Hopf Bifurcation Analysis And Bifurcation Control Research For A Van Der Pol-Duffing System

As a representative of the nonlinear vibration system, the Van der Pol-Duffing oscillatorhas the extremely rich dynamic performance. The nonlinear part of the Van der Pol-Duffingequation contains the Van der Pol system’s nonlinear damping of maintain self-excitedvibration and the third power nonlinear restoring force item of Duffing system. As acombination of two kinds of typical nonlinear system, the Van der Pol-Duffing oscillator is aresearch model in many fields, such as engineering, physics, electronics and neurology. Thusthe Van der Pol-Duffing system becomes one of the research hotspot in recent years. Basedon nonlinear science theory, the dynamic characteristics of several types of Van derPol-Duffing system has been researched and explored by using the theory and numericalmethod. The paper mainly studied the Hopf bifurcation behavior and bifurcation control ofthis class system, improved on some old theories and methods and also found some new laws.Main content of this paper are as follows:1. Summarized the research status and research purposes of Van der Pol-Duffing system,and expounds the concept and definition of stability, bifurcation, Hopf bifurcation theory,bifurcation control method, multi-scale method and Lyapunov coefficient method. Brieflysummarized and illustrates the conditions of Hopf bifurcation occurs, and Hopf bifurcationcontrol method. According to the above method, we can derive the high codimension Hopfbifurcation and also can design the corresponding Hopf bifurcation controller.2. We studied the nonlinear dynamic characteristics of a modified and uncertain chaoticVan der Pol-Duffing oscillator mainly through the theoretical and numerical methods. First ofall, we obtained the system’s mathematical equation of equivalent circuit by using theKirchhoff’s law. And we can prove the system exist a chaos attractor through the numericalsimulation method. According to the Routh-Hurwitz criterion, the stability of the system wasanalysis and the condition of the stability of the system was obtained. Secondly, thecodimension1and codimension2Hopf bifurcation of the equilibriumE0andE+were studied,respectively. And the condition for the existence of Hopf bifurcation was obtained. Besides,the linear feedback control and the feedback control based on Washout filter were applied tothis system. Finally, the circuit experiment research was carried out and the circuitimplementation of chaotic attractor was obtained through the oscilloscope.3. We mainly investigated the Hopf bifurcation and its control for a Van der Pol-Duffingoscillator which contains the squared and fifth power items. First of all, the Hopf bifurcationof the autonomous Van der Pol-Duffing system was analyzed, and the condition of Hopfbifurcation occurs was obtained. The linear and nonlinear union state feedback controller and the feedback controller based on Washout filter were designed to control the Hopf bifurcationand the amplitude of limit cycle, respectively. In addition, we designed the controller withtime-delay to control the bifurcation response of autonomous system. Finally, we derived thebifurcation response equation of Van der Pol-Duffing time-delay dynamic system bymulti-scale method. The numerical simulation method was used to study the influence ofparameters on the Hopf bifurcation.4. We mainly studied the basic nonlinear dynamic characteristics of a class of couplingand excitation Van der Pol-Duffing system. Through the nonlinear dynamics theory, thestability and Hopf bifurcation behavior of system was discussed. A feedback controller isdesigned to control the limit cycle amplitude of this system. In the end, the integral slidingmode controller was designed to control of chaotic vibrations in an autonomous system.