Computing multiple pitchfork bifurcation points

A point (x*, lambda*) is called a pitchfork bifurcation point of multiplicity p greater than or equal to 1 of the nonlinear system F(x, lambda) = 0, F:R-n x R-n --> R-n, if rank delta(x)F(x*, lambda*)= n - 1, and if the Ljapunov-Schmidt reduced equation has the normally form g(xi, mu) = +/- xi(2+p) +/- mu xi = 0. It is shown that such points satisfy a minimally extended system G(y) = 0, G:Rn+2 --> Rn+2 the dimension n + 2 of which is independent of p. For solving this system, a two-stage Newton-type method is proposed. Some numerical tests show the influence of the starting point and of the bordering vectors used in the definition of the extended system on the behavior of the iteration.