_{1}

^{*}

In this study, a discrete fractional Henon map is proposed in the Caputo discrete delta’s sense. The results show that the discrete fractional calculus is an efficient tool and the maps derived in this way have simpler forms but hold rich dynamical behaviors.

The chaotic behavior is one important aspect of dynamical systems. Much attention has been paid to the topic on fractional differential equations in the past decades [

Generally speaking, the map

This work looks similar, but the essential difference is that it adopts different definitions of fractional difference. In our research, on the basis of the Caputo-like delta difference [

Concerning nonlinear fractional differential equation of the form

here

in the sense that if a continuous function solves (2) if and only if it solves (1).

Considering the discrete fractional calculus, we can get the corresponding fractional difference equation. We start with some necessary definitions from discrete fractional calculus theory and preliminary results so that this paper is self-contained.

Definition 1. (See [

where

Definition 2. (See [

where

Theorem 1. (See [

the equivalent discrete integral equation can be obtained as

where the initial iteration reads

The complex difference equation with long-term memory is obtained here. It can reduce to the classical one when the difference order

The Henon map is given by the following pair of first-order difference equations

where

From the discrete fractional calculus, we modify the standard map as a fractional one

From (3), we can obtain the following discrete integral form from

where

For the fractional Henon map, an explicit numerical formula can be given as

when

The bifurcation diagram for fractional discrete Henon map when α = 1, x(0) = 0, y(0) = 0

Using the numerical formula (6), set the step size

Henon strange attractor for fractional discrete Henon map when α = 1, x(0) = 0

The bifurcation diagram of the fractional discrete Henon map when μ = 0.95

The bifurcation for fractional discrete Henon map diagram when μ = 0.8

The bifurcation diagram for fractional discrete Henon map when μ = 0.6

The bifurcation for fractional discrete Henon map diagram when μ = 0.4

Chaos of the fractional discrete Henon map when μ = 0.1

happens. It can be concluded that the chaos zones are clearly different when we change the difference order

In this paper, the suggested fractional Henon map demonstrates a chaotic behavior with a new type of attractors. The interesting property of the fractional map is the long term memory. Computer simulations of the fractional discrete maps with memory prove that the nonlinear dynamical systems, which are described by the equations with Caputo discrete delta’s sense, exhibit a new type of chaotic motion. Through a discrete fractional Henon map it reveals that the dynamical behavior holds the discrete memory even the difference order is very small.

The author thanks the referee for providing constructive correction suggestions. This work is financially supported by the Zhejiang Natural Science Foundation (Grant no. LQ12A01010) and Hangzhou Polytechnic (KZYZ-2009-2).