Effective Stability and KAM Theory

Delshams, Amadeu; Gutiérrez, Pere

doi:10.1006/jdeq.1996.0102

Cited by 68 publications

(77 citation statements)

References 24 publications

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“…In this case, this is exactly the result obtained in [4]. In [18], there is a similar but less flexible result, since there Vol.…”

Section: Corollary 12 For Linear Integrable Hamiltonians Rationallsupporting

confidence: 81%

“…Hence for any n ≥ 2, we shall use more classical techniques, namely general resonant normal forms as in [18] and [4], and so our proof will not be essentially new. In the special case where the frequency ω is Diophantine, that is when there exist γ > 0 and τ ≥ n−1 such that for all k ∈ Z n−1 \{0}, |k.α| Z ≥ γ|k| −τ , then, in Theorem 1.1, we can choose…”

Section: Corollary 12 For Linear Integrable Hamiltonians Rationallmentioning

confidence: 99%

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Optimal Stability and Instability for Near-Linear Hamiltonians

Bounemoura

2011

Ann. Henri Poincaré

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Abstract. In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover, in the same spirit as the notion of KAM stable integrable Hamiltonians, we will introduce a notion of effectively stable integrable Hamiltonians, conjecture a characterization of these Hamiltonians and show that our result proves this conjecture in the linear case.

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“…In this case, this is exactly the result obtained in [4]. In [18], there is a similar but less flexible result, since there Vol.…”

Section: Corollary 12 For Linear Integrable Hamiltonians Rationallsupporting

confidence: 81%

Section: Corollary 12 For Linear Integrable Hamiltonians Rationallmentioning

confidence: 99%

Optimal Stability and Instability for Near-Linear Hamiltonians

Bounemoura

2011

Ann. Henri Poincaré

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“…(0)>Ĩ + H ( x Î y) (6) where Note t hat i f w e a r e a ble to k i l l t he t ermsã, b and c;! (0) we o b t ain a lower dimensional invariant t orus with i n trinsic frequency !…”

Section: The I T Erative S C Hemementioning

confidence: 99%

On the Persistence of Lower Dimensional Invariant Tori under Quasi-Periodic Perturbations

Jorba

Villanueva

1997

Journal of Nonlinear Science

109

105

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In this work w e c o n s i d er time d ependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the e ect that t his kind o f p e r t urbations has on lower dimensional invariant t ori. Our results s h ow t hat, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the p e r t urbation to t he o n es they already have.The paper also contains estimates on the amount of surviving t ori. The w orst situation happens when the initial tori are normally elliptic. In this case, a torus (identi ed by t he v ector of intrinsic frequencies) can be continued with respect to a p e r t urbative p a r a m eter " 2 0 " 0 ], except for a set of " of measure exponentially small with " 0 . I n c a s e t hat " is xed (and su ciently small), we prove t he existence of invariant t ori for every vector of frequencies close to t he o n e o f t he initial torus, except for a set of frequencies of measure exponentially small with t he d i s t ance to the u nperturbed torus. As a particular case, if the p e r t urbation is autonomous, these results also give t he s a m e k i n d o f e s t imates on the m easure of destroyed tori.Finally, t hese results are applied to s o m e problems of celestial mechanics, in order to h elp in the d escripti o n o f t he p h ase space of some concrete m o d els.jorba@ma1.upc.es y jordi@tere.upc.es 2

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“…The selected periodic orbit belongs to the Lyapunov family associated to the vertical oscillation of the equilibrium point L 5 . The mass parameter is chosen big enough such that L 5 is unstable, but not too big in order to have the selected orbit normally elliptic (see Section 3.2).…”

Section: Introductionmentioning

confidence: 99%

Numerical computation of normal forms around some periodic orbits of the restricted three-body problem

Jorba

Villanueva

1998

Physica D: Nonlinear Phenomena

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In this paper we introduce a general methodology for computing (numerically) the normal form around a periodic orbit of an autonomous analytic Hamiltonian system. The process follows two steps. First, we expand the Hamiltonian in suitable coordinates around the orbit and second, we perform a standard normal form scheme, based on the Lie series method. This scheme is carried out up to some nite order and, neglecting the remainder, we obtain an accurate description of the dynamics in a (small enough) neighbourhood of the orbit. In particular, we obtain the invariant tori that generalize the elliptic directions of the periodic orbit. On the other hand, bounding the remainder one obtains lower estimates for the di usion time around the orbit.This procedure is applied to an elliptic periodic orbit of the spatial Restricted Three Body Problem. The selected orbit belongs to the Lyapunov family associated to the vertical oscillation of the equilibrium point L 5 . The mass parameter has been chosen such that L 5 is unstable but the periodicorbit is still stable. This allows to show the existence of regions of e ective stability near L 5 for values of bigger that the Routh critical value. The computations have been done using formal expansions with numerical coe cients.2

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Effective Stability and KAM Theory

Cited by 68 publications

References 24 publications

Optimal Stability and Instability for Near-Linear Hamiltonians

Optimal Stability and Instability for Near-Linear Hamiltonians

On the Persistence of Lower Dimensional Invariant Tori under Quasi-Periodic Perturbations

Numerical computation of normal forms around some periodic orbits of the restricted three-body problem

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