Stability And Hopf Bifurcation Of Dynamic Systems With Delay-Dependent Parameters

In this thesis, the stability and Hopf bifurcation of time-delay systems with delay-dependent parameters are investigated. The effect of the time delay on the stability of the dynamics is addressed. As the time delay increases from zero to infinite, the stationary movement (say, the equilibriums or periodic motions) may be switched from being asymptotically stable to being unstable, or from being unstable to being asymptotically stable, probably for many times. This phenomenon is usually referred to as the stability switches and is the key viewpoint of the analysis. The thesis consists of four chapters and it is arranged as follows:Firstly in Chapter 1, some fundamentals of stability and Hopf bifurcation are reviewed briefly for time-delay systems.Next in Chapter 2, the stability analysis of the Lasota-Wazewska model of single species with maturation delay is given firstly, on the basis of the graphical test of the stability switches. At the critical values of delay where the stability switches occurs, the system admits a Hopf bifurcation so that a limit cycle emerges after the equilibrium losses its stability. The pseudo-energy analysis is applied to determine the direction of bifurcation, the stability and amplitude of the bifurcating periodic solu- tions. Compared with the current methods, the pseudo-energy analysis only involves easy computation but gives prediction of the local dynamics with high accuracy.Then in Chapter 3, the stability of a time-delay system arising from visually guided movement is investigated. The peculiarity of this system is that the equilibriums and coefficients of the characteristic function depend discontinuously on the delay, and the equilibriums may change their stability at the discontinuous points. Moreover, the equilibriums may also change its stability between the two neighboring discontinuous points, which can be checked by following the graphical test of the stability switches. An important observation is that this time-delay system may change its stability for infinite times, and the equilibriums can be eventually asymptotically stable, if stability switches do occur. While for time-delay systems with delay-independent parameters, the stability of equilibrium can be switched finite many times and eventually unstable, if stability switches do occur.Finally in Chapter 4, some concluding remarks are made.