Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits
Amadeu Delshams Pere Gutiérrez
We consider an example of singular or weakly hyperbolic Hamiltonian, with $3$ degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The model consists of an integrable Hamiltonian possessing a $2$-dimensional hyperbolic invariant torus with fast frequencies $\omega/\sqrt\varepsilon$ and coincident whiskers, plus a perturbation of order $\mu=\varepsilon^p$. We choose $\omega$ as the golden vector. Our aim is to obtain asymptotic estimates for the splitting, proving the existence of transverse intersections between the perturbed whiskers for $\varepsilon$ small enough, by applying the Poincaré-Melnikov method together with a accurate control of the size of the error term.
The good arithmetic properties of the golden vector allow us to prove that the splitting function has 4 simple zeros (corresponding to nondegenerate critical points of the splitting potential), giving rise to 4 transverse homoclinic orbits. More precisely, we show that a shift of these orbits occurs when $\varepsilon$ goes across some critical values, but we establish the continuation (without bifurcations) of the 4 transverse homoclinic orbits for all values of $\varepsilon\to0$.
keywords: arithmetic properties of frequencies transverse homoclinic orbits. Poincaré-Melnikov method
Tomás Caraballo Amadeu Delshams Àngel Jorba Peter E. Kloeden Rafael Obaya
The theory of dynamical systems has undergone some spectacular and fascinating developments in the past century, as the readers of this journal are well aware, with the focus predominately on autonomous systems. There are many ways in which one could classify the work that has been done, but one that stands clearly in the forefront is the distinction between dissipative systems with their attractors and conservative systems, in particular Hamiltonian systems.
    Another classification is between autonomous and nonautonomous systems. Of course, the latter subsumes the former as special case, but with the former having special structural features, i.e., the semigroup evolution property, which has allowed an extensive and seemingly complete theory to be developed. Although not as extensive, there have also been significant developments in the past half century on nonautonomous dynamical systems, in particular the skew-product formalism involving a cocycle evolution property which generalizes the semigroup property of autonomous systems. This has been enriched in recent years by advances on random dynamical systems, which are roughly said a measure theoretic version of a skew-product flow. In particular, new concepts of random and nonautonomous attractors have been introduced and investigated.

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Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds
Amadeu Delshams Pere Gutiérrez Tere M. Seara
We consider a singular or weakly hyperbolic Hamiltonian, with $n+1$ degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The model consists of an integrable Hamiltonian possessing an $n$-dimensional hyperbolic invariant torus with fast frequencies $\omega/\sqrt\varepsilon$ and coincident whiskers, plus a perturbation of order $\mu=\varepsilon^p$. The vector $\omega$ is assumed to satisfy a Diophantine condition.
We provide a tool to study, in this singular case, the splitting of the perturbed whiskers for $\varepsilon$ small enough, as well as their homoclinic intersections, using the Poincaré--Melnikov method. Due to the exponential smallness of the Melnikov function, the size of the error term has to be carefully controlled. So we introduce flow-box coordinates in order to take advantage of the quasiperiodicity properties of the splitting. As a direct application of this approach, we obtain quite general upper bounds for the splitting.
keywords: flow-box coordinates Poincaré–Melnikov method Hyperbolic KAM theory
Amadeu Delshams Ernest Fontich Àngel Jorba
This volume is dedicated to Carles Simó on the occasion of his 60th anniversary and contains papers of the participants of the conference “International Conference on Dynamical Systems, Carles Simó Fest”, celebrated in S’Agaró, near Barcelona, from May 29th to June 3rd, 2006 and organized by former PhD students of him.
  &nbs Carles Simó was born in Barcelona in 1946. He studied Industrial Engineering and Mathematics at once, earning his Ph.D. in Mathematics in 1974.

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Computing the scattering map in the spatial Hill's problem
Amadeu Delshams Josep J. Masdemont Pablo Roldán
Let $A_1$ and $A_2$ be two normally hyperbolic invariant manifolds for a flow, such that the stable manifold of $A_1$ intersects the unstable manifold of $A_2$ transversally along a manifold Γ. The scattering map from $A_2$ to $A_1$ is the map that, given an asymptotic orbit in the past, associates the corresponding asymptotic orbit in the future through a heteroclinic orbit. It was originally introduced to prove the existence of orbits of unbounded energy in a perturbed Hamiltonian problem using a geometric approach.
    We recently computed the scattering map in the planar restricted three body problem using non-perturbative techniques, and we showed that it is a (nontrivial) integrable twist map.
    In the present paper, we compute the scattering map in a problem with three degrees of freedom using also non-perturbative techniques. Specifically, we compute the scattering map between the normally hyperbolic invariant manifolds $A_1$ and $A_2$ associated to the equilibrium points $L_1$ and $L_2$ in the spatial Hill's problem.
    In the planar problem, for each energy level (in a certain range) there is a unique Lyapunov periodic orbit around $L_{1,2}$. In the spatial problem, this periodic orbit is replaced by a three-dimensional invariant manifold practically full of invariant 2D tori. There are heteroclinic orbits between $A_1$ and $A_2$ connecting these invariant tori in rich combinations. Hence the scattering map in the spatial problem is more complicated, and it allows nontrivial transition chains.
    Scattering maps have application to e.g. mission design in Astrodynamics, and to the construction of diffusion orbits in the spatial Hill's problem.
keywords: homoclinic and heteroclinic trajectories Restricted Three-Body Problem scattering maps normally hyperbolic invariant manifolds quasi-periodic motions Arnold diffusion. normal forms invariant tori
Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies
Amadeu Delshams Marina Gonchenko Pere Gutiérrez
We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector $\omega=(1,\Omega)$, where $\Omega$ is a quadratic irrational number, or a 3-dimensional torus with a frequency vector $\omega=(1,\Omega,\Omega^2)$, where $\Omega$ is a cubic irrational number. Applying the Poincaré--Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which $\Omega$ is the so-called cubic golden number (the real root of $x^3+x-1=0$), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.
keywords: Melnikov integrals Splitting of separatrices quadratic and cubic frequencies.
Amadeu Delshams Marian Gidea Ernesto Pérez-Chavela
The material of this special issue of DCDS-A was originally dedicated in honor of the 65-th birthday of Prof. Ernesto A. Lacomba. Some of the papers in this issue reflect the joyful spirit surrounding this celebration. Sadly, shortly after the preparation of this volume was completed, Prof. Ernesto A. Lacomba passed away on June 26, 2012. Therefore this special issue is also paying a tribute to his long standing mathematical legacy.
    The work of Prof. Lacomba comprised research on geometric theory of ordinary differential equations, dynamical systems, and symplectic geometry, with applications to celestial mechanics, classical mechanics, vortex theory, thermodynamics and electrical circuits. Prof.~Lacomba was the leader of a strong research group working in these areas. In 1991 he started organizing, jointly with some members of his group and with other collaborators, the International Symposium on Hamiltonian Systems and Celestial Mechanics (HAMSYS), which became a great success over the next several years. These symposia brought together top researches from several countries, working in the aforementioned topics, as well as many graduate students who had the opportunity to learn from and connect with the experts in the field, and often get inspiration and motivation to improve and finalize their doctoral theses.
    The framework for the celebration of the 65-th birthday of Prof. Lacomba was the VI-th edition of HAMSYS, which was held in México D.F. between November 29 -- December 3, 2010.
    This symposium assembled an impressive number of highly respected researches who generated important discussions among the participants, presented new problems, and identified future research directions. The emphasis of the talks was on Hamiltonian dynamics and its relationship to several aspects of mechanics, geometric mechanics, and dynamical systems in general. The papers in this volume are an outgrowth of the themes of the symposium. All papers that were submitted to this special issue underwent a through refereeing process typical to any top mathematical journal. The accepted papers form the present issue of DCDS-A.
    The symposium received generous support from CONACYT México and UAM-I. Special thanks are due to Universidad Autónoma Metropolitana for hosting the symposium in the beautiful colonial building Casa de la primera imprenta de América. Last but not least, we thank all participants for contributing to a week-long intense and highly productive mathematical experience.
Transition map and shadowing lemma for normally hyperbolic invariant manifolds
Amadeu Delshams Marian Gidea Pablo Roldán
For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in a Hamiltonian system. This consists in the following steps: (i) computation of the scattering map and of the transition map for the Hamiltonian flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows. We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem.
keywords: Hamiltonian instability the three-body problem. Arnold diffusion

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