New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization

Abstract

In this work, we introduce a chaotic system with infinitely many equilibrium points laying on two closed curves passing the same point. The proposed system belongs to a class of systems with hidden attractors. The dynamical properties of the new system were investigated by means of phase portraits, equilibrium points, Poincaré section, bifurcation diagram, Kaplan–Yorke dimension, and Maximal Lyapunov exponents. The anti-synchronization of systems was obtained using the active control. This study broadens the current knowledge of systems with infinite equilibria.

Synchronization of a New Chaotic System Using Adaptive Control: Design and Experimental Implementation

“…Te literature shows that several types of complex chaotic systems have been proposed with the aim of solving the synchronization control problem. Te control strategies, usually developed, are active control [36][37][38][39][40][41][42], nonlinear control [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57], linear feedback control [58][59][60][61][62][63][64][65], sliding modes [66][67][68][69][70][71][72], and adaptive control [73][74][75][76][77][78][79][80][81][82][83][84][85]…”

“…In [41] Varan and Akful synchronized a hyperchaotic system and by using a Lyapunov function, achieved global asymptotic stability; numerical analysis was used to check the efectiveness of the proposed active control design. Lastly, Zhu and Du in [42] solved the antisynchronization of systems by using the active control, and the feasibility of control was verifed via numerical simulations.…”

“…Based on Pecora and Carroll's contribution, the research related to the design of controllers for synchronization of chaotic systems has been intensively studied during the last four decades . Te proposed control strategies reported in those works can be classifed as active control [36][37][38][39][40][41][42], nonlinear control [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57], linear feedback control [58][59][60][61][62][63][64][65], sliding modes [66][67][68][69][70][71][72], and adaptive control [73][74][75][76][77][78][79][80][81][82][83][...…”

A Simple Chaotic Flow with Hyperbolic Sinusoidal Function and Its Application to Voice Encryption

“…Those variations may occur even for infinitesimal changes in the parameter. Bifurcation diagram is used for the stability analysis of a dynamical system [41,42]. Moreover, the Lyapunov exponents spectrum makes it possible to qualitatively quantify a local property with respect to the attractor's stability.…”

A New DC Offset Boostable Chaotic System with Multistability, Coexisting Attractors and Its Adaptive Synchronization

“…The self-excited attractors [23][24][25] can be detected using the unstable equilibrium points while the hidden attractors can be observed in the no equilibrium system [26][27][28]. Many systems have been designed with no equilibrium [29,30], stable equilibrium [31][32][33], line and curve of equilibrium [34][35][36], non-hyperbolic equilibrium [37][38][39] and infinitely many equilibria [40,41]. The chaotic system with amplitude control and offset boosting control are reported in many papers [42][43][44][45].…”