Nonlinear Dynamics in Complexity Quantification

TL;DR

This paper reviews the evolution of chaos theory and nonlinear dynamics, highlighting the development of criteria, models, and applications in understanding and quantifying chaotic systems across various scientific fields.

Contribution

It provides a comprehensive overview of the historical phases and recent advances in nonlinear dynamics and chaos quantification, including machine learning approaches.

Findings

Development of criteria for chaos detection

Introduction of machine learning models for chaos analysis

Applications in noise reduction and control

Abstract

Chaotic systems which are due to nonlinearity have attracted a great concern in the current world and chaotic models. Systems for a wide range of operation conditions have their application in almost all branches of engineering and science. In the history of chaotic studies and nonlinearity, many different but co-existent phases can be distinguished [1]. In the initial phase, chaos was considered as a deterministic regime which, most probably, was responsible for the variations that was regarded as noise and thus was being modeled as a stochastic process. In the second phase, it was of great significance to establish criteria for detecting chaotic dynamics and thus establishing dynamical invariants which were necessary in quantifying chaos. The third step which was to develop machine learning models which could learn the dynamics and chaos from the data of the strange attractor [2]. With respect to this aspect, various model structures were developed and investigated on their ability to detect chaos from a given set of data [3]. These model structures included radial basis functions, and local linear mapping among others. The third phase is currently being investigated together with other issues surrounding nonlinear dynamics and chaos, for instance in noise reduction and control, among other issues.