Numerical Methods for Fractal-Fractional Differential Equations and Engineering

21Dynamics of 3D Chaotic Systems with Fractal-Fractional Operators

DOI: 10.1201/9781003359258-21

21.1 INTRODUCTION

The concept of chaos theory emerged from the realisation that deterministic physical systems, despite appearing to be simple, can exhibit unpredictable behaviour. There are three possible outcomes for non-linear systems: equilibrium, steady oscillation, and chaotic change. For a deterministic equation to depict non-random chaos, these three regimes—known as the logistic map—can be described by straightforward mathematical equations. Mandelbrot, however, used fractal geometry, a novel type of geometry, to model chaotic systems. On all scales, fractals exhibit irregularity, but to the same degree. The propensity for repeating self-similarity ...