Controlling chaos in high dimensional systems

Auerbach, Ditza; Grebogi, Celso; Ott, Edward; Yorke, James A.

doi:10.1103/physrevlett.69.3479

Cited by 184 publications

(55 citation statements)

References 14 publications

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“…Indeed, in the OGY method, both A and B in (84) can be extracted purely from observations of the trajectory on the chaotic attractor (Shinbrot et al, 1993). Finally, the OGY method can be extended to arbitrarily high dimensional systems, without assuming knowledge of the underlying dynamics (Auerbach et al, 1992).…”

Section: Linearization Of the Poincaré Map: Ogy Methodsmentioning

confidence: 99%

Control principles of complex systems

Liu¹,

Barabási²

2016

Rev. Mod. Phys.

543

349

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A reflection of our ultimate understanding of a complex system is our ability to control its behavior. Typically, control has multiple prerequisites: it requires an accurate map of the network that governs the interactions between the system's components, a quantitative description of the dynamical laws that govern the temporal behavior of each component, and an ability to influence the state and temporal behavior of a selected subset of the components. With deep roots in nonlinear dynamics and control theory, notions of control and controllability have taken a new life recently in the study of complex networks, inspiring several fundamental questions: What are the control principles of complex systems? How do networks organize themselves to balance control with functionality? To address these here we review recent advances on the controllability and the control of complex networks, exploring the intricate interplay between a system's structure, captured by its network topology, and the dynamical laws that govern the interactions between the components. We match the pertinent mathematical results with empirical findings and applications. We show that uncovering the control principles of complex systems can help us explore and ultimately understand the fundamental laws that govern their behavior.

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Section: Linearization Of the Poincaré Map: Ogy Methodsmentioning

confidence: 99%

Control principles of complex systems

Liu¹,

Barabási²

2016

Rev. Mod. Phys.

543

349

View full text Add to dashboard Cite

show abstract

“…In particular, this will allow the long-term behavior to be made time varying in a desired fashion; essentially, trajectories will approach a user-specified nonautonomous attractor. Such a method would extend more classical control strategies which, for example, attempt to have a chaotic system approach a fixed point [51][52][53][54][55], a periodic orbit [56][57][58][59][60][61], or even a chaotic regime [62] by taking advantage of the relevant entity's stable manifold.…”

Section: Introductionmentioning

confidence: 99%

Accurate control of hyperbolic trajectories in any dimension

Balasuriya

Padberg‐Gehle

2014

Phys. Rev. E

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The unsteady (nonautonomous) analog of a hyperbolic fixed point is a hyperbolic trajectory, whose importance is underscored by its attached stable and unstable manifolds, which have relevance in fluid flow barriers, chaotic basin boundaries, and the long-term behavior of the system. We develop a method for obtaining the unsteady control velocity which forces a hyperbolic trajectory to follow a user-prescribed variation with time. Our method is applicable in any dimension, and accuracy to any order is achievable. We demonstrate and validate our method by (1) controlling the fixed point at the origin of the Lorenz system, for example, obtaining a user-defined nonautonomous attractor, and (2) the saddle points in a droplet flow, using localized control which generates global transport.

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“…These bursts are more frequent for large noise [2]. Various modifications of the O GY method considered in [5][6][7][8][9][10][11][12] are also dis crete in time. The details of these modifications can be found in a review paper [13].…”

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confidence: 99%

Delayed Feedback Control of Chaos in an Electronic Double-Scroll Oscillator

Kittel

Parisi

Pyragas

et al. 1994

Zeitschrift Für Naturforschung A

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We present experimental results on stabilizing unstable periodic orbits of an autonomous chaos oscillator based on a simple electronic circuit. Control is achieved by applying the difference between the actual and a delayed output signal of the oscillator. The quality of chaos control can be measured via the strength of perturbation. The dependence on the delay time shows a characteristic resonancetype behavior.Since the pioneer work done by Hübler and coworkers [1], controlling chaos has become a fasci nating topic in the field of nonlinear dynamics. In particular, the method introduced by Ott, Grebogi, and Yorke (OGY) [2] develops to an important tool to overcome undesired chaotic behavior in various fields of applications. The key observation is that the typical chaotic attractor embeds an infinite number of un stable periodic orbits of distinct periods Tt. They can be stabilized by an only small, carefully chosen, feed back perturbation applied to some system parameter available for external adjustment. The OGY approach is as follows; At first, one determines some of the unstable low-periodic orbits of the strange attractor by finding the fixed points in the Poincare map of the system. Then the linear transformations close to these points and their dependence on the parameter varia tion have to be reconstructed from experiment. Fi nally, a small time-dependent perturbation is applied to the system, in order to stabilize these already exist ing periodic orbits.The O GY method is a very general one. It does not require any a priori analytical knowledge of the sys tem dynamics and has been successfully applied to various physical experiments including a magnetic ribbon [3], a spin-wave system [4], a chemical system [5], an electric diode [6], laser systems [7], and cardiac systems [8].

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Controlling chaos in high dimensional systems

Cited by 184 publications

References 14 publications

Control principles of complex systems

Control principles of complex systems

Accurate control of hyperbolic trajectories in any dimension

Delayed Feedback Control of Chaos in an Electronic Double-Scroll Oscillator

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