Numerically Unveiling Hidden Chaotic Dynamics in Nonlinear Differential Equations with Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu Fractional Derivatives

TL;DR

This paper explores how different fractional derivatives can reveal complex chaotic behaviors in nonlinear financial models, emphasizing the importance of derivative choice for accurate market analysis and prediction.

Contribution

It introduces a comparative analysis of Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu derivatives applied to financial chaos models, highlighting their impact on dynamics and computation.

Findings

Different derivatives yield varying chaotic behaviors.

Selection of derivative affects model accuracy and computation time.

Integration of AI and fractional calculus enhances financial predictions.

Abstract

In recent years, the use of variable-order differential operators has emerged as a powerful tool in the analysis of nonlinear fractional differential equations and chaotic systems. In finance, the accurate prediction of market trends and the ability to make informed investment decisions is of great importance, and the integration of artificial intelligence and mathematics has greatly improved the accuracy of these predictions. In this study, we displayed an analysis of adaptive equations produced by three fractional derivatives: the Riemann-Lioville, Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives. These fractional derivatives were employed to analyze financial models in order to gain a deeper understanding of the complex dynamics of financial markets. The models studied were the Lorenz system, Rossler system, and Shilnikov cashless model. The results showed that each fractional derivative produced varying outcomes and computation times, highlighting the importance of selecting the appropriate mathematical approach and software for financial modeling. The findings of this study underscore the continued integration of Artificial Intelligence and mathematics in financial analysis and decision-making, driving the future of investment strategies and market predictions.The application of variable-order differential operators in the analysis of nonlinear fractional differential equations and chaotic systems is an important and growing area of research that holds great promise for the field of finance.