Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians

A major result about perturbations of integrable Hamiltonian systems is the Nekhoroshev theorem, which gives exponential stability for all solutions provided the system is analytic and the integrable Hamiltonian is generic. In the particular but important case where the latter is quasi-convex, these exponential estimates have been generalized by Marco and Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by Lochak in the analytic case. In this paper, using the same approach, we investigate the situation where the Hamiltonian is assumed to be only finitely differentiable, for which it is known that exponential stability does not hold but nevertheless we prove estimates of polynomial stability.     

A Direct Proof of the Nekhoroshev Theorem for Nearly Integrable Symplectic Maps

We provide the direct proof of the Nekhoroshev theorem on the stability of nearly integrable analytic symplectic maps. Specifically, we prove the stability of the actions for a number of iterations which grows exponentially with an inverse power of the norm of the perturbation by conjugating the generating function of the map to suitable normal forms with exponentially small remainder.Communicated by Eduard Zehndersubmitted 16/06/03, accepted 31/03/04     

A Diophantine duality applied to the KAM and Nekhoroshev theorems

In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory for quasi-periodic solutions dealing only with periodic approximations and avoiding classical small divisors estimates. We obtain two results of stability, in the spirit of the KAM and Nekhoroshev theorems, in the model case of a perturbation of a constant vector field on the $n$ -dimensional torus. Our first result, which is a Nekhoroshev type theorem, is the construction of a “partial” normal form, that is a normal form with a small remainder whose size depends on the Diophantine properties of the vector. Then, assuming our vector satisfies the Bruno–Rüssmann condition, we construct an “inverted” normal form, recovering the classical KAM theorem of Kolmogorov, Arnold and Moser for constant vector fields on torus.     

Nekhoroshev estimates for commuting nearly integrable symplectomorphisms

In this paper, we prove the Nekhoroshev estimates for commuting nearly integrable symplectomorphisms. We show quantitatively how ? ^m symmetry improves the stability time. This result can be considered as a counterpart of Moser’s theorem [11] on simultaneous conjugation of commuting circle maps in the context of Nekhoroshev stability. We also discuss the possibility of Tits’ alternative for nearly integrable symplectomorphisms.     

Nekhoroshev theorem for perturbations of the central motion

In this paper we prove a Nekhoroshev type theorem for perturbations of Hamiltonians describing a particle subject to the force due to a central potential. Precisely, we prove that under an explicit condition on the potential, the Hamiltonian of the central motion is quasiconvex. Thus, when it is perturbed, two actions (the modulus of the total angular momentum and the action of the reduced radial system) are approximately conserved for times which are exponentially long with the inverse of the perturbation parameter.     

Optimal Stability and Instability for Near-Linear Hamiltonians

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.     

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Abstract. – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.?Our stability result generalizes those by Lochak-Neishtadt and P?schel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).?On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system. Manuscrit reĉu le 30 décembre 2001. In memoriam Michael R. Herman The present article is the result of a collaboration with Michael Herman, which started in October 1999. He had had the idea of studying the Nekhoroshev theory in the Gevrey category and, using a lemma of his, of producing new examples of unstable orbits for which the instability time could be compared with the distance of the system to integrability. Together we improved both the stability and instability results which he had already obtained, in view of making them match. Michael Herman’s sudden death in November 2000 prevented him from participating to the last developments and to the final writing of a work the main contributor of which he was.     

The Nekhoroshev theorem and the observation of long-term diffusion in Hamiltonian systems

The long-term diffusion properties of the action variables in real analytic quasiintegrable Hamiltonian systems is a largely open problem. The Nekhoroshev theorem provides bounds to such a diffusion as well as a set of techniques, constituting its proof, which have been used to inspect also the instability of the action variables on times longer than the Nekhoroshev stability time. In particular, the separation of the motions in a superposition of a fast drift oscillation and an extremely slow diffusion along the resonances has been observed in several numerical experiments. Global diffusion, which occurs when the range of the slow diffusion largely exceeds the range of fast drift oscillations, needs times larger than the Nekhoroshev stability times to be observed, and despite the power of modern computers, it has been detected only in a small interval of the perturbation parameter, just below the critical threshold of application of the theorem. In this paper we show through an example how sharp this phenomenon is.     

Umbilical torus bifurcations in Hamiltonian systems

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally umbilic invariant tori. These lower dimensional tori do not satisfy the usual non-degeneracy conditions that would yield persistence by an adaption of KAM theory, and there are indeed regions in parameter space with no surviving torus. We assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a (generalised) umbilical catastrophe. Combining techniques of KAM theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large Cantor sets. Applications to gyrostat dynamics are pointed out.     

Long Time Stability in Perturbations of Completely Resonant PDE's

In this paper we will present some results concerning long time stability in nonlinear perturbations of resonant linear PDE's with discrete spectrum. In particular we will prove that if the perturbation satisfies a suitable nondegeneracy condition then there exists a periodic like trajectory, i.e. a closed curve in the phase space with the property that solutions starting close to it remain close to it for very long times. Secondly, in the special case where the average of the main part of the perturbation is integrable we will prove that if the energy is initially essentially concentrated on finitely many modes, then along the corresponding solutions all the actions are approximatively constant for very long times. Applications to nonlinear wave and Schrödinger equations on a segment will also be given.     

A point-wise criterion for quasi-periodic motions in the KAM theory

We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rüssmann’s nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.     

The classical KAM theorem for Hamiltonian systems via rational approximations

In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in [4] for perturbations of constant vector fields on the torus.     

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.     

The hyperbolic invariant tori of symplectic mappings

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings. Under a Rüssmann-type non-degenerate condition, by introducing a modified KAM iteration scheme, we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough.     

Modulational Instability in the Whitham Equation for Water Waves

We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long‐wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin–Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small‐amplitude parameter. We extend the result to related, nonlinear dispersive equations.     

A generalization of Nekhoroshev’s theorem

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincaré theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.     

Exponential stability for time dependent potentials

For a classical Hamiltonian system on a torus defined by a time dependent, bounded and analytic potential we establish global and quantitative bounds for the solutions over an exponentially long interval of time by using techniques which go back to Nekhoroshev.     

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.     

Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms I - ?Birkhoff Normal Forms

. The aim of this paper (part I and II) is to explore the relationship between the effective (Nekhoroshev) stability for near-integrable Hamiltonian systems and the semi-classical asymptotics for Schrödinger operators with exponentially small error terms. Given a real analytic Hamiltonian H close to a completely integrable one and a suitable Cantor set Q \Theta defined by a Diophantine condition, we are going to find a family L_w, w ? Q \Lambda_{\omega}, \omega \in \Theta , of KAM invariant tori of H with frequencies w ? Q \omega \in \Theta which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union L \Lambda of the invariant tori which can be viewed as a simultaneous Birkhoff normal form of H around all invariant tori L_w \Lambda_{\omega} . This leads to effective stability of the quasiperiodic motion near L \Lambda . As an application we obtain in part II (semiclassical) quasimodes with exponentially small error terms which are associated with a Gevrey family of KAM tori for its principal symbol H. To do this we construct a quantum Birkhoff normal form of the Schrödinger operator around L \Lambda in suitable Gevrey classes starting from the Birkhoff normal form of H.     

Lie series method for vector fields and Hamiltonian perturbation theory

We consider a rigorous Hamiltonian perturbation theory based on the transformation of the vector field of the system, realized by the Lie method. Such a perturbative technique presents some advantages over the standard one, which uses the transformation of the Hamilton functions. Indeed, the present method is simple, and furnishes quite detailed informations on the normal form. Moreover, it leads to estimates which are better and/or simpler than those of the scalar Lie methods. The perturbation method is presented with reference to two model problems, both pertaining to the realm of the well known Nekhoroshev theorem: the confining of actions for exponentially long times in a system of coupled harmonic oscillators, and an application to the so called problem of the realization of a holonomic constraint in classical mechanics.