scholarly journals Persistence of Elliptic Lower Dimensional Invariant Tori for Small Perturbation of Degenerate Integrable Hamiltonian Systems

1997 ◽  
Vol 208 (2) ◽  
pp. 372-387 ◽  
Author(s):  
Xu Junxiang
Keyword(s):  
Hamiltonian Systems ◽  
Small Perturbation ◽  
Invariant Tori ◽  
Integrable Hamiltonian Systems ◽  
Lower Dimensional Invariant Tori ◽  
Lower Dimensional

On Invariant Tori with Prescribed Frequency in Hamiltonian Systems

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Dongfeng Zhang ◽  
Junxiang Xu ◽  
Hao Wu
Keyword(s):  
Hamiltonian Systems ◽  
Small Perturbation ◽  
Topological Degree ◽  
Invariant Torus ◽  
Invariant Tori ◽  
Rational Approximations ◽  
Integrable Hamiltonian Systems ◽  
Resonant Condition ◽  
Degeneracy Condition ◽  
Nearly Integrable Hamiltonian Systems

AbstractIn this paper we are mainly concerned with the persistence of invariant tori with prescribed frequency for analytic nearly integrable Hamiltonian systems under the Brjuno–Rüssmann non-resonant condition, when the Kolmogorov non-degeneracy condition is violated. As it is well known, the frequency of the persisting invariant tori may undergo some drifts, when the Kolmogorov non-degeneracy condition is violated. By the method of introducing external parameters and rational approximations, we prove that if the Brouwer topological degree of the frequency mapping is nonzero at some Brjuno–Rüssmann frequency, then the invariant torus with this frequency persists under small perturbation.

scholarly journals Normal form for lower dimensional elliptic tori in Hamiltonian systems

2021 ◽  
Vol 4 (6) ◽  
pp. 1-40
Author(s):  
Chiara Caracciolo ◽  
Keyword(s):  
Normal Form ◽  
Hamiltonian Systems ◽  
Invariant Tori ◽  
Integrable Hamiltonian Systems ◽  
Convergence Of An Algorithm ◽  
Planetary Problem ◽  
Order Of Magnitude ◽  
Nearly Integrable Hamiltonian Systems ◽  
Transverse Oscillations ◽  
Lower Dimensional

<abstract><p>We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, have the same order of magnitude.</p></abstract>

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