Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems
Summary (2 min read)
1. Introduction
- The present work is devoted to the study of the stability exponents for Gevrey quasi-convex near-integrable Hamiltonian systems, and to the search for an example of an unstable orbit with the highest possible speed of drift.
- It is only for n ≥ 3 that KAM tori do not a priori prevent the projection in action space of a solution from drifting arbitrarily far from its initial location; but such a drift should be exponentially slow, according to the Nekhoroshev Theorem.
- The composition of Gevrey-α functions is also Gevrey-α under appropriate assumptions (Section A.2).
- The authors think that generalizing the Nekhoroshev Theorem to Gevrey Hamiltonian functions is interesting in itself.
- The authors main motivation was to compare the exponents of stability that the authors could obtain in the Gevrey quasi-convex case with the speed of instability they could produce on specific examples.
2. An example of instability with estimate on the speed of drift
- In this part it will be more convenient to work with mappings rather than with flows (the correspondence between Nekhoroshev estimates for flows and for mappings in a rather general perturbative framework was studied in [KP94]).
- 5.3. Proposition 2.2 describes in particular the intersections of the stable and unstable manifolds of the one-parameter familly of circles C ( j) r01 with a suitable section, from which one easily deduces the existence of heteroclinic connections.
- The remaining points of the orbit drift along the heteroclinic connections between two consecutive circles.
3. Stability theorem for Gevrey classes in the quasiconvex case
- Once this normal form is obtained, the Hamiltonian character of the vector field and the quasi-convexity of the unperturbed system are used to derive stability near the torus (see Section 3.4); but they are not necessary for the reduction to the normal form itself, which amounts in fact to a one-phase averaging process.
- 1.5. The next step is the passage from the formal normal form to an approximate Gevrey normal form.
In the case of maximal rank, i.e. d = 1, one can take Q = 1 and the last inequality must be interpreted as the equality ω − ω0 = 0.
- The authors retain from the previous inequality that each element of A has absolute value less than (m − 1)pm.
- Gathering the definitions of the five thresholds and the various constants that the authors have encountered, and enlarging c, they end up with the claim of Section 1.2.
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Citations
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Cites background from "Stability and instability for Gevre..."
...See [1] and [22] for beautiful and deep examples of perturbations of fully integrable systems....
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...2) for Gevrey smooth systems has been solved affirmatively in [294,377]....
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...The literature on Arnold diffusion is immense, and we here just q uote [8,14,108,114, 116, 139, 181, 278, 280, 281, 294, 295, 299, 323, 377, 422] for results, details, and references....
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...Effective stability of the action along all the trajectories for Gevrey smooth Hamiltonians has been obtained recently by Marco and Sauzun [10]....
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References
1,257 citations
"Stability and instability for Gevre..." refers background in this paper
...[ Ca59 ]), to approach the given frequency-vector ω0 =∇ h(r0) by ar ational vectorω, close enough to it but whose period is not too large....
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726 citations
"Stability and instability for Gevre..." refers background in this paper
...In the 70s, a remarkable achievement of Hamiltonian perturbation theory was the Nekhoroshev Theorem [ Nekh77 ], which asserts that for a generic real-analytic function h and for any real-analytic perturbation f , all solutions are stable at least over exponentially long time intervals: there exist positive numbers a and b ,d epending only on h, such that for each small enough e >0 any initial condition (θ0, r0) gives rise to as ......
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360 citations
"Stability and instability for Gevre..." refers methods in this paper
...Sections 3.1–3.4 are devoted to the proof of Theorem A. The first three deal with one-phase averaging and contain a generalization of Neishadt’s Theorem [ Nei84 ] to Gevrey classes (Proposition 3.2); their use of Gevrey asymptotic expansions is directly inspired by [RS96]....
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356 citations