Universal Behaviour on the Break-up of the Spiral Mean Torus

We study numerically the critical behaviour during the break-up of the spiral mean torus in a four-dimensional symplectic map. At each point of the parameter space, the stability indices of a serial of periodic orbits are calculated with their winding numbers approaching the spiral mean torus. The critical values of the parameters when the torus breaks are determined by the criterion that the variance of the distribution on the indices reaches a minimum. Some evidence is revealed about the possible existence of a universal distribution on the stability indices of the periodic orbits at the critical. This confirms the picture given by the approximate renormalization theory of the Hamiltonian systems with three degrees of freedom.