Hopf bifurcation control for time-delay dynamical systems and applications.

摘要

This thesis is concerned with Hopf bifurcation and its control in time-delay dynamical systems.; Time-delay dynamical systems are often used to describe many phenomena of physical interest. Particular attention of this thesis is given to Hopf bifurcation and double-Hopf bifurcation of time-delay dynamical systems. Linear analysis is employed to determine the critical conditions under which Hopf and double-Hopf bifurcations may occur. Center manifold theory is applied to obtain a set of ordinary differential equations describing the center manifold of the system. Then, normal form theory is employed to consider the stability of the system. The theory and methodology are applied to study several practical time-delay systems.; Firstly, the stability problem of 2-D supersonic lifting surfaces with time-delay feedback controls is addressed. The implications of the time delay in the considered controls are discussed. The study is intended to provide some guidelines for engineering design. Then, a computational method is proposed for studying double-Hopf bifurcation of a self-sustained system with external forcing and time delay. Double-Hopf bifurcation in time-delay systems has not received much attention in the literature. In this thesis, we investigate the effect of time delay in the cascading bifurcations into double Hopf bifurcation and even more complicated motions such as chaos.; Secondly, we focus on bifurcation and chaos control, which as an emerging research field has become challenging, stimulating, and yet quite promising. Particularly, Hopf bifurcation control in both discrete and continuous time-delay dynamical systems is studied. Feedback controllers using polynomial functions are designed for controlling Hopf bifurcation in discrete maps, which have been shown to have superior properties than traditional controllers such as the washout filter controller. Hopf bifurcation control is also applied to consider time-delay dynamical systems. A time-delay feedback controller using polynomial function is proposed to delay the onset of undesirable Hopf bifurcations in an Internet congestion model, thus guaranteeing a stationary sending rate for large parameter values.; Finally, a new approach using hysteretic force is proposed for controlling chaos.; Keywords. Time delay, nonlinear dynamical system, stability, center manifold, normal form, multiple-time scales, resonance, non-resonance, Hopf bifurcation, bifurcation and chaos control, numerical simulation, hysteretic system.