Multidimensional Symplectic Separatrix Maps

Treschev, Dmitry

doi:10.1007/s00332-001-0460-2

Cited by 40 publications

(84 citation statements)

References 8 publications

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“…This implies the interior property of the minimizer of (29) and proves the first part of the lemma. We now prove the existence of the shadowing trajectory.…”

Section: Shadowing or "Diffusing" Trajectoriessupporting

confidence: 58%

“…The beautiful new idea there is to use not only classical "primary" tori, but also "secondary" tori. In [29] Treschev investigates "separatrix maps", i.e., maps along a homolinic orbit, and in [30], based on properties of these maps, he not only proves existence of diffusing orbits but also shows that the optimal diffusion speed is ε log 1/ε.…”

Section: Vadim Kaloshin and Mark Levimentioning

confidence: 99%

“…For the convenience of the reader we shall proceed in two steps. First we construct an auxiliary variational problem (29) and prove it has a solution satisfying the equations of motion (2) apart from finitely many "corners". We then modify the variational problem (see (37)) so that the corresponding solution is a true solution of the Euler-Lagrange equations and that it ε-shadows the pseudo-trajectory.…”

Section: Shadowing or "Diffusing" Trajectoriesmentioning

confidence: 99%

“…We will show, however, that in the vicinity of this pseudo-solution there exists a true solution. We will do so by Mather's method of modifying the c-action (29) so that its critical point corresponds to a true trajectory, while keeping this modification small enough to preserve the existence of an interior minimizer.…”

Section: Shadowing or "Diffusing" Trajectoriesmentioning

confidence: 99%

“…First, following ideas of Mather and Peierls, we define the Mather-Peierls' barrier function. Second, we use the barrier function to define a la Mather a variational functional (see (29)) which we then show to have a minimizer. 6 This minimizer has the desired "shadowing" properties as will be seen later; however, the solution does not satisfy the Euler-Lagrange equation of L. We then show that near this "nonsolution" there exists a solution of the Euler-Lagrange equation of L satisfying the conditions below of the main theorem (see Lemma 4).…”

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confidence: 99%

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An example of Arnold diffusion for near-integrable Hamiltonians

Калошин

Levi

2008

Bull. Amer. Math. Soc.

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Abstract. In this paper, using the ideas of Bessi and Mather, we present a simple mechanical system exhibiting Arnold diffusion. This system of a particle in a small periodic potential can be also interpreted as ray propagation in a periodic optical medium with a near-constant index of refraction. Arnold diffusion in this context manifests itself as an arbitrary finite change of direction for nearly constant index of refraction.

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“…This implies the interior property of the minimizer of (29) and proves the first part of the lemma. We now prove the existence of the shadowing trajectory.…”

Section: Shadowing or "Diffusing" Trajectoriessupporting

confidence: 58%

Section: Vadim Kaloshin and Mark Levimentioning

confidence: 99%

Section: Shadowing or "Diffusing" Trajectoriesmentioning

confidence: 99%

Section: Shadowing or "Diffusing" Trajectoriesmentioning

confidence: 99%

mentioning

confidence: 99%

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An example of Arnold diffusion for near-integrable Hamiltonians

Калошин

Levi

2008

Bull. Amer. Math. Soc.

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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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Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation

Delshams

Gidea

Llave

et al.

Hamiltonian Dynamical Systems and Applications

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We present (informally) some geometric structures that imply instability in Hamiltonian systems. We also present some finite calculations which can establish the presence of these structures in a given near integrable systems or in systems for which good numerical information is available. We also discuss some quantitative features of the diffusion mechanisms such as time of diffusion, Hausdorff dimension of diffusing orbits, etc.

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Multidimensional Symplectic Separatrix Maps

Cited by 40 publications

References 8 publications

An example of Arnold diffusion for near-integrable Hamiltonians

An example of Arnold diffusion for near-integrable Hamiltonians

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation

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