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2013
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# Junxiang Xu

Nanjing 210096, China
Department of Mathematics, Southeast University

## 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. Keywords: Hamiltonian system, KAM iteration, degenerate equilibrium, invariant tori Citation: Xu J., You J.,  Persistence of Hyperbolic-type Degenerate Lower-dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 616-650 DOI:10.1134/S1560354720060088
 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. Keywords: Hamiltonian system, KAM iteration, invariant tori, nondegeneracy condition Citation: Xu J., Lu X.,  General KAM Theorems and their Applications to Invariant Tori with Prescribed Frequencies, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 107-125 DOI:10.1134/S1560354716010068
 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. Keywords: Hamiltonian system, KAM iteration, degenerate equilibrium, invariant tori Citation: Xu J., You J.,  Persistence of Hyperbolic-type Degenerate Lower-dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems, Regular and Chaotic Dynamics, , , pp. 616-650 DOI:10.1134/S1560354720060088