Folds in Solutions of Two Parameter Systems and Their Calculation. Part I

This paper is concerned with paths of turning points in solutions of nonlinear systems having two parameters. It is well known that these paths are solutions of a particular extended system of nonlinear equations. In this paper both regular points and simple turning points in the extended system are related to the local geometry of the solution surface of the original nonlinear system. A description is given of numerical methods both for solving the extended system and for calculating certain quantities which determine the local geometry of the solution surface. Applications to perturbed bifurcation, to the formation of isolas, and to the calculation of the multiplicity of solutions are also discussed. Numerical examples are given.