Hyperchaos generated from the Lorenz chaotic system and its control

Jia, Qiang

doi:10.1016/j.physleta.2007.02.024

Cited by 221 publications

(101 citation statements)

References 15 publications

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“…As it can be seen the behavior of the system (16) is hyperchaotic. The Lorenz hyperchaotic system, which was introduced in [39], can be described as follows:…”

Section: Chaotic Systemsmentioning

confidence: 99%

Modified Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method

Tirandaz

Ahmadnia

Tavakoli

2017

IJECE

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The synchronization problem of chaotic systems using active modified projective nonlinear control method is rarely addressed. Thus the concentration of this study is to derive a modified projective controller to synchronize the two chaotic systems. Since, the parameter of the master and follower systems are considered known, so active methods are employed instead of adaptive methods. The validity of the proposed controller is studied by means of the Lyapunov stability theorem. Furthermore, some numerical simulations are shown to verify the validity of the theoretical discussions. The results demonstrate the effectiveness of the proposed method in both speed and accuracy points of views.Copyright c 2017 Institute of Advanced Engineering and Science.All rights reserved. Corresponding Author:Hamed Tirandaz Hakim Sabzevari University Electrical and Computer Engineering Faculty, Hakim Sabzevari University, Sabzevar, Iran. Phone +98051-4401288 Email: tirandaz@hsu.ac.ir INTRODUCTIONMaster-slave synchronization of chaotic systems is strikely nonlinear, since the aperiodic and nonregular behavior of chaotic systems and their sensitivity to the initial conditions. Chaotic behavior may appear in many physical systems. So, chaos synchronization subject has received a great deal of attention in the last to decades, due to its potential applications in physics, chemistry, electrical engineering, secure communication and so on [1]. Up to now, many types of controling methods are revealed and investigated for control and synchronization of chaotic systems. Active method [2,3,4,5,6], adaptive method [7,8,9] [22,23,24] are some of the introduced methods by the researchers. Among these methods, synchronization with some types of projective methods are extensively investigated in the last decades, since the faster synchronization due to its synchronization scaling factors, which master and slave chaotic systems would be synchronized up to a proportional rate. Projective lag method [25], modified projective synchronization (MPS) [26,27,28], function projective synchronization (FPS) [29], modified function projective synchronization [30,28], generalized function projective synchronization [31,32] and modified projective lag synchronization [33,34] are some generalized schemes of projective method, which utilize some type of scaling factors.When the parameters of a chaotic system are known beforehand, active related methods are preferably chosen than adaptive methods. Active synchronization problem of two chaotic systems with known parameters are vastly investigated by the researchers. For example, in [5,3,35], the active controlling method is studied for synchronization of two typical chaotic systems. And also, in [2], an active method for controling the behavior of a unified chaotic system is presented. Chaos synchronization of complex Chen and Lu chaotic systems are addressed in citeMahmoud, with designing an active control method. Furthermore, in [36] active

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“…As it can be seen the behavior of the system (16) is hyperchaotic. The Lorenz hyperchaotic system, which was introduced in [39], can be described as follows:…”

Section: Chaotic Systemsmentioning

confidence: 99%

Modified Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method

Tirandaz

Ahmadnia

Tavakoli

2017

IJECE

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“…Since the discovery of a first 4-D hyperchaotic system by Rössler in 1979 [52], many 4-D hyperchaotic systems have been found in the literature such as hyperchaotic Lorenz system [53], hyperchaotic Lü system [54], hyperchaotic Chen system [55], hyperchaotic Wang system [56], hyperchaotic Newton-Leipnik system [57], hyperchaotic Jia system [58], hyperchaotic Vaidyanathan systems [59,60,61,62,63,64,65,66,67,68], hyperchaotic Pham system [69], hyperchaotic Sampath system [70], etc.…”

Section: Introductionmentioning

confidence: 99%

Analysis, Adaptive Control and Synchronization of a Novel 4-D Hyperchaotic Hyperjerk System via Backstepping Control Method

Vaidyanathan¹

2016

Archives of Control Sciences

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Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system via backstepping control method SUNDARAPANDIAN VAIDYANATHAN A hyperjerk system is a dynamical system, which is modelled by an nth order ordinary differential equation with n 4 describing the time evolution of a single scalar variable. Equivalently, using a chain of integrators, a hyperjerk system can be modelled as a system of n first order ordinary differential equations with n 4. In this research work, a 4-D novel hyperchaotic hyperjerk system with two nonlinearities has been proposed, and its qualitative properties have been detailed. The novel hyperjerk system has a unique equilibrium at the origin, which is a saddle-focus and unstable. The Lyapunov exponents of the novel hyperjerk system are obtained as L 1 = 0.14219, L 2 = 0.04605, L 3 = 0 and L 4 = −1.39267. The Kaplan-Yorke dimension of the novel hyperjerk system is obtained as D KY = 3.1348. Next, an adaptive controller is designed via backstepping control method to stabilize the novel hyperjerk chaotic system with three unknown parameters. Moreover, an adaptive controller is designed via backstepping control method to achieve global synchronization of the identical novel hyperjerk systems with three unknown parameters. MATLAB simulations are shown to illustrate all the main results derived in this research work on a novel hyperjerk system.

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“…Since the discovery of a first 4-D hyperchaotic system by Rössler in 1979 [44], many 4-D hyperchaotic systems have been found in the literature such as hyperchaotic Lorenz system [45], hyperchaotic Lü system [46], hyperchaotic Chen system [47], hyperchaotic Wang system [48], hyperchaotic Newton-Leipnik system [49], hyperchaotic Jia system [50], hyperchaotic Vaidyanathan systems [51,52], etc. In mechanics, if the scalar x(t) represents the position of a moving object at time t, then the first derivative, x(t), represents the velocity, the second derivative,ẍ(t), represents the acceleration and the third derivative, ... x (t), represents the jerk or jolt [53].…”

Section: Introductionmentioning

confidence: 99%

Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation

Vaidyanathan¹,

Volos²,

Pham³

et al. 2015

Archives of Control Sciences

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A hyperjerk system is a dynamical system, which is modelled by an nth order ordinary differential equation with n 4 describing the time evolution of a single scalar variable. Equivalently, using a chain of integrators, a hyperjerk system can be modelled as a system of n first order ordinary differential equations with n 4. In this research work, a 4-D novel hyperchaotic hyperjerk system has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel hyperjerk system are obtained asThe Kaplan-Yorke dimension of the novel hyperjerk system is obtained as D KY = 3.1573. Next, an adaptive backstepping controller is designed to stabilize the novel hyperjerk chaotic system with three unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve global hyperchaos synchronization of the identical novel hyperjerk systems with three unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using SPICE is presented in detail to confirm the feasibility of the theoretical hyperjerk model.

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Hyperchaos generated from the Lorenz chaotic system and its control

Cited by 221 publications

References 15 publications

Modified Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method

Modified Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method

Analysis, Adaptive Control and Synchronization of a Novel 4-D Hyperchaotic Hyperjerk System via Backstepping Control Method

Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation

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