| Summary: | Many natural (biological, physical, etc.) and social systems possess power-law memory, and their mathematical modeling requires application of discrete and continuous fractional calculus. Most of these systems are nonlinear and demonstrate regular and chaotic behavior, which may be very different from the behavior of memoryless systems. Finding periodic solutions is essential for understanding regular and chaotic behavior of nonlinear systems. Fractional systems do not have periodic solutions except fixed points. Instead, they have asymptotically periodic solutions which, in the case of stable regular behavior, converge to the periodic sinks (similar to regular dissipative systems) and, in the case of unstable/chaotic behavior, act as repellers. In one of his recent papers, the first author derived equations which allow calculations of asymptotically periodic points for a wide class of discrete maps with memory. All fractional and fractional difference maps of the orders 0 < α< 1 belong to this class. In this paper we derive the equations that allow calculations of the coordinates of the asymptotically periodic points for a wider class of maps which include fractional and fractional difference maps of the arbitrary positive orders α> 0. The maps are defined as convolutions of a generating function - GK(x) , which may be the same as in a corresponding regular map xn+1= - GK(xn) + xn, with a kernel Uα(k) , which defines the type of a map. In the case of fractional maps, it is Uα(k) = kα, and it is Uα(k) = k(α), the falling factorial function, in the case of fractional difference maps. In this paper we define the space of kernel functions which allow calculations of the periodic points of the corresponding maps with memory. We also prove that in fractional maps of the orders 1 < α< 2 the total of all physical momenta of period–l points is zero. © 2022, Diogenes Co.Ltd.
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