ON FRACTIONAL ORDER MAPS AND THEIR SYNCHRONIZATION

Prashant M. Gade and Sachin Bhalekar ()
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Prashant M. Gade: Department of Physics, RTM Nagpur University, Nagpur, India
Sachin Bhalekar: ��School of Mathematics and Statistics, University of Hyderabad, Hyderabad, India

FRACTALS (fractals), 2021, vol. 29, issue 06, 1-9

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.

Keywords: Fractional Calculus; Fractional Maps; Stability and Synchronization (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21501504

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