We study the stability of linear fractional order maps. We show that in the stable region, the evolution is described by Mittag-Leffler functions and a well-defined effective Lyapunov exponent can be obtained in these cases. For one-dimensional systems, this exponent can be related to the corresponding fractional differential equation. A fractional equivalent of map f(x) = ax is stable for ac(α) < a < 1 where 0 < α < 1 is a fractional order parameter and ac(α) ≈−α. For coupled linear fractional maps, we can obtain ‘normal modes’ and reduce the evolution to an effective one-dimensional system. If the coefficient matrix has real eigenvalues, the stability of the coupled system is dictated by the stability of effective one-dimensional normal modes. If the coefficient matrix has complex eigenvalues, we obtain a much richer picture. However, in the stable region, evolution is dictated by a complex effective Lyapunov exponent. For larger α, the effective Lyapunov exponent is determined by modulus of eigenvalues. We extend these studies to fixed points of fractional nonlinear maps.

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关于分数阶图及其同步

我们研究线性分数阶映射的稳定性。我们表明，在稳定区域中，演化由 Mittag-Leffler 函数描述，并且在这些情况下可以获得明确定义的有效 Lyapunov 指数。对于一维系统，该指数可以与相应的分数微分方程相关。map 的分数等价物F(X) = 一种X是稳定的一种C(α) < 一种 < 1在哪里0 < α < 1是分数阶参数并且一种C(α) ≈-α. 对于耦合线性分数图，我们可以获得“正常模式”并将演化简化为有效的一维系统。如果系数矩阵具有实特征值，则耦合系统的稳定性由有效一维正常模式的稳定性决定。如果系数矩阵具有复杂的特征值，我们将获得更丰富的图像。然而，在稳定区域，演化是由一个复杂的有效 Lyapunov 指数决定的。对于较大的α，有效的李雅普诺夫指数由特征值的模数确定。我们将这些研究扩展到分数非线性映射的固定点。