The seminars are short lectures (25 minutes), related and complementing the main courses.

1. Arranz, Francisco Javier
   Scars in order-chaos transition and zeros of Husimi function.
   Scar phenomena is a well known quantum manifestation of classical periodic orbits. Scarred quantum states have been studied usually in position or momentum representation. However, when studied in a quantum phase space representation (coherent states representation or Husimi function), the distribution of zeros determines the regular, scarred or irregular character of the states. We will present results about the distribution of zeros of the Husimi function in order-chaos transition (from regular to irregular states through a border of scarred states) for LiCN molecular system.
2. Benet, Luis
   Integrability of interacting two-level boson systems.
   We prove that a wide class of interacting two-level bosons systems is integrable in the semiclassical limit. This class of models includes number conserving Hamiltonians, and interactions which do not conserve the particle number. Interesting consequences of this property are discussed.
3. Boatto, Stefanella
   Curvature perturbations and vortex stability.
   Abstract: [to be supplied later]
4. Borstnik, Urban
   Hierarchical Hypercube Topology of a Cluster of Personal Computers for Macromolecular Simulations.
   Clusters of personal computers are used to efficiently perform macromolecular dynamics simulations. We developed a cluster with a hierarchical hypercube topology that has links of different speeds for the dimenions of the hypercube. This topology enables an efficient algorithm for the transfer of data to be used in simulations, which shortens its running time.
5. Cruz-Pacheco, Gustavo
   Persistence of invariant tori in near integrable PDE's.
   We use a generalization of the Melnikov integral to find a necesary criteria for the persistence of invariant tori in a perturbed integrable PDE. We use as an example the nonlinear Schroedinger equation and we prove that in some interesting cases the criteria is sufficient.
6. Fontich, Ernest
   Slow manifolds.
   When one studies the dynamics of processes which converge to an equilibrium point, the observable dynamics is the one which corresponds to the directions associated to the eigenvalues of bigger modulus. This problem is important, among other fields, in chemical kinetics.
   Given a map with a stable equilibrium point, we consider the linear subspace defined as the sum of invariant subspaces associated to the eigenvalues $\lambda$ with $\lambda_0 < |\lambda| < 1$ and we study the existence and properties of an invariant manifold for the full system tangent to the given subspace. We call slow manifolds such manifolds. Usually the reduction of the system to such manifold permits to reduce significatively the dimension.
   The slow manifolds are a particular case of the so called non-resonant manifolds. We provide sufficient conditions for the existence, regularity, uniqueness and dependence on parameters for the non-resonant manifolds.
   We study the manifolds by the method of the parameterization, which has several advantages.
7. Haro, Alex
   New mechanisms for lack of equipartition of energy in chains of oscillators.
   We describe several mechanisms that prevent equipartition of energy in mechanical systems. In certain regimes, we present a quantitative prediction of the relative abundance of orbits exhibiting these mechanisms. This quantitative prediction is confirmed in numerical experiments.
8. Praprotnik, Matej
   Split Integration Symplectic Method for molecular dynamics integration.
   A combination of the analytical solution of the high-frequency harmonic part of the Hamiltonian and the numerical solution of the low-frequency remaining part forms the Split Integration Symplectic Method (SISM) for molecular dynamics (MD) integration. This method was tested on model systems of linear chain molecules, water molecules, and hydrogen peroxide molecules. The numerical results indicate that the integration time step used by the SISM is significantly larger than possible by standard methods.
9. Puig, Joaquim
   Cantor spectrum for the Almost Mathieu Operator. A model with a complex band structure.
   In this talk we will present the Almost Mathieu Operator, which is a tight binding model for the Hamiltonian of an electron ina one dimensional lattice, subject to a potential. It is also related to the Hamiltonian of an electron in a two dimensional lattice, subject to a perpendicular uniform magnetic field.
   The structure of the forbidden bands of the spectrum will be discussed. The tools will come from a dynamical analysis of the solutions of the associated eigenvalue equation.
10. Roldan, Pau
    Numerical computation of the scattering map.
    We consider perturbations of integrable Hamiltonian a priori unstable systems. The scattering map was first introduced in [DLS00] as a tool for discussing the intersection of the stable and unstable manifolds of two objects contained in the normally hyperbolic manifold $\Lambda$ of the system. It is a diffeomorphism $S:\Lambda \to \Lambda$ describing homoclinic connections to the normally hyperbolic manifold, and it is defined by orbits both forward and backwards asymptotic in time to $\Lambda$.
    We will discuss a perturbative method based on Newton iteration for the numerical computation of the scattering map. The method closely parallels the scattering map's theoretical construction. Moreover, the method is straightforward to implement because it doesn't rely on the explicit representation of the stable and unstable manifolds, nor on the construction of their intersection manifold by 'slices' (i.e. using a Poincaré section to reduce the dimension of the problem and find the intersection of the stable/unstable manifolds by interpolation). On the contrary, our method works directly with the full-dimensional numerical representation of the intersection manifold.
    The computation of the scattering map is the first step towards the numerical implementation of the mechanism for diffusion proposed in [DLS03].
11. Ross, Shane
    Transport in Dynamical Astronomy and Multibody Problems.
    The key idea in this paper is to combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power and applicability of the techniques and, in particular, we compute transport rates for the Sun-Jupiter-particle three-body system, which is appropriate for some comets and asteroids, and the Sun-Neptune-particle system, which is appropriate for Kuiper-belt objects. This is joint work with M. Dellnitz, O. Junge, W.S. Koon, F. Lekien, M.W. Lo, J.E. Marsden, K. Padberg, R. Preis, and B. Thiere.
12. Safi, Zaki
    Vibrational dynamics in HCP molecule.
    The objective of this communication is to give an idea about the behaviour of the highly excited vibrational states of a non-linear triatomic molecular system, HCP, and apply chaos theory to characterize their behavior. the first 100 highly excited states of this molecule have been calculated using the DVR-DGB method of bacic and light. then, they have been analyzed using the quantum surface of section and Poincaré surface of section. a correlation diagram has been calculated in which the variation of the level energies with some parameters is considered. with this diagram we can study the properties of some phenomena in the theory of the quantum chaos such as "scar". a complete classical study for the HCP system has been done. this study includes the calculation of the classical Poincaré surface of section (psos), classical trajectories and the bifurcation diagram.
13. Sanz, Angel
    Complexity in Bohmian Mechanics.
    To study chaos in quantum mechanics we cannot use the same kind of tools as in classical mechanics (Lyapunov's exponents, surface of sections, ... ), since we do not own any similar analogous to classical trajectories. However, within the Bohmian interpretation of quantum mechanics those tools can be used, in principle, since it translates the probabilistic formalism of the standard theory into a causal one, based on the concept of trajectory. Mathematically, these quantum trajectories are identical to those given by classical mechanics, but, physically, they are quite different. It is due to the presence of the non-local, context-depedent quantum potential.
    In this way, we shall show results which tell us that quantum dynamics (in a Bohmian sense) quite complex even in those cases which classical analogous are quite simple. This complexity is introduced by the quantum potential and to understand it could be a very important subject in future in order to understand interesting quantum phenomena related with quantum chaos, as it is the formation of scars.
14. Villegas-Blas, Carlos
    The Bargmann Transform and coherent states for the 3-sphere.
    A Bargmann transform for the space of square-integrable functions defined on the 3-sphere will be introduced. It will be shown that this transform is a coherent states transform labeled by elements of the null cuadrics in four complex variables. A description of the obatained coherent sates will be provided.
15. Yakubovich, Dmitry
    Linear dynamical systems and complex variable.
    We discuss a way how one can pass from problems about a linear stationary infinite dimensional dynamical system (such as complete controllability, for instance) to certain problems from the theory of complex variable. Examples include systems with delay, neutral systems and perturbations of diffusion semigroups.