Junxiang Xu
Publications:
Xu J., You J.
Persistence of Hyperbolic-type Degenerate Lower-dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems
2020, vol. 25, no. 6, pp. 616-650
Abstract
It is known that under Kolmogorov’s nondegeneracy condition, the nondegenerate
hyperbolic invariant torus with Diophantine frequencies will persist under small perturbations,
meaning that the perturbed system still has an invariant torus with prescribed frequencies.
However, the degenerate torus is sensitive to perturbations. In this paper, we prove the
persistence of two classes of hyperbolic-type degenerate lower-dimensional invariant tori, one
of them corrects an earlier work [34] by the second author. The proof is based on a modified
KAM iteration and analysis of stability of degenerate critical points of analytic functions.
|
Xu J., Lu X.
General KAM Theorems and their Applications to Invariant Tori with Prescribed Frequencies
2016, vol. 21, no. 1, pp. 107-125
Abstract
In this paper we develop a new KAM technique to prove two general KAM theorems for nearly integrable Hamiltonian systems without assuming any nondegeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some nondegeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.
|
Xu J., You J.
Persistence of Hyperbolic-type Degenerate Lower-dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems
, , pp. 616-650
Abstract
It is known that under Kolmogorov’s nondegeneracy condition, the nondegenerate
hyperbolic invariant torus with Diophantine frequencies will persist under small perturbations,
meaning that the perturbed system still has an invariant torus with prescribed frequencies.
However, the degenerate torus is sensitive to perturbations. In this paper, we prove the
persistence of two classes of hyperbolic-type degenerate lower-dimensional invariant tori, one
of them corrects an earlier work [34] by the second author. The proof is based on a modified
KAM iteration and analysis of stability of degenerate critical points of analytic functions.
|