We develop a method to study resonances for three-dimensional frequency vectors ω in the way that... more We develop a method to study resonances for three-dimensional frequency vectors ω in the way that was done in for the two-dimensional case. We consider the case of cubic frequency vectors of the form ω = (1, Ω, Ω 2 ), where Ω is an irrational root of a polynomial p(x) of degree 3, with integer coefficients. In the complex case (when p(x) has one real and a pair of complex conjugate roots), it is proved in [3] that the frequency vector satisfies a Diophantine condition
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of ell... more We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptichyperbolic type, having a homoclinic orbit. More precisely, we consider an (n+2)-degree-of-freedom near integrable Hamiltonian with n centres and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the centre manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel ′ nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).
We develop a method to study resonances for three-dimensional frequency vectors ω in the way that... more We develop a method to study resonances for three-dimensional frequency vectors ω in the way that was done in for the two-dimensional case. We consider the case of cubic frequency vectors of the form ω = (1, Ω, Ω 2 ), where Ω is an irrational root of a polynomial p(x) of degree 3, with integer coefficients. In the complex case (when p(x) has one real and a pair of complex conjugate roots), it is proved in [3] that the frequency vector satisfies a Diophantine condition
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of ell... more We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptichyperbolic type, having a homoclinic orbit. More precisely, we consider an (n+2)-degree-of-freedom near integrable Hamiltonian with n centres and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the centre manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel ′ nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).
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Papers by P. Gutiérrez