Stability of Fixed Points in Generalized Fractional Maps of the Orders $0< \alpha <1$

TL;DR

This paper investigates the stability conditions of fixed points in generalized fractional maps of order between 0 and 1, broadening existing stability criteria and confirming results through numerical simulations.

Contribution

It derives new, narrower stability conditions for fixed points in generalized fractional maps, extending the understanding beyond traditional fractional difference maps.

Findings

Stability conditions are narrower than for general convolution equations.

Derived conditions match previous numerical observations for fractional standard and logistic maps.

Fixed-point stability limits coincide with period-two bifurcation points.

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.