Fractional Dynamics and Discrete Maps with Memory

Abstract

The study of nonlinear dynamics in terms of discrete maps is a very important step m understanding the qualitative behavior of physical systems described by differential equations (Sagdeev et al., 1988; Zaslavsky, 2005; Chirikov, 1979; Schuster and Just, 2005; Collet and Eckman, 1980). Discrete maps lead to a much simpler formalism, which is particularly useful in computer simulations. The derivatives of non-integer orders (Samko et al., 1993; Miller and Ross, 1993; Podlubny, 1999; Kilbas et al., 2006) are a natural generalization of the ordinary differentiation of integer order. Note that the continuous limit of discrete systems with power-law long-range interactions gives differential equations with derivatives of non-integer orders with respect to coordinates (see for example, (Tarasov and Zaslavsky, 2006; Tarasov, 2006a,b)). Fractional differentiation with respect to time is characterized by long-term memory effects that correspond to intrinsic dissipative processes m the physical systems. The memory effects to discrete maps mean that the present state evolution depends on all past states. The discrete maps with memory were considered, for example, in the papers (Fulinski and Kleczkowski, 1987; Fick et al., 1981; Giona, 1991; Hartwich and Fick, 1993; Gallas, 1993; Stainslavsky, 2006) and (Tarasov, 2008d, 2009a,b; Edelman and Tarasov, 2009). The interesting question is a connection of fractional equations of motion and the discrete maps with memory.