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Branch point locator

The location of Hopf and limit points usually does not cause problems. However, the location of branch points can give problems. The region of attraction of the Newton type continuation method which is used, has the shape of a cone (see [2]). In the localisation process we cannot assume to stay in this cone. This difficulty can be avoided by introducing and and considering the extended system:

| (41)

We solve this system by Newton's method with initial data and where is the real eigenvalue with smallest norm. A branch point corresponds to a regular solution of system (41) (see [3],p. 165). We note that the second order partial derivatives (Hessian) of with respect to and are required.

The tangent vector at the singularity is also computed here. This is related to the processing of the branch point (computing the direction of the secondary branch).