A robust computational framework for analyzing fractional dynamical systems

Abstract

<p style='text-indent:20px;'>This study outlines a modified implicit finite difference method for approximating the local stable manifold near a hyperbolic equilibrium point for a nonlinear systems of fractional differential equations. The fractional derivative is described in the Caputo sense of order <inline-formula><tex-math id="M1">\begin{document}$ \alpha\; (0<\alpha \le1) $\end{document}</tex-math></inline-formula> which is approximated based on the modified trapezoidal quadrature rule of order <inline-formula><tex-math id="M2">\begin{document}$ O(\triangle t ^{2-\alpha}) $\end{document}</tex-math></inline-formula>. The solution existence, uniqueness and stability of the proposed method is discussed. Three numerical examples are presented and comparisons are made to confirm the reliability and effectiveness of the proposed method.</p>