Break-Up of Three-Frequency KAM Tori: Determination of theCritical Parameters

With a four-dimensional symplectic map we study numerically the break-up of three-frequency Kolmogorov-Arnold-Moser (KAM) tori. The locations and stabilities of a sequence of periodic orbits, whose winding numbers approach the irrational winding number of the KAM torus, are examined. The break-up of quadratic frequency tori is characterized as the exponential growth of the residue means of the convergent periodic orbits. Critical parameters of the break-up of tori with different winding numbers are calculated, which shows that the spiral mean torus is the most robust one in our model.