Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems

Abstract

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results.

Chaotic Behavior Analysis of a New Incommensurate Fractional‐Order Hopfield Neural Network System

“…For more clarifications about this property, the reader may refer to [26]. However, in order to show the bistability within system (4), we plot the bifurcation diagrams in Figure 15(a) when [q 1 , q 2 , q 3 ] � [0.8, 0.90, 0.91], a � 1, and b � − 9.…”

Multistability route in a PWL multi-scroll system through fractional-order derivatives

“…In a PWL system, linear and nonlinear components interact, leading to the special properties of generating multi-scroll attractors [6][7][8][9][10]. Various approaches to designing multi-scroll systems include adding breakpoints to Chua's function, using hysteresis functions, applying periodic functions, by using Piece-Wise Linear functions (PWL), with results ranging from 1-D to 10-D, and generating self-excited and hidden attractors.…”

Multistability Route in a PWL Multi-scroll System through Fractional-Order Derivatives

“…In a PWL system, linear and nonlinear components interact, leading to the special properties of generating multi-scroll attractors [6][7][8][9][10]. Various approaches to designing multi-scroll systems include adding breakpoints to Chua's function, using hysteresis functions, applying periodic functions, by using Piece-Wise Linear functions (PWL), with results ranging from 1-D to 10-D, and generating self-excited and hidden attractors.…”