Computation of 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(z, lambda) = 0, F : R-n x R-1 --> R-n, if rank partial derivative(s)F(x*, lambda*) = n - 1 and if the Ljapunov-Schmidt reduced equation has the normal 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. Numerical tests show the influence on the convergence behavior of the starting point and of the bordering vectors used in the definition of the extended system.