Fractional Equations of Kicked Systems and Discrete Maps

TL;DR

This paper introduces fractional discrete maps derived from kicked equations with non-integer derivatives, highlighting their long-term memory effects and generalizing classical maps with a dependence on all past states.

Contribution

It presents a novel derivation of fractional discrete maps from fractional kicked equations, emphasizing their long-term memory properties and extending classical dynamical maps.

Findings

Fractional maps incorporate long-term memory effects.

Memory weights follow power-law functions.

Generalizes standard and dissipative maps.

Abstract

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 property of the suggested fractional maps is a long-term memory. The memory effects in the fractional discrete maps mean that their present state evolution depends on all past states with special forms of weights. These forms are represented by combinations of power-law functions.