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 approac...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.展开更多
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...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.展开更多
基金Supported by Hong Kong Baptist University Faculty Research Grants,Hong Kong Grant Council Grantsthe National Natural Science Foundation of China under Grant Nos.19903001 and 19633010the Special Funds for Major State Basic Research Projects.
文摘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.
基金Supported by the National Natural Science Foundation of China under Grant No.19903001the Special Funds for Major State Basic Research Projects,and Hong Kong Research Grant Council.
文摘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.
