A new chaotic attractor coined

doi:10.1142/S0218127402004620

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A new chaotic attractor coined

Journal Article•DOI•

A new chaotic attractor coined

01 Mar 2002-International Journal of Bifurcation and Chaos (World Scientific Publishing Company)-Vol. 12, Iss: 3, pp 659-661

TL;DR: 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 attractsor and represents the transition from one to the other.

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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.

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Journal Article•DOI•

Brief paper: On pinning synchronization of complex dynamical networks

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Wenwu Yu¹, Guanrong Chen¹, Jinhu Lu²•Institutions (2)

City University of Hong Kong¹, Chinese Academy of Sciences²

01 Feb 2009-Automatica

TL;DR: Surprisingly, it is found that a network under a typical framework can realize synchronization subject to any linear feedback pinning scheme by using adaptive tuning of the coupling strength.

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962 citations

Cites background from "A new chaotic attractor coined"

...Later in 2002, Lü and Chen discovered another new chaotic system (Lü & Chen, 2002), which bridges the gap between the Lorenz system and the Chen system....
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Journal Article•DOI•

Bridge the gap between the Lorenz system and the Chen system

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Jinhu Lu, Guanrong Chen, Daizhan Cheng, Sergej Celikovsky¹•Institutions (1)

Academy of Sciences of the Czech Republic¹

01 Dec 2002-International Journal of Bifurcation and Chaos

TL;DR: A unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum and is chaotic over the entire spectrum of the key system parameter.

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Abstract: This paper introduces a unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum. The new system represents the continued transition from the Lorenz to the Chen system and is chaotic over the entire spectrum of the key system parameter. Dynamical behaviors of the unified system are investigated in somewhat detail.

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806 citations

Cites background or methods from "A new chaotic attractor coined"

...Keywords: Lorenz system; Chen system; critical system; a unified chaotic system....
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...…1996], since with these values of α in the above equation one has a12a21 > 0; when α = 0.8, it belongs to the class of chaotic systems introduced in [Lü & Chen, 2002; Lü et al., 2002a], since in this case a12a21 = 0; when 0.8 < α ≤ 1, it belongs to the generalized Chen system…...
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...5(g) is the typical chaotic attractor reported in [Lü & Chen, 2002]; Fig....
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...As mentioned, the unified system (1) contains the canonical Lorenz system [C̆elikovský & Chen, 2002], the Chen system [Chen & Ueta, 1999; Ueta & Chen, 2000], and the chaotic system recently introduced in [Lü & Chen, 2002] as special cases....
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...…12 (2002) 2917–2926 c© World Scientific Publishing Company BRIDGE THE GAP BETWEEN THE LORENZ SYSTEM AND THE CHEN SYSTEM JINHU LÜ∗ Institute of Systems Science Academy of Mathematics and System Sciences Chinese Academy of Sciences, Beijing 100080, P.R. China lvjinhu@amss.ac.cn GUANRONG…...
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Journal Article•DOI•

Adaptive synchronization of an uncertain complex dynamical network

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Jin Zhou¹, Jun-an Lu¹, Jinhu Lu²•Institutions (2)

Wuhan University¹, Chinese Academy of Sciences²

10 Apr 2006-IEEE Transactions on Automatic Control

TL;DR: This note further investigates the locally and globally adaptive synchronization of an uncertain complex dynamical network and designed adaptive controllers for network synchronization are designed.

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Abstract: This note further investigates the locally and globally adaptive synchronization of an uncertain complex dynamical network. Several network synchronization criteria are deduced. Especially, our hypotheses and designed adaptive controllers for network synchronization are rather simple in form. It is very useful for future practical engineering design. Moreover, numerical simulations are also given to show the effectiveness of our synchronization approaches.

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641 citations

Cites background from "A new chaotic attractor coined"

...Up to now, we can only estimate the boundary of very few chaotic systems [9-13], such as the Lorenz, Chen, Lü systems [14]....
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Journal Article•DOI•

Pinning adaptive synchronization of a general complex dynamical network

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Jin Zhou¹, Jun-an Lu¹, Jinhu Lu²•Institutions (2)

Wuhan University¹, Princeton University²

01 Apr 2008-Automatica

TL;DR: A simply approximate formula for estimating the detailed number of pinning nodes and the magnitude of the coupling strength for a given general complex dynamical network is provided.

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541 citations

Journal Article•DOI•

Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications

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Jinhu Lu¹, Guanrong Chen²•Institutions (2)

Chinese Academy of Sciences¹, City University of Hong Kong²

01 Apr 2006-International Journal of Bifurcation and Chaos

TL;DR: Over the last two decades, theoretical design and circuit implementation of various chaos generators have been a focal subject of increasing interest due to their promising applications in various ...

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Abstract: Over the last two decades, theoretical design and circuit implementation of various chaos generators have been a focal subject of increasing interest due to their promising applications in various ...

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456 citations

Cites background from "A new chaotic attractor coined"

...…subsection presents a 3D quadratic autonomous critical chaotic system, which can display (i) two 1-scroll chaotic attractors simultaneously, with only three equilibria, and (ii) two 2-scroll chaotic attractors simultaneously, with five equilibria [Lü & Chen, 2002; Lü et al., 2002d, 2004a]....
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Journal Article•DOI•

Yet another chaotic attractor

[...]

Guanrong Chen¹, Tetsushi Ueta²•Institutions (2)

University of Houston¹, University of Tokushima²

01 Jul 1999-International Journal of Bifurcation and Chaos

TL;DR: In this paper, the authors reported the finding of a chaotic at tractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler at tractors.

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Abstract: This Letter reports the finding of a new chaotic at tractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler at tractors.

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2,443 citations

"A new chaotic attractor coined" refers background in this paper

...…another chaotic attractor, also in a simple three-dimensional autonomous system, which nevertheless is not topologically equivalent to the Lorenz’s [Chen & Ueta, 1999; Ueta & Chen, 2000]: ẋ = a(y − x) ẏ = (c− a)x− xz + cy ż = xy − bz , (2) which is chaotic when a = 35, b = 3, c = 28 (see Fig....
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...In 1999, Chen found another chaotic attractor, also in a simple three-dimensional autonomous system, which nevertheless is not topologically equivalent to the Lorenz’s [Chen & Ueta, 1999; Ueta & Chen, 2000]: ẋ = a(y − x) ẏ = (c− a)x− xz + cy ż = xy − bz , (2) which is chaotic when a = 35, b = 3, c = 28 (see Fig....
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Book•

The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors

[...]

Colin Sparrow

01 Dec 1982

TL;DR: The first homoclinic explosion in the Lorenz equation was described in this article, where the authors proposed an approach to the problem of finding the position of the first homocalinic explosion by using the Maxima-in-z method.

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Abstract: 1. Introduction and Simple Properties.- 1.1. Introduction.- 1.2. Chaotic Ordinary Differential Equations.- 1.3. Our Approach to the Lorenz Equations.- 1.4. Simple Properties of the Lorenz Equations.- 2. Homoclinic Explosions: The First Homoclinic Explosion.- 2.1. Existence of a Homoclinic Orbit.- 2.2. The Bifurcation Associated with a Homoclinic Orbit.- 2.3. Summary and Some General Definitions.- 3. Preturbulence, Strange Attractors and Geometric Models.- 3.1. Periodic Orbits for the Hopf Bifurcation.- 3.2. Preturbulence and Return Maps.- 3.3. Strange Attractor and Homoclinic Explosions.- 3.4. Geometric Models of the Lorenz Equations.- 3.5. Summary.- 4. Period Doubling and Stable Orbits.- 4.1. Three Bifurcations Involving Periodic Orbits.- 4.2. 99.524 100.795. The x2y Period Doubling Window.- 4.3. 145 166. The x2y2 Period Doubling Window.- 4.4. Intermittent Chaos.- 4.5. 214.364 ?. The Final xy Period Doubling Window.- 4.6. Noisy Periodicity.- 4.7. Summary.- 5. From Strange Attractor to Period Doubling.- 5.1. Hooked Return Maps.- 5.2. Numerical Experiments.- 5.3. Development of Return Maps as r Increases: Homoclinic Explosions and Period Doubling.- 5.4. Numerical Experiments on Periodic Orbits.- 5.5. Period Doubling and One-Dimensional Maps.- 5.6. Global Approach and Some Conjectures.- 5.7. Summary.- 6. Symbolic Description of Orbits: The Stable Manifolds of C1 and C2.- 6.1. The Maxima-in-z Method.- 6.2. Symbolic Descriptions from the Stable Manifolds of C1 and C2.- 6.3. Summary.- 7. Large r.- 7.1. The Averaged Equations.- 7.2. Analysis and Interpretation of the Averaged Equations.- 7.3. Anomalous Periodic Orbits for Small b and Large r.- 7.4. Summary.- 8. Small b.- 8.1. Twisting Around the z-Axis.- 8.2. Homoclinic Explosions with Extra Twists.- 8.3. Periodic Orbits Without Extra Twisting Around the z-Axis.- 8.4. Heteroclinic Orbits Between C1 and C2.- 8.5. Heteroclinic Bifurcations.- 8.6. General Behaviour When b = 0.25.- 8.7. Summary.- 9. Other Approaches, Other Systems, Summary and Afterword.- 9.1. Summary of Predicted Bifurcations for Varying Parameters ?, b and r.- 9.2. Other Approaches.- 9.3. Extensions of the Lorenz System.- 9.4. Afterword - A Personal View.- Appendix A. Definitions.- Appendix B. Derivation of the Lorenz Equations from the Motion of a Laboratory Water Wheel.- Appendix C. Boundedness of the Lorenz Equations.- Appendix D. Homoclinic Explosions.- Appendix E. Numerical Methods for Studying Return Maps and for Locating Periodic Orbits.- Appendix F. Computational Difficulties Involved in Calculating Trajectories which Pass Close to the Origin.- Appendix G. Geometric Models of the Lorenz Equations.- Appendix H. One-Dimensional Maps from Successive Local Maxima in z.- Appendix I. Numerically Computed Values of k(r) for ? = 10 and b = 8/3.- Appendix J. Sequences of Homoclinic Explosions.- Appendix K. Large r the Formulae.

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1,463 citations

"A new chaotic attractor coined" refers background or methods in this paper

...Introduction In 1963, Lorenz found the first chaotic attractor in a three-dimensional autonomous system [Sparrow, 1982]:  ẋ = a(y − x) ẏ = cx− xz − y ż = xy − bz , (1)...
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...In 1963, Lorenz found the first chaotic attractor in a three-dimensional autonomous system [Sparrow, 1982]: ẋ = a(y − x) ẏ = cx− xz − y ż = xy − bz , (1) which is chaotic when a = 10, b = 8/3, c = 28 (see Fig....
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...In fact, these two intervals correspond to the three periodic solution intervals of the Lorenz system [Sparrow, 1982]....
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Book•

From Chaos To Order Methodologies, Perspectives and Applications

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Guanrong Chen, Xiaoning Dong

09 Jun 1998

TL;DR: This work focuses on chaos and order ordering chaos organization of the monograph and some applications of controlling chaos, and some approaches to controlling chaos.

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Abstract: Part 1 Prologue - chaos and order ordering chaos organization of the monograph. Part 2 Parametric variation approaches: parametric variation for control of chaos an earlier attempt the OGY method and its variants new development. Part 3 Conventional engineering control approaches: engineering perspectives on control of chaos controlling chaos via external forces open-loop control methods feedback mechanism - a control engineer's perspective controlling discrete-time chaotic systems controlling continuous-time chaotic systems a closer look at Lyapunov-type approach adaptive control of chaotic systems optimal control of chaotic systems robust control of chaotic systems a stochastic control technique new development. Part 4 Other approaches to controlling chaos: intelligent control approaches controlling chaos in DAI systems synchronization and control of chaos unification of synchronization and control new development. Part 5 Some applications of controlling chaos: chaotic synchronization for secure communication chaos and biomedical systems chaos and biochemical systems controlling chaos of laser systems controlling chaos in fluid dynamics new development. Part 6 Controlling chaos - further discussions: general perspectives to probe further toward an integrated and unified framework.

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1,227 citations

Journal Article•DOI•

Bifurcation analysis of chen's equation

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Tetsushi Ueta¹, Guanrong Chen²•Institutions (2)

University of Tokushima¹, University of Houston²

01 Aug 2000-International Journal of Bifurcation and Chaos

TL;DR: Some basic dynamical properties and various bifurcations of Chen's equation are investigated, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.

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Abstract: Anticontrol of chaos by making a nonchaotic system chaotic has led to the discovery of some new chaotic systems, particularly the continuous-time three-dimensional autonomous Chen's equation with only two quadratic terms. This paper further investigates some basic dynamical properties and various bifurcations of Chen's equation, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.

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381 citations

"A new chaotic attractor coined" refers background in this paper

...…another chaotic attractor, also in a simple three-dimensional autonomous system, which nevertheless is not topologically equivalent to the Lorenz’s [Chen & Ueta, 1999; Ueta & Chen, 2000]: ẋ = a(y − x) ẏ = (c− a)x− xz + cy ż = xy − bz , (2) which is chaotic when a = 35, b = 3, c = 28 (see Fig....
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Journal Article•DOI•

On a generalized lorenz canonical form of chaotic systems

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Sergej Čelikovský¹, Guanrong Chen²•Institutions (2)

Academy of Sciences of the Czech Republic¹, City University of Hong Kong²

01 Aug 2002-International Journal of Bifurcation and Chaos

TL;DR: The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design.

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Abstract: This paper shows that a large class of systems, introduced in [Celikovský & Vaněcek, 1994; Vaněcek & Celikovský, 1996] as the so-called generalized Lorenz system, are state-equivalent to a special canonical form that covers a broader class of chaotic systems. This canonical form, called generalized Lorenz canonical form hereafter, generalizes the one introduced and analyzed in [Celikovský & Vaněcek, 1994; Vaněcek & Celikovský, 1996], and also covers the so-called Chen system, recently introduced in [Chen & Ueta, 1999; Ueta & Chen, 2000]. Thus, this new generalized Lorenz canonical form contains as special cases the original Lorenz system, the generalized Lorenz system, and the Chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible. The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design. Some...

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314 citations