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  • 1.
    Enciso, Alberto
    et al.
    CSIC, Inst Ciencias Matemat, Madrid 28049, Spain.
    Luque, Alejandro
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Peralta-Salas, Daniel
    CSIC, Inst Ciencias Matemat, Madrid 28049, Spain.
    Stationary Phase Methods and the Splitting of Separatrices2019In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 368, no 3, p. 1297-1322Article in journal (Refereed)
    Abstract [en]

    Using stationary phase methods, we provide an explicit formula for the Melnikov function of the one and a half degrees of freedom system given by a Hamiltonian system subject to a rapidly oscillating perturbation. Remarkably, the Melnikov function turns out to be computable using very little information on the separatrix and in the case of non-analytic systems. This is related to a priori stable systems coupled with low regularity perturbations. A natural physical application is to perturbations controlled by wave-type equations, so in particular we also illustrate this result with the motion of charged particles in a rapidly oscillating electromagnetic field. Quasi-periodic perturbations are discussed too.

  • 2.
    Haro, Alex
    et al.
    Univ Barcelona, Dept Matemat & Informat, Gran Via 585, E-08007 Barcelona, Spain.
    Luque, Alejandro
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    A-posteriori KAM theory with optimal estimates for partially integrable systems2019In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 266, no 2-3, p. 1605-1674Article in journal (Refereed)
    Abstract [en]

    In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invariant tori for n-DOF Hamiltonian systems with additional n - d independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.

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