Numerical investigation of fractional order chaotic systems using a new modified Runge-Kutta method

Abstract

This article introduces a new modified two-stage fractional Runge-Kutta method for solving fractional order dynamical systems. The non-integer order derivative is considered in the Caputo sense, as it reliably captures the physical nature of the systems. A comprehensive mathematical analysis is performed, covering aspects such as consistency, convergence and error bound. The method's effectiveness is validated by comparing it with existing methods in the literature for solving linear and nonlinear fractional initial value problems. The proposed method is then utilized to investigate a wide range of commensurate fractional order continuous systems demonstrating chaotic behavior, with their phase diagrams illustrated. Parametric configurations and fractional orders for which specific fractional attractors either exhibit or lack chaotic behavior is also examined. The computation Lyapunov exponents and 0-1 test have been performed to elucidate the dynamic behaviors of the analyzed fractional order systems.