On fractional-order maps and their synchronization

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.