Study of Low-dimensional Nonlinear Fractional Difference Equations of Complex Order

TL;DR

This paper investigates complex fractional-order nonlinear maps in one and two dimensions, revealing that smooth maps tend to be regular while discontinuous maps exhibit chaos and multistability, with different bifurcation behaviors based on the generalization method.

Contribution

It introduces the study of complex fractional-order maps in 1D and 2D, analyzing their bifurcation and chaotic behavior, and compares two different generalization approaches in 2D.

Findings

Smooth maps like logistic and H{\'e}non do not show chaos.

Discontinuous maps like Lozi and Bernoulli exhibit chaos.

Complex fractional-order maps display multistability in 2D.

Abstract

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 Bernoulli maps. The generalization in $2d$ can be done in two different ways which are not equivalent for fractional-order and lead to different bifurcation diagrams. We observed that the smooth maps such as logistic, Gauss, and H{\'e}non maps do not show chaos while discontinuous maps such as Lozi, Bernoulli, and circle maps show chaos. The tent map is continuous but not differentiable and it shows chaos as well. In $2d$, we find that the complex fractional-order maps that show chaos also show multistability. Thus, it can be inferred that the smooth maps of complex fractional-order tend to show more regular behavior than the discontinuous or non-differentiable maps.