Fractional Maps and Fractional Attractors. Part I: α\alpha-Families of Maps

In this paper we present a uniform way to derive families of maps from the corresponding differential equations describing systems which experience periodic kicks. The families depend on a single parameter - the order of a differential equation $\alpha > 0$. We investigate general properties of such families and how they vary with the increase in $\alpha$ which represents increase in the space dimension and the memory of a system (increase in the weights of the earlier states). To demonstrate general properties of the $\alpha$-families we use examples from physics (Standard $\alpha$-family of maps) and population biology (Logistic $\alpha$-family of maps). We show that with the increase in $\alpha$ systems demonstrate more complex and chaotic behavior.Comment: 10 pages 8 figure

Logistic map with memory from economic model

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 values at that time, but also on their values at previous times. For the mathematical description of the memory effects, we use the theory of derivatives of non-integer order. Crises are considered as sharp splashes (bursts) of the price, which are mathematically described by the delta-functions. Using the equivalence of fractional differential equations and the Volterra integral equations, we obtain discrete maps with memory that are exact discrete analogs of fractional differential equations of economic processes. We derive logistic map with memory, its generalizations, and "economic" discrete maps with memory from the fractional differential equations, which describe the economic natural growth with competition, power-law memory and crises.Comment: 19 pages, pd