle to the time-discrete case states that all the stable modes can be explicitly expressed by the unstable modes or so-called order-parameters. Keywords." Discrete dynamics, Difference equations, Noise, Synergetics, Slaving principle, Complex systems

INTRODUCTION

This equation can be solved by

It can be rearranged to

THE ITERATION PROCEDURE FOR THE DISCRETE CASE