Controlling Hyperchaos

Yang, Ling; Liu, Zengrong; Mao, Jie

doi:10.1103/physrevlett.84.67

Cited by 65 publications

(28 citation statements)

References 11 publications

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“…On the other hand, we could take the point of view that we wish to control a "black-box" process. Time-series embedding analysis [Takens, 1980] has been shown by several groups [So & Ott, 1995;Ding et al, 1996;Yang et al, 2000] to be a valid way to sufficiently model to achieve OGYtype control of unstable periodic orbits in a chaotic system which is only known through measurement of a scalar time-series. There is hope that these techniques might be extended to allow for our modeling requirements, but such will surely be data intensive.…”

Section: Resultsmentioning

confidence: 99%

Combinatorial Control of Global Dynamics in a Chaotic Differential Equation

Bollt

2001

Int. J. Bifurcation Chaos

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Controlling chaos has been an extremely active area of research in applied dynamical systems, following the introduction of the Ott, Grebogi, Yorke (OGY) technique in 1990 , but most of this research based on parametric feedback control uses local techniques. Associated with a dynamical system which pushes forward initial conditions in time, transfer operators, including the Frobenius-Perron operator, are associated dynamical systems which push forward ensemble distributions of initial conditions. In [Bollt, 2000a[Bollt, , 2000bBollt & Kostelich, 1998], we have shown that such global representations of a discrete dynamical system are useful in controlling certain aspects of a chaotic dynamical system which could only be accessible through such a global representation. Such aspects include invariant measure targeting, as well as orbit targeting. In this paper, we develop techniques to show that our previously discrete time techniques are accessible also to a differential equation. We focus on the Duffing oscillator as an example. We also show that a recent extension of our techniques by Góra and Boyarsky [1999] can be further simplified and represented in a convenient and compact way by using a tensor product.

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Section: Resultsmentioning

confidence: 99%

Combinatorial Control of Global Dynamics in a Chaotic Differential Equation

Bollt

2001

Int. J. Bifurcation Chaos

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“…Recently, Yang et al 18 and Xu et al 19 proposed a new control method, which is called the straight-line stabilization method. This method is adopted to control the chaos in this paper…”

Section: Chaos Controlmentioning

confidence: 99%

Complexity of a Duopoly Game in the Electricity Market with Delayed Bounded Rationality

2012

Discrete Dynamics in Nature and Society

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According to a triopoly game model in the electricity market with bounded rational players, a new Cournot duopoly game model with delayed bounded rationality is established. The model is closer to the reality of the electricity market and worth spreading in oligopoly. By using the theory of bifurcations of dynamical systems, local stable region of Nash equilibrium point is obtained. Its complex dynamics is demonstrated by means of the largest Lyapunov exponent, bifurcation diagrams, phase portraits, and fractal dimensions. Since the output adjustment speed parameters are varied, the stability of Nash equilibrium gives rise to complex dynamics such as cycles of higher order and chaos. Furthermore, by using the straight-line stabilization method, the chaos can be eliminated. This paper has an important theoretical and practical significance to the electricity market under the background of developing new energy.

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“…Recently, Xu et al [31] and Yang et al [32] introduced a straight line stabilization control method. We utilize this method to control the chaos of system (9).…”

Section: Chaos Controlmentioning

confidence: 99%

Complex Dynamics and Chaos Control on a Kind of Bertrand Duopoly Game Model considering R&D Activities

Zhan

Mao

2017

Discrete Dynamics in Nature and Society

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We study a dynamic research and development two-stage input competition game model in the Bertrand duopoly oligopoly market with spillover effects on cost reduction. We investigate the stability of the Nash equilibrium point and local stable conditions and stability region of the Nash equilibrium point by the bifurcation theory. The complex dynamic behaviors of the system are shown by numerical simulations. It is demonstrated that chaos occurs for a range of managerial policies, and the associated unpredictability is solely due to the dynamics of the interaction. We show that the straight line stabilization method is the appropriate management measure to control the chaos.

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Controlling Hyperchaos

Cited by 65 publications

References 11 publications

Combinatorial Control of Global Dynamics in a Chaotic Differential Equation

Combinatorial Control of Global Dynamics in a Chaotic Differential Equation

Complexity of a Duopoly Game in the Electricity Market with Delayed Bounded Rationality

Complex Dynamics and Chaos Control on a Kind of Bertrand Duopoly Game Model considering R&D Activities

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