Strange Attractors in Fractional Differential Equations: A Topological Approach to Chaos and Stability

TL;DR

This paper investigates the chaotic dynamics of fractional differential equations using topological methods, introducing new tools to analyze strange attractors and their stability in systems with Caputo derivatives.

Contribution

It presents novel techniques for computing the fractional Conley index and Lyapunov exponents, advancing the understanding of chaos in fractional differential equations.

Findings

Identification of conditions for chaos in fractional systems

Development of methods to compute fractional Conley index

Insights into fractal properties of strange attractors

Abstract

In this work, we explore the dynamics of fractional differential equations (FDEs) through a rigorous topological analysis of strange attractors. By investigating systems with Caputo derivatives of order \( \alpha \in (0, 1) \), we identify conditions under which chaotic behavior emerges, characterized by positive topological entropy and the presence of homoclinic and heteroclinic structures. We introduce novel methods for computing the fractional Conley index and Lyapunov exponents, which allow us to distinguish between chaotic and non-chaotic attractors. Our results also provide new insights into the fractal and spectral properties of strange attractors in fractional systems, establishing a comprehensive framework for understanding chaos and stability in this context.