A New Chaotic Attractor Coined

Lü, Jinhu; Chen, Guanrong

doi:10.1142/s0218127402004620

Cited by 1,658 publications

(603 citation statements)

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“…In 2002, Lü et al 39 introduced a new chaotic system which can unify chaotic systems, such as, the Lorenz, Chen, Lü systems. The fractional-order differential equation of the unified system can be given by…”

Section: B Eigenvalues Distribution Analysismentioning

confidence: 99%

Pinning control of fractional-order weighted complex networks

Tang

Wang

Fang

2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

129

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In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The well-studied integer-order complex networks are the special cases of the fractional-order ones. The network model considered can represent both directed and undirected weighted networks. First, based on the eigenvalue analysis and fractional-order stability theory, some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated. A surprising finding is that the fractional-order complex networks can stabilize itself by reducing the fractional-order q without pinning any node. Second, numerical algorithms for fractional-order complex networks are introduced in detail. Finally, numerical simulations in scale-free complex networks are provided to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter ␤, the larger overall coupling strength c, the more capacity that the pinning scheme may possess to enhance the control performance of fractional-order complex networks. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3068350͔Recently, fractional-order differential systems have been widely investigated due to their potential applications in viscoelasticity, dielectric polarization, quantum evolution of complex systems, and many other fields. On the other hand, research of complex networks has triggered tremendous interest during the past decade. Most studies to date have concerned integer-order complex networks. In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The fractional-order complex networks generalize wellstudied integer-order complex networks. Some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated by utilizing the eigenvalue analysis and fractional-order stability theory. A surprising finding that the fractional-order networks can stabilize itself by reducing the fractional-order q without pinning any node is presented. The numerical algorithms for fractional-order networks are also presented. In the end, computer simulations in scale-free networks are given to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter ␤, the larger coupling strength c, the more capacity that the pinning strategy can possess to accelerate the control rate of fractional-order networks.

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Section: B Eigenvalues Distribution Analysismentioning

confidence: 99%

Pinning control of fractional-order weighted complex networks

Tang

Wang

Fang

2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

129

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“…Here, as an example, we state our results for the polynomial differential system in R 3 ,ẋ = a 1 x + a 2 y,ẏ = a 3 x + a 4 y + a 5 xz,ż = a 6 xy + a 7 z, see Example 3.5 in Section 3. It includes the Lorenz system [26], the Lü system [27] and the Chen system [6], and its Darboux integrability has been studied in [25,23,28], respectively. The differential system studied in [39] also belongs to this family.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Seeking Darboux Polynomials

Ferragut

Gasull

2014

Acta Appl Math

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We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in R n . We apply these techniques to some well-known systems when n = 2, 3, 4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol one, has no Darboux polynomials, giving in particular a new simple proof that the van der Pol limit cycle is not algebraic.

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“…Chaos synchronization between Zhang chaotic system [14] and the Lü chaotic system [37] is addressed in this subsection. The Zhang chaotic system is given by a three simple integer-based and nonlinear differential equations that depends on the three positive real parameters as followṡ…”

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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A New Chaotic Attractor Coined

Abstract: This letter reports the finding of a new chaotic attractor in a simple three-dimensional autonomous system, which connects the Lorenz attractor and Chen's attractor and represents the transition from one to the other.

Cited by 1,658 publications

References 5 publications

Pinning control of fractional-order weighted complex networks

Pinning control of fractional-order weighted complex networks

Seeking Darboux Polynomials

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

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