Superexponential stability of KAM tori

Morbidelli, Alessandro; Giorgilli, Antonio

doi:10.1007/bf02180145

Cited by 149 publications

(120 citation statements)

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“…Clearly the perturbation which is constructed in Theorem 1.3 is not of this type. Let us recall that it is known that, if the frequency of the torus is Diophantine, then a result of exponential stability holds true, and if one add further conditions on the Birkhoff invariants, then even super-exponential stability holds true (this was first proved in [11], see also [1] for somehow more general results). Yet there is no examples of instability showing that these assumptions on the Birkhoff invariants are indeed necessary (in fact, in the analytic case, it is a pity that there is no examples of instability at all).…”

Section: Further Commentsmentioning

confidence: 98%

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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Section: Further Commentsmentioning

confidence: 98%

Optimal Stability and Instability for Near-Linear Hamiltonians

Bounemoura

2011

Ann. Henri Poincaré

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“…Moreover, effective stability of the action, that is stability of the action in a finite but exponentially long time interval has been proved in [3]. Effective stability near an analytic KAM torus has been investigated by Morbidelli and Giorgilli in [13], [14] and [4]. Combining it with the Nekhoroshev theorem they obtained a super-exponential effective stability of the action near the torus.…”

Section: Resultsmentioning

confidence: 99%

“…In the case of KAM tori [16,18] this yields an uniform bound on the corresponding Gevrey constants with respect to ω ∈ Ω κ . It could be applied as in [13], [14] and [4] to obtain a super-exponential effective stability of the action near the torus in the case of convex Hamiltonians using the Nekhoroshev theory for Gevrey Hamiltonians developed by J.-P. Marco and D. Sauzin [12]. It seems that this method could be applied to obtain a Gevrey normal form in the case of elliptic tori and near an elliptic equilibrium point of Gevrey smooth Hamiltonians as well as in the case of hyperbolic tori and reversible systems.…”

Section: Resultsmentioning

confidence: 99%

“…In the case when the Hamiltonian and the torus are analytic a similar BNF has been obtained by Morbidelli and Giorgilli. They have proved as well effective stability of the action near analytic KAM tori and even a super exponential stability of the action [4,13,14] for convex Hamiltonians using Nekhoroshev's theory. A simultaneous normal form for a family of Gevrey KAM tori has been obtained in [16,18].…”

Section: Introductionmentioning

confidence: 99%

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Gevrey normal form and effective stability of Lagrangian tori

Mitev¹,

Popov²

2010

Discrete &Amp; Continuous Dynamical Systems - S

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Abstract. A Gevrey symplectic normal form of an analytic and more generally Gevrey smooth Hamiltonian near a Lagrangian invariant torus with a Diophantine vector of rotation is obtained. The normal form implies effective stability of the quasi-periodic motion near the torus. IntroductionThe aim of this paper is to obtain a Birkhoff Normal Form (shortly BNF) in Gevrey classes of a Gevrey smooth Hamiltonian near a Kronecker torus Λ with a Diophantine vector of rotation. Such a normal form implies "effective stability" of the quasi-periodic motion near the invariant torus, that is stability in a finite but exponentially long time interval. As in [17,19,20] it can be used to obtain a microlocal Quantum Birkhoff Normal Form for the Schrödinger operator P h = −h 2 ∆ + V (x) near Λ and to describe the semi-classical behavior of the corresponding eigenvalues (resonances).A Kronecker torus of a smooth Hamiltonian H in a symplectic manifold of dimension 2n is a smooth embedded Lagrangian submanifold Λ, diffeomorphic to the flat torus T n := R n /2πZ n , which is invariant with respect to the flow Φ t of H, and such that the restriction of Φ t to Λ is smoothly conjugated to the linear flow g t ω (ϕ) := ϕ + tω (mod 2π) on T n for some ω ∈ R n . Hereafter, we suppose that ω satisfies the usual Diophantine condition (2.4). Then there is a symplectic mapping χ from a neighborhood of the zero section T n 0 := T n × {0} to a neighborhood of Λ in X sending T n 0 to Λ and such that the Hamiltonian, where ∇H 0 (0) = ω, and the Taylor series of R 0 at I = 0 vanishes (cf.[10], Proposition 9.13). In particular, T n 0 is an invariant torus of H 0 , the restriction of the flow of H 0 to T n 0 is given by g t ω (ϕ) = ϕ + tω (mod 2π), and for any α, β ∈ N, and any N ≥ 1, we have ∂

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“…These formulae give the average times of escape. However, near every island the escape times increase superexponentially (Morbidelli and Giorgilli 1995) and tend to infinity as we approach the border of the island from outside.…”

Section: Stickiness Near a Cantorusmentioning

confidence: 96%

Stickiness effects in chaos

Contopoulos

Harsoula

2010

Celest Mech Dyn Astr

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Stickiness is a temporary confinement of orbits in a particular region of the phase space before they diffuse to a larger region. In a system of 2-degrees of freedom there are two main types of stickiness (a) stickiness around an island of stability, which is surrounded by cantori with small holes, and (b) stickiness close to the unstable asymptotic curves of unstable periodic orbits, that extend to large distances in the chaotic sea. We consider various factors that affect the time scale of stickiness due to cantori. The overall stickiness (stickiness of the second type) is maximum near the unstable asymptotic curves. An important application of stickiness is in the outer spiral arms of strong-barred spiral galaxies. These spiral arms consist mainly of sticky chaotic orbits. Such orbits may escape to large distances, or to infinity, but because of stickiness they support the spiral arms for very long times.

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Superexponential stability of KAM tori

Cited by 149 publications

References 10 publications

Optimal Stability and Instability for Near-Linear Hamiltonians

Optimal Stability and Instability for Near-Linear Hamiltonians

Gevrey normal form and effective stability of Lagrangian tori

Stickiness effects in chaos

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