Transition map and shadowing lemma for normally hyperbolic invariant manifolds

Delshams, Amadeu; Gidea, Marian; Roldán, Pablo

doi:10.3934/dcds.2013.33.1089

Cited by 29 publications

(30 citation statements)

References 16 publications

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“…Phenomena similar to the ones studied in this paper appear in the problem of Arnold's diffusion for nearly integrable Hamiltonian systems near a multiple resonance [2,11,14,20,26]. Our research is also closely related to the theory of scattering maps [13] and of separatrix maps [25].…”

Section: Introductionsupporting

confidence: 55%

Shadowing Homoclinic Chains to a Symplectic Critical Manifold

sci¹

2021

ATA

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We prove the existence of trajectories shadowing chains of heteroclinic orbits to a symplectic normally hyperbolic critical manifold of a Hamiltonian system. The results are quite different for real and complex eigenvalues. General results are applied to Hamiltonian systems depending on a parameter which slowly changes with rate ε. If the frozen autonomous system has a hyperbolic equilibrium possessing transverse homoclinic orbits, we construct trajectories shadowing homoclinic chains with energy having quasirandom jumps of order ε and changing with average rate of order ε| ln ε|. This provides a partial multidimensional extension of the results of A. Neishtadt on the destruction of adiabatic invariants for systems with one degree of freedom and a figure 8 separatrix.

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Section: Introductionsupporting

confidence: 55%

Shadowing Homoclinic Chains to a Symplectic Critical Manifold

sci¹

2021

ATA

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“…In many examples, the scattering map can be computed explicitly via perturbation theory [31,34,35] or numerically [18,19,39,40].…”

Section: Normally Hyperbolic Invariant Manifolds and Scattering Mapsmentioning

confidence: 99%

“…Remark 5.3. Statements related to Lemma 3.1 appear in [39,40,60]. The main difference is that the statements in these papers assume certain geometric conditions on the inner dynamics.…”

Section: Remarksmentioning

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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“…The proof of Theorem 2.4, given in Section 3, relies on the theory of normal hyperbolicity, on the geometric properties of scattering maps, and on the Poincaré recurrence theorem. Relevant references to our approach include [6,22,21,17,33,7].…”

Section: Methodsmentioning

confidence: 99%

“…We also note that numerical computation of the scattering map, similar to this one, in different models from celestial mechanics, have been performed in [6,22]. A related concept to that of scattering map that is more closely associated to trajectories in the phase space is that of the transition map, introduced in [17]; with the notation from above, a transition map is given by the map y − → y + . 4.5.…”

Section: 4mentioning

confidence: 99%

Arnold’s mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument

Delshams¹,

Gidea

Roldán

2016

Physica D: Nonlinear Phenomena

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We consider the spatial circular restricted three-body problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3-degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L 1 , located between Sun and Earth. Near L 1 there exists a normally hyperbolic invariant manifold, diffeomorphic to a 3-sphere. For an orbit confined to this 3-sphere, the amplitude of the motion relative to the ecliptic (the plane of the orbits of Sun and Earth) can vary only slightly. We show that we can obtain new orbits whose amplitude of motion relative to the ecliptic changes significantly, by following orbits of the flow restricted to the 3-sphere alternatively with homoclinic orbits that turn around the Earth. We provide an abstract theorem for the existence of such 'diffusing' orbits, and numerical evidence that the premises of the theorem are satisfied in the three-body problem considered here. We provide an explicit construction of diffusing orbits. The geometric mechanism underlying this construction is reminiscent of the Arnold diffusion problem for Hamiltonian systems. Our argument, however, does not involve transition chains of tori as in the classical example of Arnold. We exploit mostly the 'outer dynamics' along homoclinic orbits, and use very little information on the 'inner dynamics' restricted to the 3-sphere. As a possible application to astrodynamics, diffusing orbits as above can be used to design low cost maneuvers to change the inclination of an orbit of a satellite near L 1 from a nearly-planar orbit to a tilted orbit with respect to the ecliptic. We explore different energy levels, and estimate the largest orbital inclination that can be achieved through our construction.

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Transition map and shadowing lemma for normally hyperbolic invariant manifolds

Cited by 29 publications

References 16 publications

Shadowing Homoclinic Chains to a Symplectic Critical Manifold

Shadowing Homoclinic Chains to a Symplectic Critical Manifold

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Arnold’s mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument

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