论文标题
在分数级图及其同步上
On fractional-order maps and their synchronization
论文作者
论文摘要
我们研究线性分数图的稳定性。我们表明，在稳定区域中，Mittag-Leffler函数描述了进化，在这些情况下可以获得有效的Lyapunov指数。对于一维系统，该指数可以与相应的分数微分方程有关。 $ f（x）= ax $的分数等效于$ a_c（α）<a <a <1 $，其中$α$是分数订单参数，而$ a_c（α）\ of -ach of -α$。对于耦合的线性分数图，我们可以获得“正常模式”，并将演变减少到有效的一维系统。如果特征值是真实的，则耦合系统的稳定性由有效的一维正常模式的稳定性决定。对于复杂的特征值，我们获得了更丰富的图片。但是，在稳定区域中，模量的演变由Mittag-Leffler函数决定，有效的Lyapunov指数由特征值的模量确定。我们将这些研究扩展到分数非线性图的同步固定点。
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 $a_c(α)<a<1$ where $α$ is a fractional order parameter and $a_c(α)\approx -α$. For coupled linear fractional maps, we can obtain `normal modes' and reduce the evolution to effectively one-dimensional system. If the eigenvalues are real the stability of the coupled system is dictated by the stability of effectively one-dimensional normal modes. For complex eigenvalues, we obtain a much richer picture. However, in the stable region, the evolution of modulus is dictated by Mittag-Leffler function and the effective Lyapunov exponent is determined by modulus of eigenvalues. We extend these studies to synchronized fixed points of fractional nonlinear maps.
