Continuous-Time Difference Equations and Distributed Chaos Modelling

Abstract

The article aims to attract the attention of researchers, experts and those interested in nonlinear dynamics and chaos theory to the not well known field of continuous-time difference equations, in the hopes of opening new doors into the study of chaotic system. Deterministic chaos and related notions are used in an increasing number of scientific works. There are a lot of problems associated with the mathematical aspects of the fine structure of chaos. Just as discrete-time difference equations have proven to be excellent models of temporal (discrete) chaos, so continuous-time difference equations provide new elegant mechanisms for onset and inside reconstructions of spatio-temporal (distributed) chaos. Distributed chaos is usually described by boundary value problems for partial differential equations. A number of these boundary value problems can be reduced to continuous-time difference equations, which enable one to build new chaos scenarios arising from the properties of the equations. Whereas the emergence of deterministic chaos is usually attributed to the complex structure of attractors, these new scenarios are based on a highly complex structure of spatially extended “points” of the attractor. Examples of reducible boundary value problems are set forth in the article, but the main focus is on a very elementary overview of the principal features of solutions of the simplest nonlinear continuous-time difference equations: loss of continuity, asymptotic periodicity, gradient catastrophe, fractal geometry, space-filling property, going beyond the horizon of predictability, self-stochasticity (deterministic solutions are asymptotically described by random processes), formation of hierarchical structures (down to arbitrarily small scales). Here we have a wonderful example of how very complex phenomena can be described with very simple equations. The use of continuous-time difference equations in the study of reducible and close-to-reducible boundary value problems migh help to advance in understanding possible mathematical mechanisms for distributed chaos.