Asymptotic cycles in fractional generalizations of multidimensional maps

Abstract

In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various natural and socioeconomic systems. They are also used in engineering. Many natural and almost all socioeconomic systems possess memory which, in many cases, is power-law-like memory. Generalized fractional maps, in which memory is not exactly the power-law memory but the asymptotically power-law-like memory, are used to model and investigate general properties of these systems. In this paper we extend the definition of the notion of generalized fractional maps of arbitrary positive orders that previously was defined only for maps which, in the case of integer orders, converge to area/volume-preserving maps. Fractional generalizations of H'enon and Lozi maps belong to the newly defined class of generalized fractional maps. We derive the equations which define periodic points in generalized fractional maps. We consider applications of our results to the fractional and fractional difference H'enon and Lozi maps.