Abstract
In order to further understand a Lorenz-like system, we study the stability of the equilibrium points and the existence of Hopf bifurcation by center manifold theorem and normal form theory. More precisely, we designed a washout controller such that the equilibriumE0undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibriumE+. Besides, numerical simulation is given to illustrate the theoretical analysis. Finally, two possible electronic circuits are given to realize the uncontrolled and the controlled systems.
Highlights
Over the past decades, as we have seen, researchers have paid a great attention to the control of nonlinear dynamical systems exhibiting Hopf bifurcation phenomena, because the presence of bifurcation is very important in many physical, biological, and chemical nonlinear systems [1,2,3]
An et al [7] based on washout filter designed a state feedback controller for Hopf bifurcation of the nonlinear systems
Sotomayor et al [9] use the projection method described in [10] to calculate the first and second Lyapunov coefficients associated with Hopf bifurcations of the Watt governor system, and it was extended to the calculation of the third and fourth Lyapunov coefficients
Summary
As we have seen, researchers have paid a great attention to the control of nonlinear dynamical systems exhibiting Hopf bifurcation phenomena, because the presence of bifurcation is very important in many physical, biological, and chemical nonlinear systems [1,2,3]. Chen et al [4, 5] created a certain bifurcation at a desired location with preferred properties by appropriate control. They developed a washout-filter-aided dynamic feedback control laws for the creation of Hopf bifurcations. An et al [7] based on washout filter designed a state feedback controller for Hopf bifurcation of the nonlinear systems. Ma et al [8] designed a bifurcation controller using the method of washout filter so as to control the dynamic bifurcation in power system. Dias et al [11] studied the existence of singularly degenerated heteroclinic cycles for a suitable choice of the parameters at the equilibrium E+.
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