Continuation of the Solution Near Singular Points

Abstract

The continuation methods developed in Chapter 1, in which the unknowns and the parameter are treated on an equal basis, have a unified continuation algorithm at regular and limit points in the solution set of nonlinear system equations. From a standpoint of these forms of the continuation algorithm, it is, therefore, unnecessary to introduce the concept of a limit point. Further to the discussion in the Introduction, primary attention is given to an analysis of the behaviour of the solution in the neighbourhood of essentially singular points, i.e., points where the augmented Jacobian matrix $\bar{J}$ is singular. As a basic method of analysis we adopt a method of expansion of the solution in a Taylor series in the neighbourhood of a singular point. This enables us to construct the bifurcation equation, and, by its analysis, to find all branches of the solution. The complexity of the analysis depends on the degree of singularity of the Jacobian matrix $\bar{J}$. We shall consider the case of a simple singularity of the matrix $\bar{J}$ (rank $(\bar{J}) = m - 1$), which is the most important case for practical applications, and also a more complicated case of its double singularity (rank $(\bar{J}) = m - 2$).