A Numerical Framework for Fractional and Fractal-Fractional Analysis of the Pehlivan Chaotic System Using Caputo Derivative

The Pehlivan chaotic system, introduced by Ibrahim Pehlivan and Yilmaz Uyaroglu, is a three-dimensional autonomous chaotic model with rich dynamical properties. This paper develops a numerical framework for the fractional and fractal-fractional versions of the system using the Caputo derivative. Integrating fractal geometry with fractional calculus reveals enhanced complexity and diverse dynamical behavior. Theoretical analyses covering equilibrium points, existence and uniqueness of solutions, Ulam stability, and error bounds ensure mathematical validity. Numerical results confirm accuracy and convergence, showing how fractional and fractal-fractional parameters influence the Pehlivan attractors and demonstrate the potential of fractal-fractional calculus in chaotic system analysis.