Chaos control in a nonlinear pendulum using a semi-continuous method

Pereira-Pinto, Francisco Heitor I.; Ferreira, Armando Morado; Savi, Marcelo A.

doi:10.1016/j.chaos.2004.02.047

Cited by 34 publications

(17 citation statements)

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“…This pendulum has both torsional stiffness and damping and was previously analyzed by Franca and Savi [2001] and Pinto and Savi [2003]. Pereira-Pinto et al [2004] considered the chaos control of the cited pendulum using the close-return (CR) method [Auerbach et al, 1987] to determine the UPO embedded in the attractor, and a variation of the OGY technique called semi-continuous control (SCC) method, proposed firstly by Hübinger et al [1994] and extended by Korte et al [1995]. All signals are generated numerically by the integration of the mathematical model equations, which uses experimentally identified parameters.…”

Section: Introductionmentioning

confidence: 99%

State Space Reconstruction Using Extended State Observers to Control Chaos in a Nonlinear Pendulum

Pereira-Pinto¹,

Ferreira²,

Savi³

2005

Int. J. Bifurcation Chaos

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Chaos control may be understood as the use of tiny perturbations for the stabilization of unstable periodic orbits embedded in a chaotic attractor. Since chaos may occur in many natural processes, the idea that chaotic behavior may be controlled by small perturbations of some physical parameter allows this kind of behavior to be desirable in different applications. In general, it is not necessary to have a mathematical model to achieve the control goal since all control parameters may be resolved from time series analysis. Therefore, state space reconstruction is an important task related to chaos control. This contribution analyzes chaos control performed using a semi-continuous method based on OGY approach and proposes the use of extended state observers in order to perform state space reconstruction. The use of extended state observers allows a direct application of the control method. Comparing with the delay coordinates method, extended state observers avoids the calculation of parametric changes related to delayed Poincaré sections that influence the system dynamics. The proposed procedure is applied in the control of chaos in a nonlinear pendulum, showing that it may be used to control chaos in mechanical systems.

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Section: Introductionmentioning

confidence: 99%

State Space Reconstruction Using Extended State Observers to Control Chaos in a Nonlinear Pendulum

Pereira-Pinto¹,

Ferreira²,

Savi³

2005

Int. J. Bifurcation Chaos

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“…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%

“…On the other hand, a different approach explores the delayed feedback control proposed by Pyragas [4] that states the stabilization of chaotic systems by a feedback perturbation proportional to the difference between the present and a delayed state of the system. The original limitations of the OGY method are overcome in different references and, among others, one can cite: control of high periodic and high unstable UPO [5][6][7][8][9], control using time delay coordinates [10][11][12][13], control using different control parameters [14,15].…”

Section: Introductionmentioning

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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“…[1][2][3][4] Experimental investigations of chaotic pendulums tend to focus on either quantifiable characterizations of chaotic behavior or on controlling chaos through feedback techniques. [5][6][7][8][9][10][11] The high quality data made possible by improvements in instrumentation and driving controls have facilitated the understanding of the pendulum's chaotic behavior. Models fit to the experimental data at a particular driving condition can successfully replicate the Poincar e structures, Lyapunov exponents and fractal dimensions present at those conditions.…”

Section: Introductionmentioning

confidence: 99%

Predicting the behavior of a chaotic pendulum with a variable interaction potential

Tran

Brost

Johnston

et al. 2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The behavior of a chaotic physical pendulum is significantly modified through the addition of a magnetic interaction. The extended behavior is studied through identifying distinct characteristics in the Poincar e sections and turning point maps of the systems. The validity of our model is shown through simulated bifurcations generated from coefficients estimated at a number of different frequencies. These simulated bifurcations also demonstrate that coefficients estimated at one frequency can be used to predict the behavior of the system at a different drive frequency. A quantitative measure of the correlation dimension shows that the simulated Poincar e diagrams are in good agreement with experiment and theory. Possible sources of bias in modeled systems are identified. Although it has been the subject of many papers, books, and commercial experimental systems, the ubiquitous physical pendulum still has untapped physics. By taking advantage of magnetic dipoles, we developed a chaotic pendulum that tests our ability to model a system over a broad range of driving conditions with different interaction potentials. Notably, we find the presence of two very different attractors in our system depending on how the system is driven. This differentiates its behavior from other pendulums where the attractor is either wrapped or unwrapped depending on the drive mechanism. We introduce turning point maps that track regions of net torque, as well as Poincar e sections to help visualize the attractors due to the different interaction potentials. Comparisons between bifurcation diagrams and correlation dimension measurements indicate that large scale system behavior encompassing different attractors can be predicted using a single region of chaotic response. In particular, we show that determining correlation dimension values from a single phase slice could introduce bias and that using fitting parameters from a single frequency in simulations may underestimate the uncertainty in the model.

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Chaos control in a nonlinear pendulum using a semi-continuous method

Cited by 34 publications

References 50 publications

State Space Reconstruction Using Extended State Observers to Control Chaos in a Nonlinear Pendulum

State Space Reconstruction Using Extended State Observers to Control Chaos in a Nonlinear Pendulum

A multiparameter chaos control method applied to maps

Predicting the behavior of a chaotic pendulum with a variable interaction potential

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