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

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.