Chaos and stability analysis of the nonlinear fractional-order autonomous system

Introduction: Fractional approaches have emerged as powerful tools for modeling a wide range of phenomena in engineering and science. This study focuses on a chaotic behavior for numerical method and stability analysis to investigate the nonlinear fractional-order autonomous systems using fractional derivative operators, specifically the Atangana-Baleanu fractional derivative in the Caputo sense. Objective: The primary objective of this work is to analyze the Ulam-Hyers stability of the nonlinear fractional-order autonomous systems involving fractional derivatives. To achieve this, we develop numerical schemes based on fractional calculus principles and employ Lagrange interpolation polynomials to simulate the chaotic behavior of the proposed problem. Methods: We establish to apply Krasnoselskii's fixed-point approach to examine the existence of at least one solution and investigate the Ulam-Hyers stability results for the given problem. We obtain an approximate numerical solution using a Lagrange interpolation polynomial-based numerical scheme. Results: We examine the graphical behavior of the results obtained and show that both numerical methods are very efficient and provide precise and outstanding results to determine approximate numerical solutions of fractional differential equations. Conclusion: The graphical analysis of fractional order and parameter values reveals new insights and interesting phenomena related to chaotic systems. The findings emphasize the significant role of fractional approaches in studying nonlinear systems of scientific and physical importance. Additionally, the proposed numerical scheme is shown to be efficient and reliable for solving nonlinear fractional models.