Higher-order finite difference modeling of chaos and Turing-like patterns in space-fractional reaction–diffusion systems

Abstract

Abstract This paper explores using fractional-order methods to model chaotic dynamics and spatiotemporal Turing-type patterns in complex systems. By employing fractional calculus, which captures noninteger order dynamics, the study provides insights into the mechanisms driving pattern formation and chaos in systems such as chemical reactions, ecological models, and biological processes. A mathematical framework is developed to investigate the emergence, stability, and influence of these patterns. The integer-order spatial derivative is replaced with two-sided Riemann–Liouville fractional operators, and a fourth-order finite difference scheme is introduced for approximating fractional diffusion-like problems in one and two dimensions. Stability and convergence analyses of the proposed methods are conducted. To further explore pattern formation, the study extends its analysis to a multicomponent system, solving for spatiotemporal Turing-like patterns in both one and two dimensions. The results demonstrate the generation of several novel and existing patterns. Understanding chaotic behavior and pattern formation is essential for various scientific and engineering applications. The insights gained from this study contribute to a deeper comprehension of complex systems and may aid in controlling or utilizing these patterns across disciplines such as physics, chemistry, biology, and ecology.