A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems

In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform explicitly the computations along the proof, which contribute to present in an easily understandable way the geometric mechanism of diffusion. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifol

Arnold diffusion for a complete family of perturbations with two independent harmonics

We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps. A complete description of the different kinds of scattering maps taking place as well as the existence of piecewise smooth global scattering maps is also provided.Comment: 23 pages, 14 figure

A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies

The aim of this work is to provide asymptotic estimates for the splitting of separatrices in a perturbed 3-degree-of-freedom Hamiltonian system, associated to a 2-dimensional whiskered torus (invariant hyperbolic torus) whose frequency ratio is a quadratic irrational number. We show that the dependence of the asymptotic estimates on the perturbation parameter is described by some functions which satisfy a periodicity property, and whose behavior depends strongly on the arithmetic properties of the frequencies.Comment: 5 pages, 1 figur

Examples of integrable and non-integrable systems on singular symplectic manifolds

We present a collection of examples borrowed from celestial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization transformations, Appell's transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lowering its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with singularities which are mainly of two types: $b^m$-symplectic and $m$-folded symplectic structures. These examples comprise the three body problem as non-integrable exponent and some integrable reincarnations such as the two fixed-center problem. Given that the geometrical and dynamical properties of $b^m$-symplectic manifolds and folded symplectic manifolds are well-understood [GMP, GMP2, GMPS, KMS, Ma, CGP, GL,GLPR, MO, S, GMW], we envisage that this new point of view in this collection of examples can shed some light on classical long-standing problems concerning the study of dynamical properties of these systems seen from the Poisson viewpoint.Comment: 14 page

Psi-series of quadratic vector fields on the plane

Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields on the plane are considered. Its existence and convergence is studied, and an algorithm for the location of logarithmic singularities is developed. Moreover, the relationship between psi-series and non-integrability is stressed and in particular it is proved that quadratic systems with psi-series that are not Laurent series do not have an algebraic first integral. Besides, a criterion about non-existence of an analytic first integral is given

On bifurcations of area-preserving and nonorientable maps with quadratic homoclinic tangencies

We study bifurcations of nonorientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable two-dimensional surfaces. We consider one- and two-parameter general unfoldings and establish results related to the emergence of elliptic periodic orbitPostprint (author’s final draft

Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems

In the present paper we consider the case of a general \cont{r+2} perturbation, for $r$ large enough, of an a priori unstable Hamiltonian system of $2+1/2$ degrees of freedom, and we provide explicit conditions on it, which turn out to be \cont{2} generic and are verifiable in concrete examples, which guarantee the existence of Arnold diffusion. This is a generalization of the result in Delshams et al., \emph{Mem. Amer. Math. Soc.}, 2006, where it was considered the case of a perturbation with a finite number of harmonics in the angular variables. The method of proof is based on a careful analysis of the geography of resonances created by a generic perturbation and it contains a deep quantitative description of the invariant objects generated by the resonances therein. The scattering map is used as an essential tool to construct transition chains of objects of different topology. The combination of quantitative expressions for both the geography of resonances and the scattering map provides, in a natural way, explicit computable conditions for instability

Poincaré-Melnikov-Arnold method for twist maps

The Poincar\'e--Melnikov--Arnold method is the standard tool for detecting splitting of invariant manifolds for systems of ordinary differential equations close to ``integrable'' ones with associated separatrices. This method gives rise to an integral (continuous sum) known as the Melnikov function (or Melnikov integral). If this function is not identically zero, the separatrices split. Moreover, the non-degenerate zeros of this function are associated to transversal intersections of the perturbed invariant (stable and unstable) manifolds. There exists a similar theory for planar maps, and in this case the Melnikov function is not a continuous sum anymore, but an infinite and (a priori) analytically uncomputable (discrete) sum. In a previous work, we have given a method to compute explicitly this kind of sums in terms of elliptic functions, under hypotheses of meromorphicity over the functions in the sum. This method allows us to obtain a strong non-integrability criterion and to apply it to perturbations of elliptic billiards and integrable standard-like maps like the McMillan map. Explicit estimates of the splitting angles are also given. Our aim is extend this method to the study of the splitting of doubly asymptotic manifolds (separatrices) associated to hyperbolic fixed points of twist maps in arbitrary dimensions. We work with maps generated globally by a generating function. Using the variational principle satisfied by these maps, we associate the non-degenerated critical points of a scalar function (here called Melnikov potential) to the transversal intersections of the perturbed asymptotic manifolds. We want to stress the difference of this point of view with the usual one in the literature, that is based in the study of non-degenerated zeros of a vectorial function. The symplectic structure and the variational principle play a fundamental role in our construction. As a first example where this theory can be applied, we study standard-like perturbations of a $2d$-dimensional twist map given by~R. McLachlan, for $d\ge 2$. This map is a multidimensional generalization of the McMillan map. We prove, among other results, that any entire perturbation destroys the separatrix of the McLachlan map

On Birkhoff's conjecture about convex billiards

Birkhoff conjectured that the elliptic billiard was the only integrable convex billiard. Here we prove a local version of this conjecture: any non-trivial symmetric entire perturbation of an elliptic billiard is non-integrable