On fractional-order maps and their synchronization

Abstract

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 $a_c(\alpha )<a<1$ where $\alpha$ is a fractional order parameter and $a_c(\alpha )\approx -\alpha$. 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.

12 pages, 9 figures

References in corpus (1)

• Emergence of Order in Dynamical Phases in Coupled Fractional Gauss Map

Cited by in corpus (3)

• Stability and Dynamics of Complex Order Fractional Difference Equations
• Study of Low-dimensional Nonlinear Fractional Difference Equations of Complex Order
• Stability analysis of fixed point of fractional-order coupled map lattices