On the use of predictor-corrector continuation to trace implicitly defined curves and calculating bifurcation

We will consider the use of Predictor-corrector method to trace parameterized curves. Homotopy methods will be needed since a Newton like method cannot be used to solve the nonlinear systems involved. This is due to the fact that not much information is available about the zero point of the system. We will also consider systems which involve the presence of a natural parameter; in particular, tracing a parameter dependent curve which contains a simple turning or bifurcation point at a critical value of the parameter.