Study Of Hopf Bifurcation For Several Kinds Of Nonlinear Dynamical System

Hopf bifurcation is a kind of important dynamic bifurcation. As a leading reseach field, Hopf bifurcation control becomes very challenging. In this thesis, attention is focused on the Hopf bifurcation of dynamical systems and related analysis and control problems, which can enrich and perfect the theory results. Discussing dynamical behaviors of equilibrium, analyzing the existence of Hopf bifurcation and the bifurcation characteristics, designing bifurcation controllers, offering approaches of control and thereby achieving some desirable dynamical behaviors are the main tasks. It is mainly focused on controlling of amplitude of limit cycles, giving relationship between amplitude and controls, realizing Hopf bifurcation delay control and stabilit y control. Several types of controller are designed to achieve the desired control objective and each has its own characteristic. Some typical dynamical systems are chosen as examples to discuss.Firstly, the recent advances about the nonlinear control theory, the bifurcation control, the Hopf bifurcation control and the chaos control are summarized. Some basic concepts about nonlinear dynamics and several bifurcation control methods are introduced. A few stability theory and dynamical systems theory are proposed for the further research in this paper.A new method is presented for studying the controlling of amplitude of limit cycles in generalized Van der Pol strongly nonlinear oscillator system. The amplitude of limit cycles are obtained by the modified multip le scale method and the analytical formula regarding the amplitude o f the controlled limit cycles and the feedback gains is also obtained. The control of amplitude of limit cycles is realized by choosing suitable feedback gains and the efficiencies of different controllers are discussed and compared. The validity of the analyt ical conclusions is verified by numerical simulations and it still has high accuracy for larger parameter ?.Analysis and controlling of bifurcation for a class of chaotic Van der Pol-Duffing system with multip le unknown parame ters are conducted. The stability of the equilibria of the system is studied by using Routh- Hurwitz criterion, and the critical value of Hopf bifurcation is investigated. Based on the center manifold theory and normal form reduction, the stabilit y index of bifurcation solut ion is given. The linear control term of washout filter-aided dynamic feedback controller has been used to control Hopf bifurcation but not changing the stability. The nonlinear control term of washout filter-aided controller has been used to control the amplitude of the limit cycle but not changing the bifurcating crit ical value. The amplitude approximations in terms of control gains are derived from the center manifold theory and norma l form reduction, which can also effectively predict the amplitude of limit cycles. Nume ric a l simu la tion re sults a re prese nte d to illustrate the c orrec tness a nd effic ie nc y of the a na lytica l re sults.The strange chaotic attractor of a new chaotic system is shown with the help of the maximum Lyapunov exponent. The existence of Hopf bifurcation is obtained by the characteristic equations and corresponding dynamic behavior of the system are studied by means of the first Lyapunov coefficient. As a result, it is shown that this system has two Hopf bifurcation points, at which the Hopf bifurcation is nondegenerate and supercritical.C onsequently, the stable periodic orbits are bifurcated. N umerical simulat ions are carried out to illustrate the main theoretical results.The nonlinear dynamic property of the equilibriums of Lü system is studied. The probability and stabilit y of Hopf bifurcation of the equilibriums are investigated. Linear and nonlinear controllers are designed respective ly to delay bifurcation and realize stability control. N umerical simulation results are presented to illustrate analyt ical results found.A hybrid control strategy is applied to control the Hopf bifurcation in a new modified hyperchaotic Lü system for the first time. This method keeps the equilibrium construction of the origina l system and does not increase the dimens ion of the system. By choosing an appropriate control parameter, the control strategy can effective ly delay the Hopf bifurcation, so the stable range of the system is extended. By using the normal form theory, the stability of bifurcating solutions is analyzed. N umerical simulations show the effectiveness of the method. Bifurcation control of high dimensional nonlinear systems is much more difficult than low dimensional systems. This control strategy is simple and convenient, so it is meaningful for the study of bifurcation control of high dimensional nonlinear systems.