Bifurcation Analysis And Computation Of Several Nonlinear Problems

With the development of science and technology, investigation of more and more nonlinear problems from science, technology, and even various fields of social science, have been becoming the focus of science study. Bifurcation is a common nonlinear phenomenon and plays an important role in the nonlinear science. In this thesis, we concentrate on analysis and computation of bifurcation .It can be separated as two parts on the whole.First, we choose a class of nonlinear reaction-diffusion equations in developmental biology as the research model, because of its abundant bifurcation phenomenon and not simple expression. We introduce Liapunov-Schmidt reduction method to investigate the bifurcation of a class of nonlinear reaction-diffusion equations in developmental biology. Near the bifurcation point we obtain nontrivial solutions branch emitted from the trivial solution. The approximate analytical expressions of the nontrivial solutions are given to compare with the numerical solutions of the nonlinear problem.Second, a numerical method for computing the higher order singular points of the nonlinear problems with single parameter is considered. Based on the uniformly extended system and pseudo-arclength continuation, an uniform algorithm is given. Numerical examples are computed to show the effectiveness of our algorithm.