Extending Chaos Theory: The Role of Nonlinearity in Multiple Mappings

TL;DR

This paper broadens the understanding of chaos in dynamical systems by extending Devaney's definition to multiple mappings, highlighting the influence of nonlinearity and introducing computational tools for analysis.

Contribution

It presents a new theorem showing how nonlinearity in a single mapping can induce chaos in a system of multiple mappings, supported by computational algorithms for detection and visualization.

Findings

Nonlinear dynamics can induce chaos across multiple mappings.

Advanced algorithms effectively detect chaotic features.

Higher-dimensional systems exhibit novel chaotic behaviors.

Abstract

This work redefines the framework of chaos in dynamical systems by extending Devaney's definition to multiple mappings, emphasizing the pivotal role of nonlinearity. We propose a novel theorem demonstrating how nonlinear dynamics within a single mapping can induce chaos across a collective system, even when other components lack sensitivity. To validate these insights, we introduce advanced computational algorithms implemented in MATLAB, capable of detecting and visualizing key chaotic features such as transitivity, sensitivity, and periodic points. The results unveil new behaviors in higherdimensional systems, bridging theoretical advances with practical applications in physics, biology, and economics. By uniting rigorous mathematics with computational innovation, this study lays a foundation for future exploration of nonlinear dynamics in complex systems, offering transformative insights into the interplay between structure and chaos. Keywords: Nonlinearity, Chaos Theory, Multiple Mappings, Devaney Chaos, Computational Algorithms, Dynamical Systems.