Exploring the Role of Fractal Geometry in Understanding Nonlinear Dynamical Systems

Abstract

Fractal geometry is a key to the study of the complexity of nonlinear dynamical systems, especially those that are chaotic. The dependence on initial conditions in nonlinear systems is often very sensitive, and leads to unpredictable behavior which can be characterized by fractals including strange attractors, ifurcation diagrams, and fractal dimensions. In this paper, the overlap between fractal geometry and nonlinear dynamics is explored, with the most important mathematical terms being fractal dimension, Lyapunov exponents, and bifurcations. We discuss the use of fractal geometry in chaotic system, with examples of real-life applications in fluid dynamics, meteorology, and biology. Mathematical proofs, calculation and visualization are given to show how fractals are used to model the complexity of nonlinear systems.