Hopf Bifurcation Analysis For A Class Of Delayed Differential Equation And The Combined Multiple Shooting Pseudo-Arclength Tracing Method

With the development of science and technology, investigation of more and more nonlinearproblems from science, technology, and even various fields of social science, have been becomingthe focus of science study. Bifurcation is a common nonlinear phenomenon and plays an importantrole in the nonlinear science. In this thesis, we concentrate on analysis and computation of Hopfbifurcation. It can be be separated as three parts on the whole.Firstly, using the Lyapunov-Schmidt reduction method, we investigate the Hopf bifurcationof one class of delayed differential equations(DDEs). Near the Hopf bifurcation point, we obtainthe approximate analytic periodic solutions which bifurcated from the trivial solution.Secondly, we compare the approximate analytic periodic solution with the numerical results,which are computed by the collocation method based on piecewise Hermite polynomials. Thefact that the approximate analytic solutions nearly coincides with the numerical results shows theeffectiveness of our analysis.Finally, we consider the computation of solutions branch of two point boundary-value prob-lems of ODE with one-parameter . In general, the local continuation method is valid for calcu-lating the regular branch until it meets a singular point. Here we provide the combined multipleshooting method with the pseudo-arclength continuation method to trace successfully through thefold point.