An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

Abstract

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with “self-excited attractors.” This paper makes a contribution to the topic of fractional-order discrete-time systems with “hidden attractors” by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system.

A Fractional-Order Sinusoidal Discrete Map

“…It can be observed from the maximum Lyapunov exponent spectrums in Figure 8c that the chaotic region becomes larger and chaos intensity strengthens. Through a comparison of the map with those in [31][32][33][34][35][36], we can see that it has a typical symmetry, which may cause the symmetry-breaking bifurcation as a parameter varies. Meanwhile, the coexisting attractors also exist in the fractional chaotic map.…”

“…To this end, the study of a new fractional-order discrete map is necessary and important for the development of fractional calculus and dynamics. Recent reports discuss subjects including: the novel convenient condition for the stability of fractionalorder difference systems in the incommensurate-order case [30]; the complex dynamics in the discrete memristor-based system with fractional-order difference [31]; the chaos and projective synchronization of a fractional-order difference map with no equilibria [32]; and the rich dynamical characteristics of a new fractional-order, 2D discrete chaotic map [33]. These works mainly focus on the stability, dynamics, bifurcation, and synchronization of fractional-order discrete maps.…”

A New Fractional‐Order Map with Infinite Number of Equilibria and Its Encryption Application

“…is property makes the chaotic maps very useful in the fields of secure communication and encryption. Since such phenomenon has not received enough attention with fractional discrete-time systems [21], this paper aims to make a contribution by introducing a new fractional map that is characterized by both particular dynamic behaviors and specific properties related to the system equilibria. Namely, the proposed map possesses infinite number of equilibria in a bounded domain, being this a new feature for fractional map, not published in literature to date.…”

Extreme Multistability of a Fractional-Order Discrete-Time Neural Network

“…All of the above fractionalorder models are continuous-time systems outlined by differential equations with fractional order. For discrete-time fractional-order HNN, there is little literature on the investigation of the chaotic behaviour [19][20][21][22][23][24][25]. For example, in [19], discrete fractional complex-valued HNN was synchronized without exploring the chaotic dynamics; whereas in [20], the authors studied the chaotic behavior of a 3D discrete HNN with commensurate fractional order.…”