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Stability of fixed points in generalized fractional maps of the orders \(0< \alpha <1\)

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Abstract

Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of the asymptotic stability of the fixed points for maps of the orders \(0< \alpha <1\) that are derived in this paper are narrower than the conditions of stability for the discrete convolution equations in general and wider than the well-known conditions of stability for the fractional difference maps. The derived stability conditions for the fractional standard and logistic maps coincide with the results previously observed in numerical simulations. In nonlinear maps, one of the derived limits of the fixed-point stability coincides with the fixed-point—asymptotically period-two bifurcation point.

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Acknowledgements

The author acknowledges support from Yeshiva University’s 2021–2022 Faculty Research Fund and expresses his gratitude to the administration of Courant Institute of Mathematical Sciences at NYU for the opportunity to perform computations at Courant and to Virginia Donnelly for technical help.

Funding

This research was supported by Yeshiva University’s 2021–2022 Faculty Research Fund.

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  1. Department of Physics, Stern College at Yeshiva University, 245 Lexington Ave, New York, NY, 10016, USA

    Mark Edelman

  2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012, USA

    Mark Edelman

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Edelman, M. Stability of fixed points in generalized fractional maps of the orders \(0< \alpha <1\). Nonlinear Dyn 111, 10247–10254 (2023). https://doi.org/10.1007/s11071-023-08359-0

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