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 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.