Fractional Zaslavsky and Henon Discrete Maps

TL;DR

This paper introduces fractional generalizations of the Zaslavsky and Henon discrete maps derived from fractional differential equations, emphasizing long-term memory effects and dissipation in dynamical systems.

Contribution

It develops novel fractional discrete maps from fractional differential equations, extending classical maps with memory and dissipative properties.

Findings

Fractional maps exhibit long-term memory effects.

Derived from fractional differential equations with non-integer derivatives.

Generalize classical Zaslavsky and Henon maps.

Abstract

This paper is devoted to the memory of Professor George M. Zaslavsky passed away on November 25, 2008. In the field of discrete maps, George M. Zaslavsky introduced a dissipative standard map which is called now the Zaslavsky map. G. Zaslavsky initialized many fundamental concepts and ideas in the fractional dynamics and kinetics. In this paper, starting from kicked damped equations with derivatives of non-integer orders we derive a fractional generalization of discrete maps. These fractional maps are generalizations of the Zaslavsky map and the Henon map. The main property of the fractional differential equations and the correspondent fractional maps is a long-term memory and dissipation. The memory is realized by the fact that their present state evolution depends on all past states with special forms of weights.