Continuation of the Solution near Singular Points

The continuation methods developed in chapter 1 treat the unknowns and the parameter the same way. They use a universal continuation algorithm at regular and limit points of the solution set of nonlinear system equations. From standpoint of the continuation algorithm, it is, therefore, unnecessary to introduce the concept of a limit point. Further to the discussion in chapter 1, here the primary attention is given to analysis of the behavior of the solution in the neighborhood of essentially singular points, i.e., the points where the augmented Jacobian matrix J is singular As a basic method of analysis we adopt the method of Taylor series in the neighborhood of singular point. This enables us to construct the bifurcation equation, and, analyzing it, to find all brunches of the solution. The complexity of the analysis depends on the degree of singularity of matrix J. We shall consider the case of simple singularity of matrix J (rank(J) = n − 1), which is the most important case for applications, and also a more complicated case of its double singularity (rank(J) = n − 2).