Exploring chaotic motion through periodic orbits

Auerbach, Ditza; Cvitanović, Predrag; Eckmann, Jean‐Pierre; Gunaratne, Gemunu H.; Procaccia, Itamar

doi:10.1103/physrevlett.58.2387

Cited by 501 publications

(219 citation statements)

References 10 publications

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“…Many alternative approaches to the OGY method have been proposed for the stabilization of the unstable periodic orbits (UPO) [47] of a chaotic dynamics. In general the strategies for the control of chaos can be classi"ed into two main classes, namely: closed loop or feedback methods and open loop or non feedback methods.…”

Section: The Basic Ideamentioning

confidence: 99%

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The control of chaos: theory and applications

et al. 2000

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Section: The Basic Ideamentioning

confidence: 99%

“…(51) can usually be extracted directly from the measurement [9]. The location of the periodic orbits are obtained by looking at recurrences in the embedded space [46,47]. The matrix A in Eq.…”

Section: Use Of Delay Coordinatesmentioning

confidence: 99%

The control of chaos: theory and applications

et al. 2000

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“…The close return method [23] is employed with this aim. The UPO identification is done by considering a number of map iterations and a prescribed tolerance as follows:…”

Section: Controlling Mapsmentioning

confidence: 99%

“…After the UPO identification, the local dynamics expressed by the Jacobian matrix and the sensitivity matrix of the transition maps in a neighborhood of the control points are determined using the least-square fit method [6,8,13,23,24]. The sensitivity matrices are evaluated allowing trajectories to come close to a control point and then one perturbs the parameters by proportional value of the maximum permissible.…”

Section: Controlling Mapsmentioning

confidence: 99%

“…Here, the close-return (CR) method [23] is employed to indentify the UPOs embedded in the attractor. Afterwards, the local dynamics expressed by the Jacobian matrix and the sensitivity matrix of the transition maps in a neighborhood of the control points are determined using the least-square fit method [6,8,13,23,24]. Moreover, the singular value decomposition (SVD) technique is employed for determining the stable and unstable directions near the control point.…”

Section: Introductionmentioning

confidence: 99%

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A multiparameter chaos control method applied to maps

Paula¹,

Savi²

2008

Braz. J. Phys.

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Chaos is a kind of nonlinear system response that has a dense set of unstable periodic orbits (UPOs) embedded in a chaotic attractor. The idea of the chaos control is to explore the UPO stabilization obtaining dynamical systems that may quickly react to some new situation, changing conditions and their response. The OGY (Ott-Grebogi-Yorke) method achieves system stabilization by using small perturbations promoted in the neighborhood of the desired orbit when the trajectory crosses a specific surface, such as a Poincaré section. This contribution proposes a multiparameter (MP) method based on OGY approach in order to control chaotic behavior using different control parameters. As an application of the proposed multiparameter general formulation it is presented an uncoupled approach where the control parameters do not influence the system dynamics when they are not active. This method is applied to control chaos in maps using two control parameters. The two-dimensional Hénon and Ikeda maps are of concern. Results show that the proposed procedure can be a good alternative for chaos control since it provides a more effective UPO stabilization than the classical singleparameter OGY approach.

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Chaos in protein dynamics

et al. 1997

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MD simulations, currently the most detailed description of the dynamic evolution of proteins, are based on the repeated solution of a set of differential equations implementing Newton's second law. Many such systems are known to exhibit chaotic behavior, i.e., very small changes in initial conditions are amplified exponentially and lead to vastly different, inherently unpredictable behavior. We have investigated the response of a protein fragment in an explicit solvent environment to very small perturbations of the atomic positions (10(-3)-10(-9) A). Independent of the starting conformation (native-like, compact, extended), perturbed dynamics trajectories deviated rapidly, leading to conformations that differ by approximately 1 A RMSD within 1-2 ps. Furthermore, introducing the perturbation more than 1-2 ps before a significant conformational transition leads to a loss of the transition in the perturbed trajectories. We present evidence that the observed chaotic behavior reflects physical properties of the system rather than numerical instabilities of the calculation and discuss the implications for models of protein folding and the use of MD as a tool to analyze protein folding pathways.

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Exploring chaotic motion through periodic orbits

Cited by 501 publications

References 10 publications

The control of chaos: theory and applications

The control of chaos: theory and applications

A multiparameter chaos control method applied to maps

Chaos in protein dynamics

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