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The study of systems with memory requires methods which are different from the methods used in regular dynamics. Systems with power-law memory in many cases can be described by fractional differential equations, which are…
Using kicked differential equations of motion with derivatives of noninteger orders, we obtain generalizations of the dissipative standard map. The main property of these generalized maps, which are called fractional maps, is long-term…
In this paper we extend the notion of an $\alpha$-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling…
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…
Dynamical behaviour of discrete dynamical systems has been investigated extensively in the past few decades. However, in several applications, long term memory plays an important role in the evolution of dynamical variables. The definition…
Starting from kicked equations of motion with derivatives of non-integer orders, we obtain "fractional" discrete maps. These maps are generalizations of well-known universal, standard, dissipative, kicked damped rotator maps. The main…
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…
The presence of the power-law memory is a significant feature of many natural (biological, physical, etc.) and social systems. Continuous and discrete fractional calculus is the instrument to describe the behavior of systems with the…
Derivatives of fractional order with respect to time describe long-term memory effects. Using nonlinear differential equation with Caputo fractional derivative of arbitrary order $\alpha>0$, we obtain discrete maps with power-law memory.…
There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of…
Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of…
We study the fractional maps of complex order, $\alpha_0e^{i r \pi/2}$ for $0<\alpha_0<1$ and $0\le r<1$ in 1 and 2 dimensions. In two dimensions, we study H{\'e}non and Lozi map and in $1d$, we study logistic, tent, Gauss, circle, and…
Discrete maps with long-term memory are obtained from nonlinear differential equations with Riemann-Liouville and Caputo fractional derivatives. These maps are generalizations of the well-known universal map. The memory means that their…
Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by using integrations and…
General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the non-local properties of linear and nonlinear dynamical systems are studied by using of general fractional calculus, equations with general…
A generalization of the economic model of logistic growth, which takes into account the effects of memory and crises, is suggested. Memory effect means that the economic factors and parameters at any given time depend not only on their…
This article is a short review of the recent results on properties of nonlinear fractional maps which are maps with power- or asymptotically power-law memory. These maps demonstrate the new type of attractors - cascade of bifurcations type…
The regular logistic map was introduced in 1960s, served as an example of a complex system, and was used as an instrument to demonstrate and investigate the period doubling cascade of bifurcations scenario of transition to chaos. In this…
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 $\alpha$-Families of Maps as…
Properties of the phase space of the standard maps with memory obtained from the differential equations with the Riemann-Liouville and Caputo derivatives are considered. Properties of the attractors which these fractional dynamical systems…