No abstract

The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a torus with a fast frequency vector ω/ √ ε, with ω = (1, Ω) where the frequency ratio Ω is a quadratic irrational number. Applying the Poincaré-Melnikov method, we carry out a careful study of the dominant harmonics of the Melnikov potential. This allows us to provide an asymptotic estimate for the maximal splitting distance, and show the existence of transverse homoclinic orbits to the whiskered tori with an asymptotic estimate for the transversality of the splitting. Both estimates are exponentially small in ε, with the functions in the exponents being periodic with respect to ln ε, and can be explicitly constructed from the continued fraction of Ω. In this way, we emphasize the strong dependence of our results on the arithmetic properties of Ω. In particular, for quadratic ratios Ω with a 1-periodic or 2-periodic continued fraction (called metallic and metallic-colored ratios respectively), we provide accurate upper and lower bounds for the splitting. The estimate for the maximal splitting distance is valid for all sufficiently small values of ε, and the transversality can be established for a majority of values of ε, excluding small intervals around some transition values where changes in the dominance of the harmonics take place, and bifurcations could occur. Applied Math cluster system for research computing (see http://www.ma1.upc.edu/eixam/), and in particular Pau Roldán and Albert Granados for their support in the use of this cluster.

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.Newhouse Theorem [33]. Let M 2 be a C ∞ compact 2-dimensional manifold and let r ≥ 2. Assume that f ∈ Diff r (M 2 ) has a hyperbolic set whose stable and unstable

We study the exponentially small splitting of invariant manifolds of whiskered\ud (hyperbolic) tori with two fast frequencies in nearly integrable Hamiltonian systems whose\ud hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver\ud number Ω =\ud √\ud 2\ud −\ud 1. We show that the Poincar ́\ud e – Melnikov method can be applied to establish\ud the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic\ud estimates for the transversality of the splitting\ud whose dependence on the perturbation parameter\ud ε\ud satisfies a periodicity property. We also prove\ud the continuation of the transversality of the\ud homoclinic orbits for all the sufficiently small values of\ud ε\ud , generalizing the results previously\ud known for the golden numberPostprint (published version

We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Hénon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Hénon maps.