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.
- Publication:
-
Fractals
- Pub Date:
- 2021
- DOI:
- 10.1142/S0218348X21501504
- arXiv:
- arXiv:2007.04822
- Bibcode:
- 2021Fract..2950150G
- Keywords:
-
- Fractional Calculus;
- Fractional Maps;
- Stability and Synchronization;
- Nonlinear Sciences - Chaotic Dynamics;
- Mathematics - Dynamical Systems;
- 26A33;
- 39A30
- E-Print:
- 12 pages, 9 figures