Dynamics Of Duffing Systems Under Delayed State Feedback Control

A detailed study is made in this dissertation on the applicability of the classical perturbation methods to the dynamical systems with time delays, and on the global view of local bifurcations and the complex dynamical behaviors of several Duffing systems under different delayed state feedback control.First of all, the method of Fredholm alternative is used to obtain the approximate solution of the first order for the bifurcating periodic solutions of a nonlinear system with delayed velocity feedback. The bifurcation direction and the stability of the bifurcating periodic solution are also determined. It is shown that with the increase of time delay from zero to infinity, the equilibrium of system may experience stability switches at the critical values of time delay, where the Hopf bifurcation will necessarily occur with the bifurcating periodic solution having alternative stability on their two-dimensional local center manifolds. This study can serve as a comparison with the study in later chapters.Secondly, the effectiveness and limitation of the classical perturbation, such as the method of multiple scales and the Poincare-Lindstedt method, are discussed in detail through a Duffing oscillator with delayed velocity feedback. It is shown that the two perturbation methods are effective only in solving the approximate solution of the first two orders. An ambiguity or paradox will be encountered when they are used to seeking for the third or higher order approximation of solution.Then a shooting scheme is successfully designed to seek for periodic solutions of autonomous delay differential equations. The numerical results obtained by the shooting method coincide very well with those obtained by the method of multiple scales, which demonstrates that although these perturbation methods have some shortcomings, they are effective in revealing the primary dynamics of systems with delays.Afterwards, a Duffing system with delayed displacement feedback is studied by the method of multiple scales and other numerical methods. The uniform formulas for computing the critical values of time delay are given and the global diagrams of bifurcation for the periodic solutions with respect to the time delay are obtained under different parametric combinations. It is shown that the Hopf bifurcation and the saddle-node bifurcation are the only two types of bifurcation observed in such a system. It is also proved that there exist an infinite number of solution branches in the bifurcation diagram by using a simple property of delay system, referred to as "periodicity in delay", although the periodic solutions on most of the solution branches cannot be verified throughnumerical integration. The reason why the periodic solutions on some of the bifurcation branches cannot be located by using direct integration is also discussed from the viewpoint of the finite dimensional basin of attraction. In addition, the special route of tori sequence to chaos is also illustrated.At last, the stability switches of equilibrium and the global diagrams of bifurcation for a Duffing oscillator with both delayed velocity feedback and delayed displacement feedback are studied by using the similar method as above. Formulas for computing the critical values of time delay are also obtained after a detailed analysis for the system parameters. It is surprisingly shown that the combination of both displacement feedback and velocity feedback seems to induce no more bifurcation characteristics than one type of the feedback only. Through such a study, it is also shown that when the time delay is long enough such that the stability switches of the equilibrium of system halt, the bifurcation branches originated from the critical time delays cannot be surely stable or unstable in the whole solution space, that is, their stability depends on cases. This phenomenon also implies that the stability of the periodic solutions on the bifurcation diagrams indicates only the stability on the local center manifold. Some distinguished features of delay dyn...