Global chaos synchronization of the parametrically excited Duffing oscillators by linear state error feedback control

Wu, Xiaofeng; Cai, Jianping; Wang, Muhong

doi:10.1016/j.chaos.2006.06.014

Cited by 32 publications

(15 citation statements)

References 23 publications

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“…The first example presents chaos synchronization of the parametrically excited gyroscope [6] and the parametrically excited Duffing oscillator [15]. The second example presents anti-synchronization of the same two systems.…”

Section: Numerical Examplesmentioning

confidence: 99%

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An integral control for synchronization of a class of unknown non-autonomous chaotic systems

2010

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Section: Numerical Examplesmentioning

confidence: 99%

“…Many techniques that use linear or nonlinear feedback control have been proposed to synchronize chaotic systems [5][6][7][8][9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

An integral control for synchronization of a class of unknown non-autonomous chaotic systems

2010

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“…In the same way, we compute the invariant coordinates and the kneading increment of c 2 and by equalities (14), we may write…”

Section: Symmetry Effect On the Kneading Theory For Bimodal Mapsmentioning

confidence: 99%

“…There are applications in neurology, ecology, secure communications, cryptography, chaotic synchronization, and so on. Due to the rich behaviour of these equations, recently there has been also several studies on the synchronization of two coupled Duffing equations [13,14]. The most general forced form of the Duffing equation is …”

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confidence: 99%

Kneading theory analysis of the Duffing equation

Caneco

Grácio

Rocha

2009

Chaos, Solitons & Fractals

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Dedicated to the memory of J. Sousa Ramos a b s t r a c tThe purpose of this paper is to study the symmetry effect on the kneading theory for symmetric unimodal maps and for symmetric bimodal maps. We obtain some properties about the kneading determinant for these maps, that implies some simplifications in the usual formula to compute, explicitly, the topological entropy. As an application, we study the chaotic behaviour of the two-well Duffing equation with forcing.Ó 2009 Elsevier Ltd. All rights reserved. Motivation and introductionThe Duffing equation has been used to model the nonlinear dynamics of special types of mechanical and electrical systems. This differential equation has been named after the studies of Duffing in 1918 [1], has a cubic nonlinearity and describes an oscillator. It is the simplest oscillator displaying catastrophic jumps of amplitude and phase when the frequency of the forcing term is taken as a gradually changing parameter. It has drawn extensive attention due to the richness of its chaotic behaviour with a variety of interesting bifurcations, torus and Arnold's tongues. The main applications have been in electronics, but it can also have applications in mechanics and in biology. For example, the brain is full of oscillators at micro and macro level [15]. There are applications in neurology, ecology, secure communications, cryptography, chaotic synchronization, and so on. Due to the rich behaviour of these equations, recently there has been also several studies on the synchronization of two coupled Duffing equations [13,14]. The most general forced form of the Duffing equation is In [2], Xie et al. used symbolic dynamics to study the behaviour of chaotic attractors and to analyze different periodic windows inside a closed bifurcation region in the parameter plane. Symbolic dynamics is a rigorous tool to understand chaotic motions in dynamical systems.In this work, we will use techniques of kneading theory due essentially to Milnor and Thurston [8] and Sousa Ramos [5][6][7], applying these techniques to the study of the chaotic behaviour. We will use kneading theory to evaluate the topological entropy, which measures the chaoticity of the system. Generally, the graphic of the return map is a very complicated set of points. Therefore, in order to be able to apply these techniques, we show, in Section 2, regions in the parameter plane where 0960-0779/$ -see front matter Ó

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“…For this reason, investigation of the synchronization behaviour of parametrically excited oscillators is relevant for a variety of applications. Till now, only a few works have been devoted to the study of synchronization of parametrically modulated systems [28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Global chaos synchronization of coupled parametrically excited pendula

2009

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In this paper, we study the synchronization behaviour of two linearly coupled parametrically excited chaotic pendula. The stability of the synchronized state is examined using Lyapunov stability theory and linear matrix inequality (LMI); and some sufficient criteria for global asymptotic synchronization are derived from which an estimated critical coupling is determined. Numerical solutions are presented to verify the theoretical analysis. We also examined the transition to stable synchronous state and show that this corresponds to a boundary crisis of the chaotic attractor.

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Global chaos synchronization of the parametrically excited Duffing oscillators by linear state error feedback control

Cited by 32 publications

References 23 publications

An integral control for synchronization of a class of unknown non-autonomous chaotic systems

An integral control for synchronization of a class of unknown non-autonomous chaotic systems

Kneading theory analysis of the Duffing equation

Global chaos synchronization of coupled parametrically excited pendula

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