Topological methods in the instability problem of Hamiltonian systems

Gidea, Marian; Llave, Rafael de la

doi:10.3934/dcds.2006.14.295

Cited by 54 publications

(82 citation statements)

References 30 publications

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“…The approaches used in these works range from variational to geometric/topological. This paper is more in line with the geometric method developed in [21,32,31,33].…”

Section: Methodssupporting

confidence: 59%

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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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“…The approaches used in these works range from variational to geometric/topological. This paper is more in line with the geometric method developed in [21,32,31,33].…”

Section: Methodssupporting

confidence: 59%

“…We mention that transition chains of level sets of the action also appear in [32], but they have not been numerically implemented before. Now we briefly explain the construction of transition chains of action level sets.…”

Section: Methodsmentioning

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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“…An interesting particular case is when the motion given by f 0 is integrable when restricted to the invariant manifold Λ = k 0 (N ). This situation occurs in the problems considered in [DLS00,DLS03,DLS06a,DLS06b,GL06a,GL06b]. In such a case, the dynamics by f 0 on Λ has a simple expression and one can carry the perturbation theory to all orders in ε less or equal than r. In those papers, one can find detailed perturbative formulas to order m ≤ r with error estimates.…”

Section: A Geometric Framework For a Perturbative Calculation Of The mentioning

confidence: 99%

“…The use of the scattering map was crucial for the applications in [DLS03,DLS06a,GL06a,GL06b]. In those papers, it was shown that the perturbative computations of the scattering map are a convenient improvement of the Melnikov method since they are geometrically natural.…”

Section: Introductionmentioning

confidence: 99%

Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Delshams

Llave

Seara

2008

Advances in Mathematics

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104

273

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Given a normally hyperbolic invariant manifold Λ for a map f , whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future. We show that when f and Λ are symplectic (resp. exact symplectic) then, the scattering map is symplectic (resp. exact symplectic). Furthermore, we show that, in the exact symplectic case, there are extremely easy formulas for the primitive function, which have a variational interpretation as difference of actions. We use this geometric information to obtain efficient perturbative calculations of the scattering map using deformation theory. This perturbation theory generalizes and extends several results already obtained using the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained by geometric natural methods and do not involve the use of particular coordinate systems, hence the results can be used to obtain intersection properties of objects of any type. We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodic potential. We show that the geometric theory reproduces the results obtained in [DLS06b] using methods of fast-slow systems. Moreover, the geometric theory allows to compute perturbatively the dependence on the slow variables, which does not seem to be accessible to the previous methods.

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“…Topological methods were previously applied to the Arnold diffusion problem in [35,28,21,23]. Transition chains of tori formed with topologically crossing heteroclinic connections were considered in [22].…”

Section: Marian Gidea and Clark Robinsonmentioning

confidence: 99%

Obstruction argument for transition chains of tori interspersed with gaps

Gidea

Robinson²

2009

Discrete &Amp; Continuous Dynamical Systems - S

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We consider a dynamical system whose phase space contains a two-dimensional normally hyperbolic invariant manifold diffeomorphic to an annulus. We assume that the dynamics restricted to the annulus is given by an area preserving monotone twist map. We assume that in the annulus there exist finite sequences of primary invariant Lipschitz tori of dimension 1, with the property that the unstable manifold of each torus has a topologically crossing intersection with the stable manifold of the next torus in the sequence. We assume that the dynamics along these tori is topologically transitive. We assume that the tori in these sequences, with the exception of the tori at the ends of the sequences, can be C 0-approximated from both sides by other primary invariant tori in the annulus. We assume that the region in the annulus between two successive sequences of tori is a Birkhoff zone of instability. We prove the existence of orbits that follow the sequences of invariant tori and cross the Birkhoff zones of instability.

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Topological methods in the instability problem of Hamiltonian systems

Cited by 54 publications

References 30 publications

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

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

Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Obstruction argument for transition chains of tori interspersed with gaps

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