Bifurcations, Chaos And Related Controls In Nonlinear Oscillatory Systems

In this thesis,the bifurcations,chaos and related control problems in nonlinear oscillatory systems are studied.Attention is focused on the successive bifurcations of periodic solutions,occurrence of chaos and several control problems of bifurcations and chaos.The thesis consists of nine chapters.Chapter 1 is the introduction.The foundations of bifurcations and chaos,the theory of stability and bifurcation periodic solutions of dynamical systems,the definite qualitative analysis methods,the developments of studying bifurcations and chaos in nonlinear oscillatory systems,and the contributions of the thesis are introduced briefly.In Chapter 2,a new perturbation procedure for determining limit cycles in threedimensional nonlinear autonomous systems is presented.The main idea of the modified L-P method is adopted and a new nonlinear parametric transformation and a suitable nonlinear frequency expansion are introduced in this procedure.The analytical approximate expression and amplitude and frequency of limit cycled can be obtained.A typical exampie shows that,by comparing with the classical methods of multiple scales and normal form,the present perturbation method process several merits such as simple deductions, higher accuracy and applicable to higher-dimensional nonlinear systems.In Chapter 3,a semi-analytical procedure for analyzing stability and bifurcations of limit cycles of higher-dimensional nonlinear autonomous systems is presented.The procedure is the combination of the incremental harmonics balance method and the theory of stability and bifurcation of periodic solutions of dynamical systems.It can be applied to investigate the changes of some dynamical characteristics including the amplitude and frequency,number,symmetry and periodicity of limit cycles.Applying the procedure to the three-dimensional system discussed in chapter 2,the critical parameter values for symmetry-breaking bifurcation,the first and secondary period-doubling bifurcations of the system and the analytical approximate expressions of the limit cycles(including the symmetry,asymmetry and period-doubled limit cycles) are obtained.The results obtained by the present method are in pretty good agreement with those of numerical integration. It demonstrates that the proposed procedure has higher accuracy.In Chapter 4,the Mathieu-Duffing oscillator is studied.The course of infinite perioddoubling bifurcations route to chaos are analyzed.The series of critical values of perioddoubling bifurcations,the threshold value for the occurrence of chaos and the approximations of the limit cycles(including the period-doubled limit cycles) are obtained.The present results are in good agreement with those of numerical integration.In Chapter 5,the control of chaotic orbit of Mathieu-Duffing oscillator is studied. Based on the idea of open-plus-closed loop control,a controller consisting of periodic external excitation and linear error state feedback is designed.The control of the chaotic orbit to the periodic and higher periodic orbits is investigated.Moreoore,using a comparison theorem of second order ordinary differential equations,global entrainment of the present open-plus-closed loop control is proved under certain conditions.In Chapter 6,chaos synchronization of coupled Mathieu-Duffing oscillators is studied. By replacing the chaotic orbit by the virtual periodic orbit containing multiple harmonics and applying the stability theory of Hill equation,under the couplings of the unidirectional and bidirectional linear error state feedback controllers,the synchronization regions are obtained and the behaviors of the synchronization time in the synchronization regions are analyzed.Comparing the results of some predecessors where the chaotic orbit is replaced by a periodic orbit with one harmonics,the results of the present method is more accurate.In chapter 7,chaos synchronization criteria of two identical transducer systems is discussed.Based on the stability theory of linear time-varying system and Gerschgorin theorem,one chaos synchronization criterion for two identical transducer systems is obtained. Then,by utilizing the property of similar matrixes and Lyapunov stability theory respectively,the criterion mentioned above is optimized in the sense of reducing the lower bounds of coupling coefficients.In chapter 8,the controllability of amplitude and frequency of limit cycles in nonlinear dynamical systems is discussed.The multiple scales method is used to calculate the normal forms of dynamical systems and hence the analytical relationship between the amplitude or frequency of the controlled limit cycle and the feedback gains.Thus,the amplitude and frequency of limit cycles can be controlled by setting suitable feedback gains.A three-dimensional system is chosen as an illustrated example.The efficiency of some types of linear or nonlinear state feedback controllers are illustrated and compared.In chapter 9,several conclusions of this thesis are given and some prospects of future research work are proposed.