Control of Rössler system to periodic motions using impulsive control methods

Yang, Tao; Yang, Chun-Mei; Yang, Lin-Bao

doi:10.1016/s0375-9601(97)00408-8

Cited by 58 publications

(27 citation statements)

References 4 publications

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“…The theory of impulsive ordinary differential equations and its applications to the fields of science and engineering have been very active research topics [28][29][30][43][44][45], since the theory provides a natural framework for mathematical modeling of many physical phenomena. Furthermore, impulsive control, which is based on the theory of impulsive differential equations, has gained renewed interests recently for its promising applications towards controlling systems exhibiting chaotic behaviour.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, it was realized that, such a control method allows the stabilization of a chaotic system using only small control impulses, even though the chaotic behaviour may follow unpredictable patterns (in general, chaotic signals are broadband, noise like and difficult to predict). Examples include the impulsive control of autonomous systems of ODEÕs such as Lorenz system and ChuaÕs oscillator [29,43,45] and non-autonomous chaotic systems of ODEÕs, such as the DuffingÕs oscillator [42], where the stabilization of the chaotic system is achieved in a small region of the phase space using the notion of practical stability (instead of controlling the non-autonomous chaotic system to an equilibrium position).…”

Section: Introductionmentioning

confidence: 99%

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Impulsive control and synchronization of spatiotemporal chaos

Khadra

Liu

Shen

2005

Chaos, Solitons & Fractals

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The impulsive control of spatiotemporal chaos of a particular type of non-linear partial differential equations has been investigated. A criterion for the solutions of these partial differential equations to be equi-attractive in the large is determined and an estimate for the basin of attraction is given in terms of the impulse durations and the magnitude of the impulses. Extending these results to impulsively synchronize spatiotemporal chaos of the same type of partial differential equations is explored. A proof for the existence of a certain kind of impulses for synchronization such that the error dynamics is equi-attractive in the large, is established. A comparison of the developed theoretical model with other existent numerical models available in the literature has been studied. Several simulation results are given to confirm the theoretical results. Moreover, an investigation of the Lyapunov exponents of the error dynamics between impulsively synchronized spatiotemporal chaotic systems, is done to further confirm the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Impulsive control and synchronization of spatiotemporal chaos

Khadra

Liu

Shen

2005

Chaos, Solitons & Fractals

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“…I MPULSIVE differential equations have gained considerable attention in science and engineering [14], [15], [17], [24], [31], [32] in recent years, since they provide a natural framework for mathematical modeling of many physical phenomena. Examples include population-growth models [14] and maneuvers of spacecraft [17].…”

Section: Introductionmentioning

confidence: 99%

Application of impulsive synchronization to communication security

Khadra

Liu

Shen

2003

IEEE Trans. Circuits Syst. I

120

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Abstract-In this paper, criteria on uniform equi-boundedness and equi-Lagrange stability for impulsive systems are derived. These criteria are used to synchronize two nonidentical chaotic systems by impulsively controlling a nonautonomous second order system, which leads to the development of an induced-message scheme for communication system security. With the scheme, message signals are not transmitted across public channels, but induced at the receiver end. The scheme overcomes the transmission time-frame congestion in impulsive cryptosystems discussed in the literature and improves system security. Simulation results are given to demonstrate the performance of the proposed scheme.

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“…Impulsive methods for dynamical systems' control and synchronization are some known approaches in the field of chaos [Yang et al 1997, Osipov et al 1998a, Osipov et al 1998b, Khadra et al 2005. It was used successfully for controlling Rössler system [Yang et al 1997] and the Duffing oscillator [Osipov et al 1998a] to periodic motions.…”

mentioning

confidence: 99%

“…It was used successfully for controlling Rössler system [Yang et al 1997] and the Duffing oscillator [Osipov et al 1998a] to periodic motions. More recent paper about impulsive control was more successful in establishing more conservative and sufficient conditions for the stabilization and synchronization of Lorenz systems via impulsive control.…”

mentioning

confidence: 99%