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| The Anti-control of the Hopf Bifurcation of the Cable-stayed Beam |
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Abstract In recent year, there has been increasing interest in bifurcation control. Anti-control of bifurcation, as opposed to the bifurcation control, which refers to design a controller to reduce some existing bifurcation dynamics of a given nonlinear system, means to create a needed bifurcation at a needed location with preferred properties by appropriate control. In this paper, we investigate the anti-control of Hopf bifurcations. The washout-filter-aided controller has great application value in the application. In this study, we want to introduce a Hopf bifurcation at an designed stable equilibrium point. Following, a washout-filter-aided controller is employed together to deal with the problem. At last, according to the Poincare map and expansion method of power series, we can prove the advantage of using washout-filter-aided controller to obtain a certain Hopf bifurcation at the needed location. By defining a mapping, we can make the system to increase to (2+1)-dimensional system. In this case, the Poincare section can be chosen suitably. Following, the numbers of the fixed points can be determined by using the second derivative of the function. In addition, the fixed point can be solved by using the Poincare map. There are some conclusions can be obtained. In order to unchange the location of the equilibrium points, we can modify the parameters of the feedback controller. Then, by using washout-filter-aided dynamic feedback controller, a Hopf bifurcation is created with needed location and preferable properties. According to the calculation of the Poincare map, there is one fixed point of the system at most. Moreover, the equilibrium is the same one at which the system is designed to operate with the use of washout-filter-aided dynamic feedback controller. In addition, the period solution near the bifurcation point can be obtained with Poincare map. According to the proof of the Poincare map in theory, a Hopf bifurcation can be created at a needed location with preferred properties by the dynamic feedback control indeed. In the future, a greater action may be induced in more applications by the work on anti-control of bifurcations.
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Received: 10 July 2015
Published: 24 February 2016
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2016, Vol. 37 
