Degenerate Lower Dimensional Tori in Hamiltonian Systems

By Yuecai Han, Yong Li, Y. Yi. 2006

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Abstract We study the persistence of lower-dimensional tori in Hamiltonian systems of the form H ( x , y , z ) = 〈 ω , y 〉 + 1 2 〈 z , M ( ω ) z 〉 + e P ( x , y , z , ω ) , where ( x , y , z ) ∈ T n × R n × R 2 m , e is a small parameter, and M ( ω ) can be singular. We show under a weak Melnikov nonresonant condition and certain singularity-removing conditions on the perturbation that the majority of unperturbed n -tori can still survive from the small perturbation. As an application, we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable, properly degenerate Hamiltonian system for which neither the Kolmogorov nor the g -nondegenerate condition is satisfied.

Subfields: Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7857725f2db8f7a5f80195eee30e715ce6efa5ab