A multiparameter chaos control method based on OGY approach
Introduction
A chaotic attractor has a dense set of unstable periodic orbits (UPOs) and the system often visits the neighborhood of each one of them. Moreover, chaotic response has sensitive dependence to initial condition, which implies that the system’s evolution may be altered by small perturbations. Chaos control explores all this richness employing tiny perturbations for the stabilization of a UPO embedded in a chaotic attractor. This makes this kind of behavior to be desirable in a variety of applications, since one of these UPO can provide better performance than others in a particular situation.
Chaos control methods may be classified as discrete or continuous techniques. The first chaos control method was proposed by Ott et al. [9], nowadays known as the OGY method as a tribute of their authors (Ott–Grebogi–Yorke). This is a discrete technique that considers small perturbations applied in the neighborhood of the desired orbit when the trajectory crosses a specific surface, such as some Poincaré section [5], [16]. On the other hand, continuous methods are exemplified by the so called delayed feedback control, proposed by Pyragas [12], which states that chaotic systems can be stabilized by a feedback perturbation proportional to the difference between the present and a delayed state of the system. There are many improvements of the OGY method that aim to overcome some of its original limitations, as for example: control of high periodic and high unstable UPO [6], [8], [13], control using time delay coordinates [4], [7], [17] and control using multiparameter approach based on pole placement formalism [2].
The semi-continuous (SC) control method lies between the continuous and the discrete time control because one can introduce as many intermediate Poincaré sections, viewed as control stations, as it is necessary to achieve stabilization of a desirable UPO. Therefore, the SC method provides a more effective control since it allows a great number of actuation during a period.
This contribution considers a semi-continuous multiparameter chaos control method built upon the OGY method [10], [11], [14]. The idea is to use different control parameters in order to perform the UPO stabilization and, because of that, the map that establishes the relation between the system responses in two subsequent control stations depends on all control parameters. The proposed method assumes that only one control parameter actuates in each control station, defining active and passive parameters. On this basis, two different approaches may be adopted for the general formulation: coupled and uncoupled approaches. The coupled approach considers that all control parameters influence the system dynamics although they are not active. The uncoupled approach, on the other hand, is a particular case where control parameters return to the reference value when they become passive and therefore, they are not influencing the system dynamics. As an application of the general formulation a two-parameter control of a nonlinear pendulum is carried out considering coupled and uncoupled approaches. All signals are numerically generated by the integration of the mathematical model equations, using experimentally identified parameters. The close-return (CR) method [1] is employed to determine the UPO 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 [1], [8], [10], [11], [14]. Moreover, the singular value decomposition (SVD) technique is employed for determining the stable and unstable directions near the control point. Results show that the multiparameter approaches can be good alternatives for chaos control since it provides a more effective UPO stabilization when compared to those obtained from the classical single parameter approach.
Section snippets
Multiparameter chaos control method
A chaos control method may be understood as a two stage technique. The first step is known as learning stage where the unstable periodic orbits are identified and some system characteristics are evaluated. After that, there is the control stage where the desirable UPOs are stabilized.
The OGY approach is described considering a discrete system of the form of a map , where is an accessible parameter for control. This is equivalent to a parameter dependent map associated with a
Nonlinear pendulum
As a mechanical application of the general semi-continuous multiparameter chaos control procedure, a nonlinear pendulum actuated by two different parameters is considered. The motivation of the proposed pendulum is an experimental set up discussed in De Paula et al. [3]. A mathematical model is developed to describe the pendulum dynamical behavior while the corresponding parameters are obtained from the experimental apparatus. Numerical simulations of such model are employed in order to obtain
Numerical simulations
Nonlinear pendulum numerical simulations are carried out in order to evaluate the capability of the proposed multiparameter chaos control method to stabilized desirable UPOs. System characteristics are evaluated from time series generated by numerical integration of the mathematical model equations, using experimentally identified parameters.
In the first stage of the control strategy UPOs embedded in the chaotic attractor are identified. The close return method [1], [10] is employed with this
Conclusions
This contribution presents a multiparameter semi-continuous chaos control method built upon the OGY technique. The procedure assumes that only one control parameter actuates in each control station, defining active and passive control parameters. On this basis, two different approaches may be adopted for the general formulation: coupled and uncoupled actuations. The coupled approach is the case where all control parameters influences system dynamics although they are not active, since they have
Acknowledgement
The authors acknowledge the support of Brazilian Research Council (CNPq).
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