Caputo standard α-family of maps: Fractional difference vs. fractional

Abstract

In this paper, the author compares behaviors of systems which can be described by fractional differential and fractional difference equations using the fractional and fractional difference Caputo standard α-Families of maps as examples. The author shows that properties of fractional difference maps (systems with falling factorial-law memory) are similar to the properties of fractional maps (systems with power-law memory). The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations type trajectories, and dependence of properties on the memory parameter α) and differences in properties of falling factorial- and power-law memory maps are investigated. Unlike fractional calculus, whose history is more than three hundred years old, fractional difference calculus is relatively young—it is approximately thirty years old. This is probably the result of the fact that, despite the beautiful mathematics which arises during the development of fractional difference calculus, it does not have too many applications in nature and engineering. As it has been recently demonstrated, the simplest fractional difference equations (when a fractional difference on the left is equal to a nonlinear function on the right) are equivalent to maps with falling factorial-law memory. Falling factorial-law memory is asymptotically power-law memory with the rate of convergence proportional to the inverse of time (or number of iterations in discrete cases). It is difficult to distinguish power-law from asymptotically power-law memory which frequently appears in investigation of noisy natural systems. This is the major motivation for the presented work in which we study the simplest fractional difference equations with sine nonlinearity and compare their properties with properties of the corresponding systems with power-law memory.