Extreme Multistability in Simple Area-Preserving Map

Abstract

Initial condition-relied extreme multistability has been recently found in many continuous dynamical systems. However, such a specific phenomenon has not yet been discovered in a discrete iterative map. To investigate this phenomenon, this paper proposes a two-dimensional conservative map only with one sine nonlinearity. The proposed simple discrete map is area-preserving in the phase space and displays the coexistence of infinite chaotic and quasi-periodic orbits caused by infinite fixed points. Multiple numerical results indicate that the area-preserving chaotic and quasi-periodic orbits have the initial condition-relied quasi-periodic route to chaos and initial condition-boosting bifurcation dynamics, which allow the simple area-preserving map to emerge the complex phenomenon of extreme multistability. Furthermore, a microcontroller-based hardware platform is developed to implement the initial conditionboosting chaotic signals.

An amplitude-controllable 3-D hyperchaotic map with homogenous multistability

“…Amplitude control and polarity adjustment of the chaotic sequence obtain much flexibility for chaos-based a) Corresponding author. E-mail address: goontry@126.com; chunbiaolee@nuist.edu.cn applications and therefore introduce much more value in information engineering [13][14][15][16][17]. Although the amplitude and offset control of continuous chaotic systems have been systematically explored and reported [18][19], the geometric control of discrete chaotic maps is still in the beginning stages.…”

“…Although the amplitude and offset control of continuous chaotic systems have been systematically explored and reported [18][19], the geometric control of discrete chaotic maps is still in the beginning stages. Trigonometric function nonlinearity brings great convenience for chaos production [20], but it also intensifies the difficulty of amplitude control and poses a great risk for multistability [21][22][23][24][25]. It is of great theoretical significance and engineering values to design chaotic maps based on trigonometric function nonlinearity and study their controllability.…”

A 2D Hyperchaotic Map: Amplitude Control, Coexisting Symmetrical Attractors and Circuit Implementation

“…In continuous systems, amplitude control with independent control knobs and coexisting symmetric attractors have been widely studied, but in discrete mapping the amplitude control with a single parameter has not received enough attention. For example [29][30][31][32], the multistable phenomenon of the map has been discussed in detail, and the position of the phase trajectory through the multistability to achieve the purpose of amplitude control was given, but this ignored a method of directly adjusting the signal amplitude with a single knob. Besides, in many discrete maps like [33], although there are abundant attractor coexistence phenomena, there are no symmetrical coexisting attractors.…”

A New Circumscribed Self‐Excited Spherical Strange Attractor

“…Since the occurrence of chaotic behavior in the system's dynamic has always been taken into consideration, numerous investigations have been performed on particular characteristics of chaotic or nonlinear systems, and diverse systems have been introduced. In other words, some researchers have introduced multistable [6][7][8][9], megastable [10][11][12], extreme multistable [13][14][15], variable-boostable [16,17], memristor-based [18][19][20], conservative [21][22][23], multicluster [24], or any kinds of symmetrical systems [25][26][27][28]. For instance, the paper proposed by Bao et al has introduced a nonautonomous 2D neural system and has also studied the multistability of the defined model [7].…”