| In this thesis, it consists of two parts. First we concentrate on the nonlinear problem with a single parameter. There are many kinds of bifurcation points, e.g. folds, pitchfork bifurcation points, transcritical bifurcation points, etc. We propose a new approach to detection of bifurcation in one single parameter nonlinear problem.In order to detect these singular points, we usually use regular extended system, the main idea is extending the original equations by adding new equations, then the extended system is not singular any more at these points. But generally there are difficulties for detecting the singularities in it. First, we do not know what kind of bifurcation will happen in advance,it is difficult to choose what kind of regular extended system for detecting them. Second, if there are higher order singularities, even the multi-dimensional null space, one parameter is not enough for establishing a regular extended system. So we propose another approach to detection of bifurcation in one single parameter nonlinear problem. Our idea is to introduce a uniformly extended system, which usually is not regular. Based on the uniformly extended system and pseudo-arclength continuation, an uniform algorithm is given.Before showing the numerical examples we discuss the relation between the singularities in the original problem and the uniformly extended system, which is very important for our algorithm. Numerical examples are computed to show the effectiveness of our algorithm.In the second part, we consider a class of reaction-diffusion equations in developmental biology. Near the bifurcation points, using the Liapunove-Schmidt reduction process,we obtain the nontrivial solution branches which are bifurcated from the trival solution when the parameter changes. The approximate analytical expressions of the nontrivial solutions are given to compare with the numerical solutions of the nonlinear problem.
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