Generation of a Four-Wing Chaotic Attractor by Two Weakly-Coupled Lorenz Systems

Grassi, Giuseppe; Severance, Frank L.; Mashev, Emil D.; Bazuin, Bradley J.; Miller, D.A.

doi:10.1142/s0218127408021580

Cited by 19 publications

(8 citation statements)

References 16 publications

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“…Applying the above method (AdamsBashford-Moulton algorithm), system (18) can be discretized as follows:…”

Section: Numerical Algorithm For Simulation Of Fractional-order Systemsmentioning

confidence: 99%

“…In these systems, the basic technique to generate different number of scrolls is increasing the number of equilibrium points and the number of scrolls equals to that of the equilibria. Although it is more difficult to obtain a chaotic attractor with more than double wings from a smooth dynamical system, some four-wing Lorenz-like chaotic systems have been introduced in recent years [12,13,[17][18][19][20][21][22]. All these systems have five equilibrium points and each wing wonders around a nonzero equilibria.…”

Section: Introductionmentioning

confidence: 99%

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Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form

Dadras

Momeni

et al. 2011

Nonlinear Dyn

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In this paper, a new simple 4D smooth autonomous system is proposed, which illustrates two interesting rare phenomena: first, this system can generate a four-wing hyperchaotic and a four-wing chaotic attractor and second, this generation occurs under condition that the system has only one equilibrium point at the origin. The dynamic analysis approach in the paper involves time series, phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps, to investigate some basic dynamical behaviors of the proposed 4D system. The physical existence of the fourwing hyperchaotic attractor is verified by an electronic circuit. Finally, it is shown that the fractional-order S. Dadras · H.R. Momeni ( ) Automation and Instruments Lab, form of the system can also generate a chaotic fourwing attractor.

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“…Applying the above method (AdamsBashford-Moulton algorithm), system (18) can be discretized as follows:…”

Section: Numerical Algorithm For Simulation Of Fractional-order Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form

Dadras

Momeni

et al. 2011

Nonlinear Dyn

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“…Similarly, the generations of autonomous chaotic systems with multi-scroll or multi-wing attractor [13][14][15] were sometimes considered a key issue for many engineering applications. Thus, creating a memristive system with a multi-scroll or multi-wing attractor has a practical significance, which is the motivation of this paper.…”

Section: Introductionmentioning

confidence: 99%

“…Besides, the new system has a line equilibrium, which makes the new memristive system harder to study than the classical dynamical system with only a limit number of equilibria. It can be seen from the enumerated characteristics that the new proposed system has richer dynamical behavior than that of most of the known memristive systems [14][15][16][17][18]. Therefore, it is necessary to explore its dynamical behaviors by theory and experiment.…”

Section: Introductionmentioning

confidence: 99%

A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium

et al. 2015

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A new hyper-chaotic system is presented in this paper by adding a smooth flux-controlled memristor and a cross-product item into a three-dimensional autonomous chaotic system. It is exciting that this new memristive system can show a four-wing hyperchaotic attractor with a line equilibrium. The dynamical behaviors of the proposed system are analyzed by Lyapunov exponents, bifurcation diagram and Poincaré maps. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of hyperchaos in the system is verified theoretically. Finally, an electronic circuit is designed to implement the hyperchaotic memristive system.

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“…Chaos attractor with four-wing has been introduced in [24][25][26][27], however, to the best of our knowledge, there are no reports on four-wing hyperchaotic attractor. In this letter, a simple and yet interesting four-dimensional (4D) continuous-time autonomous hyperchaotic system, with a complicated four-wing attractor, is firstly introduced and analyzed.…”

mentioning

confidence: 99%

A new 4D four-wing hyperchaotic attractor and its circuit implementation

Cao

Huang

et al. 2010

2010 International Conference on Communications, Circuits and Systems (ICCCAS)

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In this paper, a new simple four-dimensional continuous-time autonomous hyperchaotic system is introduced, which displays a complicated four-wing attractor. The existence of the hyperchaos is verified by bifurcation analysis, and in the meantime bifurcation routes from period to quasi-period, then to chaos and finally to hyperchaos is determined. Different configurations of the hyperchaotic attractor are illustrated not only by computer simulation but also by electronic circuit realization. 1. INTR ODUCT I O N H yperchaos was first reported by Rossler in 1979 [1]. Since then, it has been studied with increasing interests due to its plentiful applications in applied mathematics and physics, especially in engineering and technology such as chaos-based encryption [2], secure communications [3], lasers [4], Colpitts oscillators [5], biological networks [6], nonlinear circuits [7], coupled map lattices [8], chaos control [9-1 0] and chaos synchronization [11-1 2]. The generation of different hyperchaotic systems has attracted more and more attention in the last few years [1 3 -23]. However, it is now known that generating a hyperchaotic attractor from an originally chaotic but not hyperchaotic system by means of control is a theoretically attractive and yet technically challenging task [1 7]. Some effective methods have been developed, e.g., by adding a simple state-feedback controller [1 7-18] or sinusoidal parametric perturbations [19][20], to some typical chaotic systems, which include the generalized Lorenz systems, Lorenz system, Chen system, Lu system, and a unified chaotic system to generate hyperchaotic attractor [1 7-23].Chaos attractor with four-wing has been introduced in [24-27], however, to the best of our knowledge, there are no reports on four-wing hyperchaotic attractor. In this letter, a simple and yet interesting four-dimensional (4D) continuous-time autonomous hyperchaotic system, with a complicated four-wing attractor, is firstly introduced and analyzed. "T aishan Scholarship" Construction Engineering . bifurcation analysis. Bifurcation routes: from period to quasi-period, then to chaos, and finally to hyperchaos are determined. The new-proposed system is implemented via an electronic circuit in laboratory, which shows perfect agreement with the numerical simulations. II. T HE NEW HYP ER CHA OT I C SY ST EMConsider the following 4D continuous-time autonomous system:where a,b,c,d,k,m E R+ are constant parameters.For system (1 ), one has (1 ) ax ay ai aw VV =-+-+-+-= a-b-c< °(2) ax ay az awHence, for all b + C > a , the system is dissipative. The equilibrium of system (1 ) can be found by letting the right-hand side of (1 ) be equal to zero. Obviously, the system has only one equilibrium point So = (0,0,0,0) .By linearizing system (1 ) at So = (0,0,0,0) , one obtains the Jacobian

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Generation of a Four-Wing Chaotic Attractor by Two Weakly-Coupled Lorenz Systems

Cited by 19 publications

References 16 publications

Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form

Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form

A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium

A new 4D four-wing hyperchaotic attractor and its circuit implementation

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