(abstracts at the end of the document)

 Monday, September 22 

 Tuesday, September 23 

 Wednesday, September 24 (La Mercè holiday in Barcelona) 

 Thursday, September 25

 Friday, September 26 

Abstracts

Massimiliano Berti 

Title

Nash Moser theory and Periodic solutions of Hamiltonian PDEs in higher dimensions with finite regularity

Abstract

We present new existence results of periodic solutions of Hamiltonian PDEs defined on higher dimensional tori, compact Lie groups, Zoll manifolds. In the case of tori we extend previous results of Bourgain under weaker non-resonance conditions and for only differentiable nonlinearities. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash-Moser iteration scheme, where we just need estimates of interpolation type for the inverse linearized operators.

Alessandra Celletti 

Title

Some results on the dynamics of nearly-integrable, dissipative systems"

Abstract

We present some results on the dynamics of nearly-integrable systems which are subject to a dissipation. These systems are ruled by two parameters: the perturbing parameter, measuring the strength of the nonlinearity, and the dissipative parameter. By means of a suitable KAM theory we investigate the existence of invariant attractors. Furthermore we extend converse KAM theory to study the non-existence of invariant attractors. Two models have been analyzed, namely the dissipative standard map and the spin--orbit problem for the study of rotational dynamics of celestial bodies.

The relation between invariant attractors and periodic orbits provides an estimate of the break-down threshold as a function of the perturbing and dissipative parameters. Close to this limit we study the existence of cantori, providing a proof for the sawtooth map and numerical evidence for the dissipative standard map.

Alain Chenciner 

Title

Unchained polygons and the N-body problem

Abstract

The simplest solutions of the N-body problem --regular N-gon relative equilibria-- are shown to be organizing centers from which stem some recently studied classes of periodic solutions.

The paradigmatic examples are the ``Eight'' families for an odd number of bodies and the ``Hip-Hop'' families for an even number. Global continuation is sought for via action minimization under symmetry constraints in rotating frames (collaboration with Jacques Féjoz). 

Chong-Qing Cheng 

Title

Connecting Orbits of Autonomous Lagrangian Systems

Abstract

We show how to construct orbits in autonomous Lagrangian systems which connecting different Aubry sets in a given Energy surface. These orbits are constructed along two kinds of local connecting orbits, one is of modified C-equivalence, another one is heteroclinic orbit. 

Luigi Chierchia 

Title

Maximal Tori for the Planetary N Body Problem

Abstract

Proving the existence of regular, bounded motions for the dynamics of N bodies (point masses) interacting only through gravitational attraction is an extremely rich and difficult mathematical problem. A major breakthrough was given by V.I. Arnold in 1963 [1] who proved, using the newly born KAM theory, the existence of a positive measure set of initial conditions giving rise to quasi-periodic motion spanning "maximal" tori in phase space for the planar, planetary 3 body problem (one big mass and 2 small masses moving on a plane). The full extension to the general planetary N body problem took more than fourty year to be completed [2] and still presents interesting open problems.

In this lecture, I will review some of the main contributions to this problem and discuss recent developments.

References

 [1] V.I. Arnold. Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics. Usephi Mat. Nauk, 18(6 (114)):91-192, 1963

[2] J. Fejoz. Demonstration du thoreme de Arnold sur la stabilite du systeme planetaire (d'apres M. Herman). Ergod. Th. & Dynam.

Sys, 24(5):1521-1582, 2004

[3] P. Robutel. Stability of the Planetary Three-body Problem. Celestial Mech. Dynam. Astronom., 62(3):219-261, 1995. II. KAM theory and existence of quasiperiodic motions

[4] H. Russmann. Invariant Tori in Non-Degenerate Nearly Integrable Hamiltonian Systems. R. & C. Dynamics, 2(6):119-203, March 2001

[5] L. Chierchia and F. Pusateri.  Analytic Lagrangian tori for the planetary many-body problem. Ergod. Th. & Dynam. Sys. To appear.

[6] G. Pinzari and L. Chierchia. Full symplectic reduction of the spatial many-body problem. Preprint 2008 

Hakan Eliasson 

Title

Dynamical localization for the discrete one-dimensional quasi-periodic

Schrödinger equation

Abstract

The quasi-periodic Schrödinger equation in one spacedimension has been intensively studied both as a finite-dimensional and as an infinite-dimensional dynamical system since the work of Dinaburg & Sinai in the middle 70's and of Fröhlich&Spencer&Wittwer and Sinai in the late 80's.

We shall discuss the property of dynamical localization for this equation in the strong coupling regime.

Vassili Gelfreich 

Title

Unbounded Energy Growth in Hamiltonian Systems with a Slowly Varying Parameter

Abstract

We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly  hyperbolic invariant set with chaotic behaviour, then the full system  has orbits with unbounded energy growth (under very mild genericity  assumptions). We also provide formulas for the calculation of the rate of the fastest energy growth. We apply our general theory to non-autonomous perturbations of geodesic flows, Hamiltonian systems with billiard-like and homogeneous potentials. In these examples, we show the existence of orbits with the rates of energy growth that range, depending on the type of perturbation, from  linear to exponential in time.

 Marian Gidea 

Title

Geometria Situs of Instability of Dynamical Systems with Several Degrees of Freedom

Abstract

The topic of this talk is the instability problem of dynamical systems with several degrees of freedom that are close to integrable systems. The instability of Hamiltonian system was considered by Poincare as `the fundamental problem of dynamics'. The origins of this problem reside with the question on the instability of the solar system. A new impetus in this direction is owed to the seminal paper of Arnold [1], where he conjectured that instability is generic in the whole of Hamiltonian system.

In this talk we will present an ensemble of geometrical and topological techniques that can be used to prove instability in a large class of dynamical systems that are close to integrable. We liberally use the term `geometria situs' (originally introduced by Leibniz and later adopted by Euler in the early developments of topology) to describe this type of geometrical-topological approach (see [3] for an

informal presentation).

We will discuss instability in the following type of systems: perturbations of the geodesic flow, the large gap problem of Hamiltonian systems, perturbations of volume preserving maps, and the spatial restricted three-body problem.

In these systems one can identify a normally hyperbolic invariant manifold con-taining a sequence of invariant tori that possesses heteroclinic connections between

nearby tori. These chains of tori are in general interspersed with gaps. Then one uses the dynamics along the transverse heteroclinic connections together with the dynamics intrinsic to the normally hyperbolic manifold to prove that there exists orbits that travel arbitrary far, as well as chaotic orbits (see [2]). This type of argument can be generalized in several ways: Instead of using invariant tori one can use certain `almost invariant sets' near where an orbit spends some long time (see [5]). Instead of transverse heteroclinic connections one can use topologically crossing ones (see [6]). To show the existence of orbits with prescribed itineraries, one can use the windowing method (see [7]). Furthermore, the windowing method can be used in rigorous numerical experiments, such as in the study of the dynamics of homoclinic excursions in the spatial restricted three-body problem (see [4]).

References

[1] V.I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5 (1964), 581{585.

[2] A. Delshams, R. de la Llave and T. M. Seara, A geometric mechanism for di-
fusion in hamiltonian systems overcoming the large gap problem: heuristics and rigorous verications on a model, Memoirs of the American Mathematical Society, 179 (2006),1-141.1

[3] A. Delshams, M. Gidea, R. de la Llave, and T.M. Seara, Geometric approaches to the problem of instability in Hamilto- nian systems. An informal presentation, in Hamiltonian dynamical systems and applications (Eds. W. Craig), Springer, Dordrecht, 2008.

[4] A. Delshams, J.J. Masdemont and P. Roldan, Computing the Scattering Map in the Spatial HillS Problem, Discrete and Continuous Dynamical Systems { Series B, 10 (2008), 455{483.

[5] M. Gidea and R. de la Llave, Topological Methods in the Instability Problem of Hamiltonian Systems, Discrete and Continuous Dynamical Systems - Series A, 14 (2006),295{328.

[6] M. Gidea and C. Robinson, Topologically crossing heteroclinic connections to invarianttori, Jour. Di. Equat., 193 (2003), 49{74.

[7] P. Zgliczynski and M.  Gidea, Covering relations for multidimensional dynamical

systems-I, Journal of Diferential Equations, 202 (2004), 32{58. 

George Haller 

Title

Stability and instability in inertial particle dynamics

Abstract

The dynamics of inertial (i.e., finite-size) particles in fluid flows may differ significantly from infinitesimal fluid particle dynamics. Inertial particles turn out to be attracted to a lower-dimensional slow manifold on which the equations of motion are dissipative. In certain flow regions, the slow manifold becomes unstable and leads to an unexpected departure of inertial particle motion from fluid motion. Here I discuss exact analytic results for the inertial slow manifold and its instabilities. I also show applications to hurricane dynamics and atmospheric contamination problems.

Alex Haro 

Title

Non-twist KAM Theory

Abstract

In this talk we present some rigorous results on the persistence of degenerate tori in families of non-twist symplectomorphisms. The main tools are a translated torus theorem with parameters and results from singularity theory.

The proofs also lead to numerical algoritms, and we present some preliminary computations.

Gemma Huguet 

Title

Existence of Arnold diffusion in a-priori unstable Hamiltonian systems by means of geometric methods. An example

Abstract

In this talk, we describe the geometric features of a mechanism for detecting global instability in a priori-unstable Hamiltonian systems. The mechanism presented is based on decomposing the motion in two types of dynamics, one called inner that takes place inside a normally hyperbolic invariant manifold, where a lot of regular objects (i.e., invariant tori both primary and secondary) live, and another one called outer (or scattering map), that takes into account the asymptotic motions to the normally hyperbolic invariant manifold (NHIM). The combination of both types of dynamics gives rise to chaotic dynamics and instability. 

This mechanism was introduced by Delshams, de la Llave and Seara and it was first applied to prove the existence of orbits of unbounded energy in generic geodesic flows with a periodic and quasi-periodic time-dependent potential [DLS00,DLS06b]. In [DLS06a] it was used to overcome the large gap problem in Arnold diffusion for

the case of an a-priori unstable nearly integrable Hamiltonian system, under the assumption that the perturbation was a trigonometric polynomial in the angular variables. 

We refer to [DGLS08] for a survey of this method and also the topological method of correctly aligned windows due to Easton, which was used by de la Llave and Gidea to avoid the use of KAM theory in constructing some invariant tori in the NHIM [GL06b]. 

In this talk we consider the case of a general perturbation, regular 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 generic, which guarantee the existence of Arnold diffusion [DH08]. We will check these conditions on a particular example and we will show that it admits diffusing orbits that follow this geometric mechanism. 

The method of proof is based on a careful analysis of the resonant domains and it contains a deeper quantitative description of the invariant objects generated by the resonances therein. Finally, we use the scattering map to construct transition chains of objects of different topology.

Regarding the scattering map it is worth mentioning that the paper [DLS08] contains an analytic study of its geometric properties and Delshams, Masdemont and Roldan develop in [CDMR06,DMR08] a method to compute the scattering map associated to heteroclinic trajectories connecting bounded orbits around unstable equilibria, in a non-trivial time-independent three degrees of freedom Hamiltonian system. 

References 

[CDMR06] E. Canalias, A. Delshams, J. Masdemont and P. Roldán. The scattering map in the planar restricted three body problem. Celestial Mech. Dynam. Astronom., 95:155–-171, 2006.

[DLS00] A. Delshams, R. de la Llave, and T.M. Seara.. A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T2. Comm. Math. Phys., 209(2):353--392, 2000.

[DLS06a] Amadeu Delshams, Rafael de la Llave, and Tere M. Seara. A geometric

mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. Mem. Amer. Math. Soc., 179(844):viii+141, 2006.

[DLS06b] Amadeu Delshams, Rafael de la Llave, and Tere M. Seara. Orbits of

unbounded energy in quasi-periodic perturbations of geodesic flows. Adv. Math., 202(1):64--188, 2006.

[DLS08] Amadeu Delshams, Rafael de~la Llave, and Tere~M. Seara. Geometric properties of the scattering map of a normally hyperbolic  invariant manifold. Adv.  Math., 217(3):1096--1153, 2008.

[DGLS08] A. Delshams, M. Gidea, R. de la Llave and T. M. Seara. Geometric approaches to the problem of instability in Hamiltonian systems: an informal presentation. Hamiltonian dynamical systems and  applications. Proceedings of the NATO Advanced Study Institute on Hamiltonian dynamical systems and applications, Montreal, Canada, June, 18-29, 2007. Berlin: Springer. NATO Science for Peace and Security Series B: Physics and Biophysics, 285--336, 2008.

[DH08] A. Delshams and G. Huguet. The large gap problem in Arnold diffusion

 for non polynomial perturbations of an a priori unstable Hamiltonian system. Preprint, 2008.

[DMR08] Amadeu Delshams, Josep Masdemont and Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete Contin. Dyn. Syst. Series B,  10(2/3):455--483, 2008.

[GL06b] Marian Gidea and Rafael de la Llave. Topological methods in the instability

problem of Hamiltonian systems. Discrete Contin. Dyn. Syst., 14(2):295--328, 2006. 

Robert Sinclair MacKay 

Title

The interaction of two charges in a uniform magnetic field

Abstract

We obtain a variety of results for this basic problem of plasma physics, notably existence of second species chaos for charges of opposite sign provided the gyrofrequencies do not sum to zero, an extra integral for the case of equal gyrofrequencies, and partial analysis of the scattering in the cases of same and opposite signs of charge.

Joint work with Diogo Pinheiro. 

Anatoly Neishtadt 

Title

Ratchet phenomena in a spatially periodic harmonic potential

Abstract

Transport of a particle in a spatially periodic harmonic potential under the influence of a slowly time-dependent unbiased periodic external force is studied. The equations of motion are the same as in the problem of a slowly forced nonlinear pendulum. Using methods of the adiabatic perturbation theory we show that for a periodic external force of general kind the system demonstrates directed (ratchet) transport and obtain a formula for the average velocity of this transport.

Ernesto Pérez-Chavela 

Title

The n-Body Problem in Spaces of Constant Curvature

Abstract

In this work we generalize the Newtonian n-body problem to

spaces of constant curvature _, we derive the new equations of motion

and study the 2-dimensional case. For _ > 0, the motion takes place

on spheres. For _ < 0, we use the Weierstrass hyperboloidal model of

hyperbolic geometry. We will illustrate several results concerning central

configurations, relative equilibria and we will prove Saari’s conjecture for

“moving geodesics”. Our results also shed some new light on the classical

n-body problem in Euclidean space. The material presented in this

talk is contained in the paper The N–body problem in spaces of constant

curvature, a preprint (with an extensive bibliography on this subject) is

available at http://arXiv.org/abs/0807.1747 

Joaquim Puig 

Title

One-Dimensional Quasi-Periodic Schrödinger Operators. Spectral Theory and Dynamics

Abstract

One-dimensional quasi-periodic Schrödinger operators arise naturally in several models of mathematical physics and through the linearization around quasi-periodic

orbits in dynamical systems. In this talk we will see how many spectral properties of these operators can be derived through an analysis of the dynamics of the corresponding eigenvalue equations and the skew-products they define (and vice-versa). A central tool will be the concept of reducibility of quasi-periodic skew-products.

There are several questions in spectral theory which can be studied by a fruitful interaction of these two points of view. We will consider, in particular, the issue of Cantor spectrum, which has been described for the Almost Mathieu and other models, together with some of its implications for the corresponding dynamics.

Vered Rom-Kedar 

Title

The parabolic resonance instability

Abstract    

The parabolic resonance instability was first identified a decade ago when we  studied near-integrable two degrees of freedom Hamiltonian systems in which the angular momentum is nearly preserved [1,2].  We then showed that it appears persistently in near integrable n d.o.f. Hamiltonian families depending on p parameters provided n+p≥3 , namely, that it is a ubiquitous instability [3]. Since then, we observed parabolic resonances in various applications, such as  the forced periodic 1D NLS equation and the driven surface waves. For the forced 1D NLS we have recently shown that this instability can lead to spatial decoherence of small amplitude nearly flat solutions [4,5]. The analysis of the parabolic resonance instability turns out to be elegant and

revealing: we propose that the phase-space volume of its chaotic zone should scale differently from both the elliptic and the hyperbolic cases [6].

      [1] V. Rom-Kedar; Parabolic resonances and instabilities, Chaos, 7(1):148-158, 1997.

     [2]  V. Rom-Kedar and N. Paldor; From the tropic to the poles in forty days. Bull. Amer. Mete. Soc., 78(12):2779-2784, 1997.

     [3] A. Litvak-Hinenzon and V. Rom-Kedar; On energy surfaces and the resonance web; SIAM J. Appl. Dyn. Syst. 3(4), 525---573, 2004.

     [4]  E. Shlizermann and V. Rom-Kedar;  Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos, Submitted, 2008.

      [5] E. Shlizermann and V Rom-Kedar; Three types of chaos in the

forced nonlinear Schrodinger equation.   Physical Review Letters, 96,

024104, 2006.

      [6] V. Rom-Kedar and D. Turaev; The Parabolic resonance instability, in preparation, 2008. 

David Sauzin 

Title

Resurgent splitting in a perturbed McMillan map

Abstract

This is a joint work with Pau Martin and Tere Seara. We consider a two-parameter family of analytic symplectic maps of the plane, obtained as a perturbation of the McMillan map, which is a one-parameter integrable map with a homoclinic loop. Generically, the loop splits but this phenomenon is exponentially small with respect to the relevant parameter. Using ideas from Écalle’s resurgence theory, we provide an asymptotic formula which generalizes a previous result by Delshams and Ramirez-Ros.

Dmitry Treschev 

Title

Polymorphisms and adiabatic chaos.

Abstract

Abstract.  In the end of the last century Vershik introduced some dynamical systems, called polymorphisms. These systems are multivalued self-maps of an interval, where (roughly speaking) each branch has some probability. By definition the standard Lebesgue measure should be invariant.

Unexpectedly polymorphisms appeared in the problem of destruction of an adiabatic invariant after a multiple passage through a separatrix.

We plan to discuss ergodic properties of polymorphisms. 

Turgay Uzer 

Title

The classical mechanics of intramolecular energy flow

Abstract

How does vibrational energy travel in molecules? Answering this question succinctly seems a hopeless task considering the complexity of interatomic interactions in a molecule. Yet even before scientists were burdened by this knowledge, the so-called statistical theories posited the answer: Vibrational energy travels ``very fast'' and distributes itself statistically among the vibrational modes of a molecule, assumed to resemble an assembly of coupled oscillators, well before a reaction takes place. Reaction rate theories based on these assumptions -- known collectively as statistical or RRKM theories remain reliable working tools of the practicing chemist because they have been vindicated in an overwhelming number of chemical reactions.  

However, numerical studies of Hamiltonian systems have provided solid evidence that the approach to equilibrium usually proceeds more slowly than predicted by statistical

Theories and it is also nonuniform, showing intriguing fits and starts.

In particular, for Hamiltonian systems with two degrees of freedom, the familiar picture of chaotic seas, rigid boundaries in terms of noble tori, leaky barriers in terms

of cantori has been well-established in the literature, and these structures are found to be the source of anomalous transport in such systems.  

Beyond two degrees of freedom, the transport picture in terms of phase space structures is less clear. However, the phase space of higher dimensional systems shows similar features like the abundance of periodic orbits, and a mixture of chaotic and regular regions, the latter being characterized (under some hypothesis) by invariant tori of various dimensions. The KAM theorem states that these structures are in general robust with respect to an increase of the perturbation or equivalently to an increase of energy. The comprehension of transport properties has to rely on these robust structures that are encountered by any typical trajectory. Roughly speaking, the presence of so many periodic orbits explains why generic trajectories, even when the system is strongly chaotic, display long intervals of near-regular behavior alternating with fits of chaos--a hallmark of anomalous diffusion.

In my lecture, I will focus on the dynamics of a model for the planar OCS, represented by a Hamiltonian system with three strongly coupled degrees of freedom. My aim will be to identify the relevant structures in phase space which are responsible for trappings and escapes, strongly influencing the transport properties (most prominently, the redistribution of intramolecular energy among the three modes). 

A brief report of the subject of my lecture has appeared in R. Paskauskas, C. Chandre, and T. Uzer, Physical Review Letters  100, 083001 (2008), 29 February 2008.

Yingfei Yi

Title

Quasi-periodic orbits in properly degenerate Hamiltonian systems

Abstract

We consider the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding

Hamiltonian only depends on part of the action variables. Such a Hamiltonian system arises frequently in  celestial mechanics, for instance,  in perturbed Kepler problems like the restricted and non-restricted $3$-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. We will show  under certain  natural non-degenerate conditions that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.

This is a joint work with Y. Han and Y. Li. 

Poster session by

 Nikklas Brännström

Title to be confirmed 

Marina Gonchenko

Diophantine conditions for cubic frequency vectors

Masayoshi Saito

Non-Schubart Periodic Orbits in the Rectilinear Three-Body Problem Masaya

Masayoshi SEKIGUCHI

Stability bounded Collisions in the Restricted

Three-Body Problem — a limiting case —