Study of low-dimensional nonlinear fractional difference equations of complex order

Abstract

We study the fractional maps of complex order, [Formula: see text], for [Formula: see text] and [Formula: see text] in one and two dimensions. In two dimensions, we study Hénon, Duffing, and Lozi maps, and in [Formula: see text], we study logistic, tent, Gauss, circle, and Bernoulli maps. The generalization in [Formula: see text] 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, Duffing, and Hénon maps, do not show chaos, while discontinuous maps, such as Bernoulli and circle maps,show chaos. The tent and Lozi map are continuous but not differentiable, and they show chaos as well. In [Formula: see text], 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.

Survey of Recent Applications of the Chaotic Lozi Map

“…Moreover, they noted that extremely strange bifurcations are observed in model L 1 , showing bifurcation diagrams for ν = 0.4 + 0.3i and ν = 0.4 + 0.5i (see Figure 7a,b of [42]) which clearly indicate the possibility of very large periods and a rich bifurcation structure that is not generally seen in integer-order systems.…”

“…Other authors, such as Joshi et al [42], have considered the definition of a fractional map (13) for the case where the order of derivation ν is a complex number with 0 < Re(ν) < 1. They used L 1 to denote the model defined by (13) and L 2 to denote the one defined by…”

“…Joshi et al [42] studied the control of a complex fractional-ordered Lozi map (see Section 2.5.2). They used the Pyragas method [78], which is a feedback method that can be attempted without any detailed knowledge of the dynamical system under consideration.…”

“…Joshi et al [42] studied the synchronization of complex fractional-ordered Lozi maps (see Section 2.5.3). They defined a system of two symmetrically coupled maps…”