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

Lochak, Pierre; Marco, Jean-Pierre; Sauzin, David

doi:10.1090/memo/0775

Cited by 68 publications

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“…hyperbolic part, a Mel nikov potential was defined in [DG00], without using generating functions. Later, the approach giving a potential from the generating functions was developed in [LMS03] (see also [Sau01]). Here, we are considering a 2-d.o.f.…”

Section: Mel Nikov Integralsmentioning

confidence: 99%

Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

Delshams

Gutiérrez

Pacha

2013

Physica D: Nonlinear Phenomena

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We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and sufficient condition for the transversality, in terms of a Riccati equation whose solutions give the slope of the invariant manifolds in a direction transverse to the loop. The key point of our approach is to write the invariant manifolds in terms of generating functions, which are solutions of the Hamilton-Jacobi equation. In some examples, we show that it is enough to analyse the phase portrait of the Riccati equation without solving it explicitly. Finally, we consider an analogous problem in a perturbative situation. If the invariant manifolds of the unperturbed loop coincide, we have a problem of splitting of separatrices. In this case, the Riccati equation is replaced by a Mel nikov potential defined as an integral, providing a condition for the existence of a perturbed loop and its transversality. This is also illustrated with a concrete example.

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Section: Mel Nikov Integralsmentioning

confidence: 99%

Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

Delshams

Gutiérrez

Pacha

2013

Physica D: Nonlinear Phenomena

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“…We now give a result, due to Lazutkin in it's original form, [Laz03], and extended by [DGJS97a], [Sau01], and [LMS03] to the quasiperiodic setting. See Appendix A for the proof.…”

Section: Size Of the Homoclinic Splittingmentioning

confidence: 98%

“…The motivation for doing this stems from studying the splitting of the manifolds W s,u λ in the vicinity of the homoclinic trajectory (1.17). The general ideology that, being able to extend "splitting related functions" to a large complex domain yields good estimates, is due to Lazutkin [Laz03], as is emphasized in [LMS03].In order to study the intersection more closely, we express the actions as functions of the original angle variables (φ, ψ) = X s,u (z, θ) + (0, θ) appearing in the Hamiltonian (1.1). To this end, let F s,u : (z, θ) → (φ, ψ) be the above coordinate transformations, and write Y s,u =Ȳ s,u • F s,u .…”

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confidence: 99%

An expansion of the homoclinic splitting matrix for the rapidly, quasiperiodically, forced pendulum

Stenlund

2010

Journal of Mathematical Physics

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Abstract. We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, devising an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing hyperbolicity, by a shift-of-contour argument. Hence, we infer a similar upper bound on the splitting itself. Main Concepts and Results 1.1.Background and history. The study of "separatrix splitting" dates back to Poincaré's classic Les Méthodes Nouvelles de la Mécanique Céleste [Poi93].Starting with Kolmogorov's 1954 note [Kol54], it was proved in a series of papers over a period of twenty years that quasiperiodic motions (invariant tori) are typical for nearly integrable Hamiltonians [Mos62,Arn63,Mos66a,Mos66b,Mos67], and that motions which become quasiperiodic asymptotically in time (stable/unstable manifolds) are stable under small perturbations [Mos67,Gra74]. Arnold [Arn64] described a mechanism how a chain of such "whiskered" tori could provide a way of escape for special trajectories, resulting in instability in the system. (A trajectory would typically lie on a torus and therefore stay eternally within a bounded region in phase space.) The latter is often called Arnold mechanism and the general idea of instability goes by the name Arnold diffusion. It is conjectured in [AA68] that Arnold diffusion due to Arnold mechanism is present quite generically, for instance in the three body problem.Arnold mechanism is based on Poincaré's concept of biasymptotic solutions, discussed in the last chapter of [Poi93], that are formed at intersections of whiskers of tori. Following such intersections, a trajectory can "diffuse" in a finite time from a neighbourhood of one torus to a neighbourhood of another, and so on.Chirikov's report [Chi79] is a very nice physical account on Arnold diffusion, while Lochak's compendium [Loc99] discusses more recent developments in a readable fashion and is a good point to start learning about diffusion. Gelfreich's introduction [Gel01] to splitting of separatrices is excellent, and we recommend it to anyone intending to study the topic. From there one should advance to [GL01], which covers more topics with more details. The extensive memoir [LMS03] by Lochak, Marco, and Sauzin is written from the geometric point of view. It has a historical flavor, making it interesting and accessible to virtually anyone.For the separatrix splitting of the periodically forced pendulum, see [HMS88,SMH91,DS92, EKS93]. In the case of the standard map, an asymptotic expression of the splitting has been obtained in [Gel99] of a pendulum coupled to d rotators, with φ ∈ S 1 := R/2πZ and I ∈ R the coordinate and momentum of the pendulum, and ψ ∈ T d := (S 1 ) d and A ∈ R d the angles and actions of the rotators, respectively. The perturbation f is assumed to be real-valued and real-analytic in its arguments, and λ is a (small) real number, whereas g > 0. This Hamiltonian is sometimes called the generalized Ar...

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“…Discussions on such a definition are available in [8]. Using methods of Poincaré and Melnikov, Arnold showed that this splitting can be estimated, for sufficiently small ǫ, by…”

Section: The Mechanism Of Arnoldmentioning

confidence: 99%

Hamiltonian Systems: Stability and Instability Theory

Bernard

2006

Encyclopedia of Mathematical Physics

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The solar system has long appeared to astronomers and mathematicians as a model of stability. On the other hand, statistical mechanics relies on the assumption that large assemblies of particles form highly unstable systems (at the microscopic scale). Yet all these physical situations are described, at least to a certain degree of approximation, by Hamiltonian systems.One may hope that Hamiltonian systems can be classified in two different categories, stable and unstable ones. However, the situation is much more complicated and both stable and unstable behaviors cohabit in typical systems. Even our examples are not perfect paradigms of stability and instability. Indeed, it is now clear from numerical as well as theoretical points of views that some instability is present over long time-scales in the solar systems, so that for example future collisions between planets can not be completely ruled out in view of our present understanding. On the other hand, unexpected patterns of stability have been discovered in systems involving a large number of particles.Understanding the impact of stable and unstable effects in Hamiltonian systems has been considered since Poincaré as one of the most important questions in dynamical systems. In the present text, we will discuss model Hamiltonian systems of the formwhere (q, p) ∈ T d × U , with U a bounded open subset of R n . Recall that the equations of motion areThe textbook [1] is a good general introduction on Hamiltonian systems. We will always denote by ω(p) the frequency map ∂ p h(p), which plays a crucial role. Here, as is obvious in (2), the action variables p are preserved under the evolution in the unperturbed case ǫ = 0. We will try to explain what is known on the evolution of these action variables for the perturbed system. As we will see, in many situations, these variables are extremely stable. For example, KAM theorem implies that, for a positive measure of initial conditions (q 0 , p 0 ) the trajectory (q(t), p(t)) satisfies p(t) − p (0) Cǫ for all times. Examples show that some initial conditions may lead to unstable trajectories, that is trajectories such that p(t) − p(0) 1/C for some t (depending on ǫ) and some fixed constant C independent of ǫ. However, this is, as we will see, possible only for very large time t (meaning that t as a function of ǫ has to go to infinity very quickly when ǫ −→ 0). The main questions here are to understand in what situation instability is or is not possible, and what kind of evolutions can have the actions variable p. Another important question is to estimate the speed (as a function of the parameter ǫ) of the evolutions of p.

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

Cited by 68 publications

References 0 publications

Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

An expansion of the homoclinic splitting matrix for the rapidly, quasiperiodically, forced pendulum

Hamiltonian Systems: Stability and Instability Theory

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