On fractional order maps and their synchronization. (English) Zbl 1489.39028
Summary: 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 \(0<\alpha<1\) 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 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 \(\alpha\), the effective Lyapunov exponent is determined by modulus of eigenvalues. We extend these studies to fixed points of fractional nonlinear maps.
MSC:
| 39B12 | Iteration theory, iterative and composite equations |
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 33E12 | Mittag-Leffler functions and generalizations |
| 37C75 | Stability theory for smooth dynamical systems |
| 26A33 | Fractional derivatives and integrals |
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