On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization

Liao, Xiaoxin; Fu, Yuli; Xie, Shengli

doi:10.1360/04yf0087

Cited by 49 publications

(64 citation statements)

References 15 publications

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“…In our papers [9,10] , a new ellipsoid estimation for the globally attractive set of the system is given, which improves and generalizes the results in refs. [7,8].…”

Section: Introductionsupporting

confidence: 68%

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Globally exponentially attractive sets of the family of Lorenz systems

Liao

Xie

et al. 2008

Sci. China Ser. F-Inf. Sci.

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In this paper, the concept of globally exponentially attractive set is proposed and used to consider the ultimate bounds of the family of Lorenz systems with varying parameters. Explicit estimations of the ultimate bounds are derived. The results presented in this paper contain all the existing results as special cases. In particular, the critical cases, b → 1 + and a → 0 + , for which the previous methods failed, have been solved using a unified formula.

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“…In our papers [9,10] , a new ellipsoid estimation for the globally attractive set of the system is given, which improves and generalizes the results in refs. [7,8].…”

Section: Introductionsupporting

confidence: 68%

“…Here, the generalization means that the globally attractive set [7,[9][10][11] is extended to the globally exponentially attractive set.…”

Section: ) Takingmentioning

confidence: 99%

Globally exponentially attractive sets of the family of Lorenz systems

Liao

Xie

et al. 2008

Sci. China Ser. F-Inf. Sci.

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“…The Russian scholar, Leonov [12,13] was the first one to investigate the estimation of the global attractive set of the Lorenz system. Liao et al [14,15] discussed the stability and global attractive set of the Lorenz system. Li et al [16] applied the method of Lagrange multiplier to obtain an estimation of the global attractive set for the Lorenz system.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of stochastic Lorenz family of chaotic systems with jump

Huang¹,

Cao

Jiang³

2013

J Math Chem

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This paper discusses the Lorenz family of chaotic systems under the influence of poisson noise. We obtain the sufficient and necessary conditions for stochastic stability under some assumptions for stochastic Lorenz family of chaotic systems with jump. We then investigate the estimation of the global attractive set and stochastic bifurcation behavior of the family of stochastic Lorenz system. Numerical experiments illustrate the results.

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“…This will greatly simplify the dynamic analysis of the system. By virtue of Lyapunov function method, the ellipsoidal estimates of the ultimate bound and positively invariant set for the Lorenz system were proposed in [18,19], and a butterfly-shaped localization set for the Lorenz attractor was given in [20]. Based on a similar method, bound of the hyperchaotic Lorenz-Stenflo system was presented in [21].…”

mentioning

confidence: 99%

A novel bounded 4D chaotic system

Zhang

2011

Nonlinear Dyn

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This paper presents a novel bounded fourdimensional (4D) chaotic system which can display hyperchaos, chaos, quasiperiodic and periodic behaviors, and may have a unique equilibrium, three equilibria and five equilibria for the different system parameters. Numerical simulation shows that the chaotic attractors of the new system exhibit very strange shapes which are distinctly different from those of the existing chaotic attractors. In addition, we investigate the ultimate bound and positively invariant set for the new system based on the Lyapunov function method, and obtain a hyperelliptic estimate of it for the system with certain parameters.

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On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization

Cited by 49 publications

References 15 publications

Globally exponentially attractive sets of the family of Lorenz systems

Globally exponentially attractive sets of the family of Lorenz systems

Dynamics of stochastic Lorenz family of chaotic systems with jump

A novel bounded 4D chaotic system

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