Near-integrable Hamiltonian Systems Research Articles

Moser's theorem for lower dimensional tori

Moser's C ℓ -version of Kolmogorov's theorem on the persistence of maximal quasi-periodic solutions for nearly-integrable Hamiltonian system is extended to the persistence of non-maximal quasi-periodic solutions corresponding to lower-dimensional elliptic tori of any dimension n between one and the number of degrees of freedom. The theorem is proved for Hamiltonian functions of class C ℓ for any ℓ > 6 n + 5 and the quasi-periodic solutions are proved to be of class C p for any p with 2 < p < p * for a suitable p * = p * ( n , ℓ ) > 2 (which tends to infinity when ℓ → ∞ ).

Explicit calculations of homoclinic tangles in tokamaks

Explicit numerical calculations of homoclinic tangles are presented for a physically realistic model of a resonantly perturbed magnetic field in a tokamak. The structure of these tangles is consistent with that expected from the general theory of near-integrable Hamiltonian systems commonly studied with simple algebraic twist map models. In addition, understanding the structure of homoclinic tangles corresponding to the primary separatrix of a poloidally diverted tokamak allows one to make predictions of the locations and structure of magnetic footprints and heat buildup on the tokamak wall. These separatrix tangles undergo an interesting bifurcation sequence as the current through a set of error field correction coils is increased. Since this model of the magnetic field is very realistic, these features are expected to be experimentally verifiable.

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 Poschel, 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.

On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems

Introduction and some salient features of the model Hamiltonian Symplectic geometry and the splitting of invariant manifolds Estimating the splitting matrix using normal forms The Hamilton-Jacobi method for a simple resonance Appendix. Invariant tori with vanishing or zero torsion Bibliography.

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.

Trajectories in a neighbourhood of asymptotic surfaces of a priori unstable Hamiltonian systems

We consider a priori unstable (initially hyperbolic) near-integrable Hamiltonian systems in a neighbourhood of asymptotic surfaces of a family of hyperbolic tori. Applying the idea of the anti-integrable limit, we construct a large class of chaotic trajectories.

Numerical evidence of fast diffusion in a three-degree-of-freedom Hamiltonian system with a saddle-center

Arnold diffusion is a complicated phenomenon resulting from the presence of a transition chain of whiskered invariant tori. It is very slow and subtle behavior but it is believed to give a central mechanism for long-time instability in multi-degree-of-freedom, near-integrable Hamiltonian systems. Recently, it was shown that similar but fast behavior can also occur in Hamiltonian systems which are not near-integrable but possess saddle-centers. We provide a numerical evidence of this fast diffusion for a three-degree-of-freedom system.

Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems

Perturbing an integrable 3 degree of freedom (d.o.f.) Hamiltonian system containing a normally parabolic 2-torus which is m-resonant ( m=1 or 2) creates a parabolic m-resonance ( m-PR). PRs of different types are either persistent or of low co-dimension, hence they appear robustly in many applications. Energy–momenta bifurcation diagram is constructed as a tool for studying the global structure of 3 d.o.f. near-integrable systems. A link between the diagram shape, PR and the resonance structure is found. The differences between the dynamics appearing in 2 and 3 d.o.f. systems exhibiting PRs are studied analytically and numerically. The numerical study demonstrates that PRs are an unavoidable source of large and fast instabilities in typical 3 d.o.f. systems.

Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms II –¶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 Schrodinger operators with exponentially small error terms. Given a real analytic Hamiltonian H close to a completely integrable one and a suitable Cantor set \( \Theta \) defined by a Diophantine condition, we are going to find a family \( \Lambda_{\omega}, \omega \in \Theta \), of KAM invariant tori of H with frequencies \( \omega \in \Theta\) 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 \( \Lambda \) of the invariant tori which can be viewed as a simultaneous Birkhoff normal form of H around all invariant tori \( \Lambda_{\omega} \). This leads to effective stability of the quasiperiodic motion near \( \Lambda \). 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 Schrodinger operator around \( \Lambda \) in suitable Gevrey classes starting from the Birkhoff normal form of H.

On homoclinic splitting problems

Counterexamples to the homoclinic splitting claimed in the paper “Existence of exponentially small separatrix settings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems” by Rudnev and Wiggins [Physica D 114 (1998) 3–80] are given. These counterexamples invalidate the proofs of the main results of that paper.

Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems

In this paper we study a generalization of Arnold's original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori, that has subsequently been referred to as “ Arnold diffusion” in case when the number of degrees of freedom n ≥ 3. Namely, we consider a widely studied model of a pendulum weakly coupled with n − 1 rotors with the degeneracies in the unperturbed Hamiltonian, corresponding to different timescales, existing in the problem. Using an alloy of the iterative and direct methods developed within the last years we give exponentially small upper bounds for the splitting measure of transversality for the case of an even, analytic perturbation, thus improving the estimate of Gallavotti (1994), which he calls quasiflat, and generalizing the analogous recent estimate of Delshams et al. (1996) for the rapidly quasiperiodically forced pendulum to a much larger class of Hamiltonian systems. In particular, the exponentially small upper bound for the transversality measure of the splitting applies when the Hamiltonian has extra degeneracies, namely when the frequencies on a torus become near-resonant. In fact, we show that in such a case the quantity in question becomes smaller, which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger Nekhoroshev exponent. Nekhoroshev exponent. Nevertheless, we emphasize that getting uniform estimates for an arbitrary n ≥ 3 is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector. Although recent developments show that the first order of canonical perturbation theory, given by Melnikov integrals, generally cannot be accepted as the leading-order answer for the splitting distance for the case of more than two degrees of freedom, including the rapidly quasiperiodically forced pendulum problem, we suggest an analytic perturbation, the majority of whose Fourier components are strictly nondashzero, for which Melnikov integrals can be vindicated as the leading-order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions. This allows us to bound the splitting distance from below. Furthermore, having such a perturbation, for the case of three degrees of freedom, we use a simple number-theoretical argument to find the asymptotics of the Fourier series with exponentially small coefficients involved. This enables us to compute the numerous homoclinic orbits for the whiskered tori of asymptotically full measure, and by proving the domineering contribution of the first order of perturbation theory for the transversality measure, to suggest a leading-order answer for this quantity, which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies. We elucidate the numerous arithmetic issues that obstruct getting a compact leading-order approximation for the splitting size, most of which can be overcome in the case of three degrees of freedom, as our example shows. These obstacles can be also possibly avoided in the same fashion for an arbitrary n ≥ 3 if one treats the case when the frequencies of the rotors are near a resonance of multiplicity n − 3 or n − 2.

Letter singularities of integrable and near-integrable hamiltonian systems

The aim of this Letter is to show that the singularities of integrable Hamiltonian systems, besides being important for such systems themselves, also have many applications in the study of near-integrable systems. In particular, we will show how they are related to Kolmogorov’s nondegeneracy condition (in the famous KAM theorem), the Poincare-Melnikov function and its generalizations, topological entropy, and nonintegrability criteria.

LetterSingularities of Integrable and Near-Integrable Hamiltonian Systems

LetterSingularities of Integrable and Near-Integrable Hamiltonian Systems

A λ-Lemma for Partially Hyperbolic Tori and the Obstruction Property

Using a scheme given by Marco, we prove that partially hyperbolic tori along resonant surfaces of near-integrable Hamiltonian systems possess the obstruction property in Arnold's terminology. The proof is based on a specific lambda lemma for these tori.

Studies of Melnikov method and transversal homoclinic orbits in the circular planar restricted three-body problem

Non-Hamiltonian systems containing degenerate fixed points obtained from two degrees of freedom near-integrable Hamiltonian systems through non-canonical transformations are dealt with in this paper. Two criteria for determining the existence of transversal hontoclinic and heteroclinic orbits are presented. By exploiting these criteria the existence of the transversal homoclinic orbits and so, of the transversal homoclinic tangle phenomenon in the near-integrable circular planar restricted three-body problem with sufficiently small mass ratio of the two primaries is proven. Under some assumptions, the existence of the transversal heteroclinic orbits is proven. The global qualitative phase diagram is also illustrated.

Effective Stability and KAM Theory

The two main stability results for nearly-integrable Hamiltonian systems are revisited: Nekhoroshev theorem, concerning exponential lower bounds for the stability time (effective stability), and KAM theorem, concerning the preservation of a majority of the nonresonant invariant tori (perpetual stability). To stress the relationship between both theorems, a common approach is given to their proof, consisting of bringing the system to a normal form constructed through the Lie series method. The estimates obtained for the size of the remainder rely on bounds of the associated vectorfields, allowing one to get the “optimal” stability exponent in Nekhoroshev theorem for quasiconvex systems. On the other hand, a direct and complete proof of the isoenergetic KAM theorem is obtained. Moreover, a modification of the proof leads to the notion of nearly-invariant torus, which constitutes a bridge between KAM and Nekhoroshev theorems.

Compensations in small divisor problems

A general direct method, alternative to KAM theory, apt to deal with small divisor problems in the real-analytic category, is presented and tested on several small divisor problems including the construction of maximal quasi-periodic solutions for nearly-integrable non-degenerate Hamiltonian or Lagrangian systems and the construction of lower dimensional resonant tori for nearly-integrable Hamiltonian systems. The method is based on an explicit graph theoretical representation of the formal power series solutions, which allows to prove compensations among the monomials forming such representation.

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.

On the spectral analysis of trajectories in near-integrable Hamiltonian systems

Spectral properties associated with the deformation of tori and the transition to chaos in near-integrable Hamiltonian systems are studied. Information about the construction of tori is provided by studying the evolution of the integrals of the unperturbed system when a perturbation is added. The authors show that the low band of the power spectrum converges exponentially for regular trajectories but they pass abruptly to 1/f alpha divergence when chaos occurs. These results are valid for systems of two or more degrees of freedom and provide a clear distinction between regular and chaotic motion.

Instability-driven energy transport in near-integrable, many degrees-of-freedom, Hamiltonian systems.

We report a numerical study of near-integrable Hamiltonian systems with many degrees of freedom. Recently reported transition phenomena for a coarse discrete sine-Gordon system are reexamined in the near-integrable limit and with the aid of statistical and integrable diagnostics. We show that the critical energy of transition from recurrence to chaotic transport of energy coincides with the onset, in the initial data, of integrable exponential instabilities and associated homoclinic orbits. This Letter aims to tie together classical and recent numerical experiments and theory of lattice instabilities and integrable systems.