A λ-lemma for normally hyperbolic invariant manifolds

Cresson, Jacky; Wiggins, Stephen

doi:10.1134/s1560354715010074

Cited by 6 publications

(7 citation statements)

References 29 publications

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“…We apply two versions of the lambda lemma [26,27,50,65,85], and derive two transversality properties. The first version is concerned with the asymptotic behavior of the backwards iterates of an .n s /-dimensional manifold transverse to W u .ƒ/.…”

Section: 11bmentioning

confidence: 99%

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Gidea

Llave

Seara

2019

Comm Pure Appl Math

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We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three‐body problem and the spatial circular restricted three‐body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well‐known “two‐dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower‐dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry‐Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.

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Section: 11bmentioning

confidence: 99%

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Gidea

Llave

Seara

2019

Comm Pure Appl Math

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“…Thus, by contraction mapping principle, given any (u 0 , v 0 , z k ) such that u 0 , z k ≤ δ there exists indeed a unique length-k orbit with the given values of u 0 , v 0 and z k . We already proved that this orbit must satisfy (20) and (21). Since v 0 ∈ A implies F i 0 v 0 ∈ A for all i = 0, .…”

Section: Moreover Asmentioning

confidence: 81%

“…. , k by the invariance of A with respect to F 0 , estimates (20) and (21) imply that the orbit lies in Z δ as required.…”

Section: Moreover Asmentioning

confidence: 93%

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Arnold Diffusion in a priory chaotic Hamiltonian systems

Vassily¹,

Turaev²

2014

Preprint

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“…Then, the local stable and unstable manifolds are propagated backward and forward in time to extend them to the global stable and unstable manifolds. These two manifolds are no longer cylindrical, but trajectories in the vicinity of these manifolds are expected to follow these manifold due to the λ lemma [22]. Figure 1(b) shows that these stable and unstable manifolds emanate toward the x 3 direction, which is perpendicular to the original ξ direction.…”

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confidence: 99%

Mechanism and Experimental Observability of Global Switching Between Reactive and Nonreactive Coordinates at High Total Energies

et al. 2015

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We present a mechanism of global reaction coordinate switching, namely, a phenomenon in which the reaction coordinate dynamically switches to another coordinate as the total energy of the system increases. The mechanism is based on global changes in the underlying phase space geometry caused by a switching of dominant unstable modes from the original reactive mode to another nonreactive mode in systems with more than 2 degrees of freedom. We demonstrate an experimental observability to detect a reaction coordinate switching in an ionization reaction of a hydrogen atom in crossed electric and magnetic fields. For this reaction, the reaction coordinate is a coordinate along which electrons escape and its switching changes the escaping direction from the direction of the electric field to that of the magnetic field and, thus, the switching can be detected experimentally by measuring the angle-resolved momentum distribution of escaping electrons. DOI: 10.1103/PhysRevLett.115.093003 PACS numbers: 32.80.Rm, 05.45.Ac, 45.20.Jj, 82.20.-w In the conventional picture of chemical reactions, a system starts from a reactant and ends up with a product by overcoming a first-rank saddle in between the two. If the total energy of a system is close enough to the potential energy of the saddle, a harmonic approximation of the potential energy surface at the saddle is valid during the time interval in which the system traverses through the saddle. Then, the direction toward which the reaction occurs is determined by the unstable mode at the saddle. As the total energy increases, the nonlinearity of the potential energy starts to play an important role even in the vicinity of the saddle, and the nonlinear couplings between the unstable mode and the other modes become no longer negligible.At the total energy in which the harmonic terms dominate the nonlinear terms, one can still take them into account in a perturbative manner [1][2][3][4][5] so that the resulting perturbed unstable mode is decoupled from the other perturbed modes and the perturbed unstable mode is hyperbolic. A key geometrical object behind the perturbed modes is a normally hyperbolic invariant manifold (NHIM) [6,7], defined as a zero level set of the perturbed unstable mode and its conjugate momentum on an equienergy surface [5]. The NHIM has a pair of stable and unstable normal directions that are linear combinations of this perturbed unstable mode and its conjugate momentum. There, the stable and unstable manifolds emanating in the normal directions determine the directions toward which the reaction occurs. Specifically, if the NHIM has a topology of a hypersphere, these manifolds are cylindrical and, thus, are called a cylindrical stable manifold and a cylindrical unstable manifold, respectively. In that case, every reactive trajectory should run through the cylindrical stable manifold and then run through the cylindrical unstable manifold [5,[8][9][10][11][12].If the total energy increases even further, the perturbative construction of the NHIM starts to fai...

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A λ-lemma for normally hyperbolic invariant manifolds

Abstract: Abstract. Let N be a smooth manifold and f : N → N be a C ℓ , ℓ ≥ 2 diffeomorphism. Let M be a normally hyperbolic invariant manifold, not necessarily compact. We prove an analogue of the λ-lemma in this case.

Cited by 6 publications

References 29 publications

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Arnold Diffusion in a priory chaotic Hamiltonian systems

Mechanism and Experimental Observability of Global Switching Between Reactive and Nonreactive Coordinates at High Total Energies

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