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 $ \alpha\; (0<\alpha \le1) $ which is approximated based on the modified trapezoidal quadrature rule of order $ O(\triangle t ^{2-\alpha}) $. 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.

中文翻译：

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用于分析分数动态系统的强大计算框架

本研究概述了一种改进的隐式有限差分方法，用于逼近分数阶微分方程非线性系统双曲平衡点附近的局部稳定流形。分数阶导数用卡普托级 $\alpha\ 来描述；(0<\alpha \le1) $ 是基于 $ O(\triangle t ^{2-\alpha}) $ 阶的修正梯形求积规则近似的。讨论了所提方法的解存在性、唯一性和稳定性。给出了三个数值例子，并进行了比较以确认所提出方法的可靠性和有效性。