On universality in fractional dynamics

Abstract

In this paper the author presents the results of the preliminary investigation of fractional dynamical systems based on the results of numerical simulations of fractional maps. Fractional maps are equivalent to fractional differential equations describing systems experiencing periodic kicks. Their properties depend on the value of two parameters: the nonlinearity parameter, which arises from the corresponding regular dynamical systems; and the memory parameter which is the order of the fractional derivative in the corresponding non-linear fractional differential equations. The examples of the fractional Standard and Logistic maps demonstrate that phase space of non-linear fractional dynamical systems may contain periodic sinks, attracting slow diverging trajectories, attracting accelerator mode trajectories, chaotic attractors, and cascade of bifurcations type trajectories whose properties are different from properties of attractors in regular dynamical systems. The author argues that discovered properties should be evident in the natural (biological, psychological, physical, etc.) and engineering systems with powerlaw memory.

Logistic map with memory from economic model

“…Some properties of the fractional logistic maps, which can be represented in the form (28), are investigated by computer simulation in [49,50,51,52,53,54,55]. In this paper, we proved that the logistic map with memory (28) and (32), and the economics maps (16) describe a very special case of economic dynamics, when price is close to zero between bursts.…”

“…The discrete maps with memory, which are exact discrete analogues of the fractional differential equations, were first proposed in works [28,29,30,31]. Then, this approach, which is based on the equivalence of the fractional differential equations and the discrete maps with memory, has been applied in works [46,47,48,49,50,51,52,53,54,55] to describe properties of the discrete maps with memory. Computer simulations of some discrete maps with memory were realized in [46,47,48,49,50,51,52,53,54,55].…”

Universality in Systems with Power-Law Memory and Fractional Dynamics

“…In this section we consider various ways to introduce maps with power-law memory and fractional maps following [11,12,13,14,15,16,17,18,19,20,48,49,50,51,57,58].…”

“…Investigation of the T = 2 n -sinks' stability with n > 2 by analytic methods is complicated. In papers [11,12,13,14,15,16,17,18,19,20] this is done by numerical simulations on individual trajectories with various values of parameters (K and α) and initial conditions. As in the case of the fixed point and T = 2-sink, stability of the high order sinks is asymptotic.…”

On Nonlinear Fractional Maps: Nonlinear Maps with Power-Law Memory

“…(13)- (16) with G K (x) = x − Kx(1 − x), which for α = 1 yield the regular logistic map, are called the fractional logistic α-families of maps (Caputo, Riemann-Liouville, and Caputo difference, correspondingly). Initial investigation of the general properties of fractional dynamical systems (systems with power-or asymptotically power-law memory) was performed using the fractional standard and logistic α-families of maps 16,23,31,[39][40][41][42][43][44][45][46] . In spite of some differences, bifurcations with changes in the memory parameter, intersection of trajectories and overlapping of chaotic attractors, power-law convergence/divergence of trajectories, and the new type of attractors -cascade of bifurcations type trajectories (CBTT) were demonstrated in all α-families of maps.…”