Imperial College London, South Kensington Campus, London SW7 2AZ
Department of Mathematics, Imperial College London
Delshams A., Gutierrez P., Koltsova O. Y., Pacha J. R.
Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré–Mel’nikov method
2010, vol. 15, no. 2-3, pp. 222-236
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic–hyperbolic type, having a homoclinic orbit. More precisely, we consider an $(n+2)$-degree-of-freedom near integrable Hamiltonian with $n$ centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center 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).
Koltsova O. Y.
Families of multi-round homoclinic and periodic orbits near a saddle-center equilibrium
2003, vol. 8, no. 2, pp. 191-200
We consider a real analytic two degrees of freedom Hamiltonian system possessing a homoclinic orbit to a saddle-center equilibrium $p$ (two nonzero real and two nonzero imaginary eigenvalues). We take a two-parameter unfolding for such a system and show that in the case of nonresonance there are countable sets of multi-round homoclinic orbits to $p$. We also find families of periodic orbits, accumulating a the homoclinic orbits.