Asymptotic cycles in fractional maps of arbitrary positive orders

Abstract

Many natural and social systems possess power-law memory, and their mathematical modeling requires the application of discrete and continuous fractional calculus. Most of these systems are nonlinear and demonstrate regular and chaotic behavior, different from the behavior of memoryless systems. Finding periodic solutions is essential for understanding the 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 that 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 < \alpha < 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 \alpha > 0. The maps are defined as convolutions of a generating function -G_K(x), which may be the same as in a corresponding regular map x_{n+1} = -G_K(x_n)+x_n, with a kernel U_{\alpha}(k), which defines the type of a map. In the case of fractional maps, it is U{\alpha}(k) = k^{\alpha}, and it is U{\alpha}(k) = k^{(\alpha)}, the falling factorial function, in the case of fractional difference maps. In this paper, we define the space of kernel functions that allow calculations of the periodic points of the corresponding maps with memory. We also prove that in fractional maps of the orders 1 < \alpha < 2 the total of all physical momenta of period-l points is zero.