Hidden Hyperchaotic Attractors in a New 5D System Based on Chaotic System with Two Stable Node-Foci

2019 ◽  
Vol 29 (07) ◽  
pp. 1950092 ◽  
Author(s):  
Qigui Yang ◽  
Lingbing Yang ◽  
Bin Ou
Keyword(s):  
Chaotic System ◽  
Stable Equilibrium ◽  
Complex Dynamics ◽  
Dynamical Evolution ◽  
Stable Node ◽  
Hyperchaotic System ◽  
Hidden Attractors ◽  
Circuit Realization ◽  
Switching Algorithm ◽  
Linear Controllers

This paper reports some hidden hyperchaotic attractors and complex dynamics in a new five-dimensional (5D) system with only two nonlinear terms. The system is generated by adding two linear controllers to an unusual 3D autonomous quadratic chaotic system with two stable node-foci. In particular, the hyperchaotic system without equilibrium or with only one stable equilibrium can generate two kinds of hidden hyperchaotic attractors with three positive Lyapunov exponents. Numerical methods not only verify the existence of such attractors and hyperchaotic attractors, but also show the dynamical evolution of this system. The 5D system has self-excited attractors and two types of hidden attractors with the change of its parameter. The parameter switching algorithm is further utilized to numerically approximate the attractor. Specifically, the hidden hyperchaotic attractor can be approximated by switching between two self-excited chaotic attractors. Finally, the circuit realization results are consistent with the numerical results.

Bursting, Dynamics, and Circuit Implementation of a New Fractional-Order Chaotic System With Coexisting Hidden Attractors

2019 ◽  
Vol 14 (7) ◽  
Author(s):  
Meng Jiao Wang ◽  
Xiao Han Liao ◽  
Yong Deng ◽  
Zhi Jun Li ◽  
Yi Ceng Zeng ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Neuroendocrine Cells ◽  
Order System ◽  
Hidden Attractors ◽  
Main Mode ◽  
Circuit Realization

Systems with hidden attractors have been the hot research topic of recent years because of their striking features. Fractional-order systems with hidden attractors are newly introduced and barely investigated. In this paper, a new 4D fractional-order chaotic system with hidden attractors is proposed. The abundant and complex hidden dynamical behaviors are studied by nonlinear theory, numerical simulation, and circuit realization. As the main mode of electrical behavior in many neuroendocrine cells, bursting oscillations (BOs) exist in this system. This complicated phenomenon is seldom found in the chaotic systems, especially in the fractional-order chaotic systems without equilibrium points. With the view of practical application, the spectral entropy (SE) algorithm is chosen to estimate the complexity of this fractional-order system for selecting more appropriate parameters. Interestingly, there is a state variable correlated with offset boosting that can adjust the amplitude of the variable conveniently. In addition, the circuit of this fractional-order chaotic system is designed and verified by analog as well as hardware circuit. All the results are very consistent with those of numerical simulation.

scholarly journals Chaotic Dynamics by Some Quadratic Jerk Systems

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 227
Author(s):  
Mei Liu ◽  
Bo Sang ◽  
Ning Wang ◽  
Irfan Ahmad
Keyword(s):  
Hopf Bifurcation ◽  
Cross Sections ◽  
Chaotic Dynamics ◽  
Stable Equilibrium ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Chaotic Attractors ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
Subcritical Hopf Bifurcation

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

Four-Wing Hidden Attractors with One Stable Equilibrium Point

2020 ◽  
Vol 30 (06) ◽  
pp. 2050086 ◽  
Author(s):  
Quanli Deng ◽  
Chunhua Wang ◽  
Linmao Yang
Keyword(s):  
Equilibrium Point ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Stable Node ◽  
Stable Equilibrium Point ◽  
The Novel ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
Attraction Basin

Although multiwing hidden attractor chaotic systems have attracted a lot of interest, the currently reported multiwing hidden attractor chaotic systems are either with no equilibrium point or with an infinite number of equilibrium points. The multiwing hidden attractor chaotic systems with stable equilibrium points have not been reported. This paper reports a four-wing hidden attractor chaotic system, which has only one stable node-focus equilibrium point. The novel system can also generate a hidden attractor with one-wing and hidden attractors with quasi-periodic and periodic coexistence. In addition, a self-excited attractor with one-wing can be generated by adjusting the parameters of the novel system. The hidden attractors of the novel system are verified by the cross-section of attraction basins. And the hidden behavior is investigated by choosing different initial states. Moreover, the coexisting transient four-wing phenomenon of the self-excited one-wing attractor system is studied by the time domain waveforms and attraction basin. The dynamical characteristics of the novel system are studied by Lyapunov exponents spectrum, bifurcation diagram and Poincaré map. Furthermore, the novel hidden attractor system with four-wing and one-wing are implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations.

Generating a Chaotic System with One Stable Equilibrium

2017 ◽  
Vol 27 (04) ◽  
pp. 1750053 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Sajad Jafari ◽  
Tomasz Kapitaniak ◽  
Christos Volos ◽  
Sifeu Takougang Kingni
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Hidden Attractors ◽  
Chaotic Flow ◽  
Line Equilibrium

Although chaotic systems with hidden attractors have been discovered recently, there are few investigations about relationships among them. This brief work introduces a novel chaotic system with only one stable equilibrium that is constructed by adding a tiny perturbation into a known chaotic flow having hidden attractors with a line equilibrium.

Constructing a New 3D Chaotic System with Any Number of Equilibria

2019 ◽  
Vol 29 (05) ◽  
pp. 1950060 ◽  
Author(s):  
Qigui Yang ◽  
Xinmei Qiao
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Complex Dynamics ◽  
Type Systems ◽  
Dynamical Behaviors ◽  
Stable Equilibria ◽  
Relationship Of ◽  
Number Of Equilibria

In the chaotic polynomial Lorenz-type systems (including Lorenz, Chen, Lü and Yang systems) and Rössler system, their equilibria are unstable and the number of the hyperbolic equilibria are no more than three. This paper shows how to construct a simple analytic (nonpolynomial) chaotic system that can have any preassigned number of equilibria. A special 3D chaotic system with no equilibrium is first presented and discussed. Using a methodology of adding a constant controller to the third equation of such a chaotic system, it is shown that a chaotic system with any preassigned number of equilibria can be generated. Two complete mathematical characterizations for the number and stability of their equilibria are further rigorously derived and studied. This system is very interesting in the sense that some complex dynamics are found, revealing many amazing properties: (i) a hidden chaotic attractor exists with no equilibria or only one stable equilibrium; (ii) the chaotic attractor coexists with unstable equilibria, including two/five unstable equilibria; (iii) the chaotic attractor coexists with stable equilibria and unstable equilibria, including one stable and two unstable equilibria/94 stable and 93 unstable equilibria; (iv) the chaotic attractor coexists with infinitely many nonhyperbolic isolated equilibria. These results reveal an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of some unusual chaotic systems.

A Novel Cubic–Equilibrium Chaotic System with Coexisting Hidden Attractors: Analysis, and Circuit Implementation

2017 ◽  
Vol 27 (04) ◽  
pp. 1850066 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Tomasz Kapitaniak
Keyword(s):  
Chaotic System ◽  
Autonomous System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
Hidden Attractors ◽  
Dynamical Properties

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium.

A CHAOTIC SYSTEM WITH ONE SADDLE AND TWO STABLE NODE-FOCI

2008 ◽  
Vol 18 (05) ◽  
pp. 1393-1414 ◽  
Author(s):  
QIGUI YANG ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Autonomous System ◽  
Lorenz System ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Type Systems ◽  
Stable Node ◽  
Compound Structure ◽  
Chen System

This paper reports the finding of a chaotic system with one saddle and two stable node-foci in a simple three-dimensional (3D) autonomous system. The system connects the original Lorenz system and the original Chen system and represents a transition from one to the other. The algebraical form of the chaotic attractor is very similar to the Lorenz-type systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the chaotic system has a chaotic attractor, one saddle and two stable node-foci. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions, and the compound structure of the system are analyzed and demonstrated with careful numerical simulations.

One-to-four-wing hyperchaotic fractional-order system and its circuit realization

Circuit World ◽  
2020 ◽  
Vol 46 (2) ◽  
pp. 107-115
Author(s):  
Xiang Li ◽  
Zhijun Li ◽  
Zihao Wen
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Complexity Analysis ◽  
Initial Conditions ◽  
Fractional Order System ◽  
Order System ◽  
Hyperchaotic System ◽  
Circuit Realization ◽  
Content Type

Purpose This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems with variable-wing attractors, although some chaotic systems can generate one-to-four-wing attractors, none of them are hyperchaotic attractors, which is incomplete for the dynamic characteristics of chaotic systems. Design/methodology/approach A novel 4D fractional-order hyperchaotic system is proposed based on the classical three-dimensional Lü system. The complex and abundant dynamic behaviors of the fractional-order system are analyzed by phase diagrams, bifurcation diagrams and the corresponding Lyapunov exponents. In addition, SE and C0 algorithms are used to analyze the complexity of the fractional-order system. Then, the influence of order q on the system is also investigated. Finally, the circuit is implemented using physical components. Findings The most particular interest is that the system can generate one-to-four-wing hyperchaotic attractors with only one parameter variation. Then, the hardware circuit experimental results tally with the numerical simulations, which proves the validity and feasibility of the fractional-order hyperchaotic system. Besides, under different initial conditions, coexisting attractors can be obtained by changing the parameter d or the order q. Then, the complexity analysis of the system shows that the fractional-order chaotic system has higher complexity than the corresponding integer-order chaotic system. Originality/value The circuit structure of the fractional-order hyperchaotic system is simple and easy to implement, and one-to-four-wing hyperchaotic attractors can be observed in the circuit. To the best of the knowledge, this unique phenomenon has not been reported in any literature. It is of great reference value to analysis and circuit realization of fractional-order chaotic systems.

Multiscroll Hyperchaotic System with Hidden Attractors and Its Circuit Implementation

2019 ◽  
Vol 29 (09) ◽  
pp. 1950117 ◽  
Author(s):  
Xin Zhang ◽  
Chunhua Wang
Keyword(s):  
Numerical Simulation ◽  
Adaptive Control ◽  
Theoretical Analysis ◽  
Chaotic System ◽  
Control Method ◽  
Hyperchaotic System ◽  
Circuit Implementation ◽  
Hidden Attractors ◽  
System A ◽  
Simulation Results

Based on the study on Jerk chaotic system, a multiscroll hyperchaotic system with hidden attractors is proposed in this paper, which has infinite number of equilibriums. The chaotic system can generate [Formula: see text] scroll hyperchaotic hidden attractors. The dynamic characteristics of the multiscroll hyperchaotic system with hidden attractors are analyzed by means of dynamic analysis methods such as Lyapunov exponents and bifurcation diagram. In addition, we have studied the synchronization of the system by applying an adaptive control method. The hardware experiment of the proposed multiscroll hyperchaotic system with hidden attractors is carried out using discrete components. The hardware experimental results are consistent with the numerical simulation results of MATLAB and the theoretical analysis results.

A Chaotic System with Two Stable Equilibrium Points: Dynamics, Circuit Realization and Communication Application

2017 ◽  
Vol 27 (08) ◽  
pp. 1750130 ◽  
Author(s):  
Xiong Wang ◽  
Akif Akgul ◽  
Serdar Cicek ◽  
Viet-Thanh Pham ◽  
Duy Vo Hoang
Keyword(s):  
Chaotic System ◽  
Communication System ◽  
Secure Communication ◽  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Circuit Realization ◽  
Shift Keying ◽  
Stable Equilibria ◽  
Chaos Shift Keying

Recent evidence suggests that a system with only stable equilibria can generate chaotic behavior. In this work, we study a chaotic system with two stable equilibrium points. The dynamics of the system is investigated via phase portrait, bifurcation diagram and Lyapunov exponents. The feasibility of the system is introducing its electronic realization. Moreover, the chaotic system is used in Symmetric Chaos Shift Keying (SCSK) and Chaotic ON-OFF Keying (COOK) modulated communication designs for secure communication. It is determined that the SCSK modulated communication system implemented with the chaotic system is more successful than COOK modulation for secure communication.

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