Lloc: Aula 1, Facultat de Matemàtiques, UB.
A càrrec de: Janina Kotus, Warsaw University of Technology.
Títol: Measure theoretic properties of the Julia set of transcendental functions of the complex plane (*).
A càrrec de: Pascale Roesch, University of Lille.
Títol: Local connectivity for cubic polynomials with a parbolic point (**).
In the second part of the talk we will show how the thermodynamics' formalism can be applied to construct a conformal measure supported on the Julia set, and to estimate the Hausdorf dimension of the Julia set if this is hyperbolic and has zero Lebesgue measure.
(**)Resum:
For cubic poynomials having a fixed parabolic point, we prove that the
basin of attraction of this point has a locally connected boundary. The
method uses puzzles in order to transfer the result to the parameter plane.
Lloc: Aula 5, Facultat de Matemàtiques, UB.
A càrrec de: Florentino Borondo, Dpt. de Química, UAM.
Títol:Trayectorias cl´sicas, caos y trayectorias cu´nticas (*).
A càrrec de:Alberto Verjovsky, Instituto de Matemáticas,UNAM, Cuernavaca, Mexico.
Títol: Acciones ortogonales, involuciones e hipercuádricas en espacios proyectivos.
Por su propia naturaleza estos estudios se restringen al ambito de la Mecanica Clasica, que resulta a menudo suficiente, sobre todo si solo se esta interesado en magnitudes promedios.
Sin embargo, los fenomenos netamentemente cuanticos, tales como tunneling, interferencias o resonancias, aguardan todavia una explicacion intuitiva satisfactoria.
Un enfoque, que esta ganando auge son las llamadas trayectorias cuanticas que aparecen en la formulacion de Bohm-de Broglie de la Mecanica Cuantica. A diferencia de sus homologas clasicas estan trayectorias evolucionan de forma colectiva, en el marco de una teoria no local.
En el seminario se presentaran algunas aplicaciones numericas recientes.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Dmitry Turaev, Weierstrass Institute, Berlin.
Títol:Multi-pulse homoclinic loops and superhomoclinic orbits in Hamiltonian systems(*).
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Patricia Dominguez, Univ. Autonoma de Puebla, Mexico i Dept. de Matematica Aplicada i Analisi, UB.
Títol:Un estudio de la familia f_C(z) = C sen(z) (*).
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de:Guillermo Sienra, Depto. de Matematicas, Fac. de Ciencias, UNAM, Mexico i Dept. de Matematica Aplicada i Analisi, UB.
Títol: Atractores y estabilidad en dinamica holomorfa(*).
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Ugo Locatelli, Univ. Roma II Tor Vergata i Univ. de Milano Bicocca.
Títol: Nekhorosev vs KAM: A match played near to equilibria points systems.
A càrrec de: Carles Simó, Dept. Mat. Aplicada i Analisi, UB.
Títol: Simple Choreographic Motions of N Bodies: A Preliminary Study.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Joaquim Puig, Dept. Mat. Aplicada i Analisi, UB.
Títol: Analytic families of reducible linear quasi-periodic equations in Lie algebras.
Resum: This talk deals with the existence of analytic families of reducible systems in linear equations with quasi-periodic coefficients depending on parameters. The matrices defining the linear equation lie in a matrix Lie algebra, and are perturbations of a time-independent matrix. The dependence on the parameters and the angles is assumed to be real analytic and the frequencies to be Diophantine. A theorem is given which states that one can modify a system of this kind adding a time-independent matrix and obtain systems which are reducible to the original unperturbed matrix. This modifying term depends on the geometric properties of the unperturbed matrix and the Lie algebra $g$, and its dependence on the external parameters is real analytic. The proof is based on classical KAM techniques adapted to the context of the Lie algebra $g$. Some applications are included for the cases of $sl(2,\mR)$ and $so(3,\mR)$.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Raphael Krikorian, Centre de Mathematiques, Ecole Polytechnique Palaiseau, France.
Títol: Reducibility of quasi-periodic cocycles: nonperturbative results .
Resum: Linear differential equations with periodic coefficients are well understood via Floquet theory which asserts that one can reduce such an ODE to an equation with constant coefficients. The situation in the case where the coefficients are quasi-periodic is much more complicated due to the appearance of small divisors and in fact Floquet theory is no longer true. However in perturbative situations (when the coefficients are close to constants) it is expected that for most systems one can generalize Floquet type results. In this talk, we shall be interested in the nonperturbative case
Dia: Dimecres 6 de novembre de 2002.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Bosco Garcia-Archilla, Depto. Matematica Aplicada II, ETSIE, Universidad de Sevilla.
Títol: Metodos de Krylov, determinantes y calculo de bifurcaciones en ecuaciones en derivadas parciales .
Resum: Es bien conocido que el determinante de una matriz se obtiene sin costo alguno como subproducto de la eliminacion gaussiana. Por ello, se utiliza para la deteccion de bifurcaciones en problemas dependientes de un parametro.
En problemas de ecuaciones en derivadas parciales, es frecuente el recurso a metodos iterativos para la resolucion de sistemas lineales, ante la inviabilidad de la eliminacion gaussiana. Abordaremos como se puede obtener el determinante de una matriz como subproducto de la familia de metodos iterativos conocidos como metodos de Krylov, que incluyen al popular metodo GMRES.
Mostraremos las dificultades que pueden presentarse asi como procedimientos para superarlas. En los experimentos numericos mostraremos la eficiencia de esta nueva tecnica para la localizacion de bifurcaciones tipo pitchfork y transcriticas.
A càrrec de: Joan Sanchez, Dept de Fisica Aplicada, UPC.
Títol:Calcul d'orbites periodiques en sistemes dissipatius de dimensio elevada: aplicacio a un problema de conveccio termica .
Resum:El calcul de varietats invariants en sistemes de dimensio elevada, provinents de la discretitzacio d'EDP, presenta dificultats afegides a las que ja es tenen en dimensio baixa. Consisteixen, essencialment, en que els problemes d'algebra lineal que apareixen no es poden resoldre per metodes directes. Es presentaran alguns resultats preliminars en el calcul d'orbites periodiques en un problema de conveccio termica bidimensional en un recinte anular. S'ha intentat trobar-les de la manera mes semblant possible a com es fa per a sistemes baix-dimensionals; substituint unicament l'algebra lineal directa per iterativa tant per a resoldre sistemes lineals com per a trobar valors i vectors propis. Es comentaran alguns aspectes relatius a l'eficiencia del metodes que s'han emprat per a trobar orbites periodiques, punts fixes i les seves estabilitats.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Luca Biasco, SISSA, Trieste, Italia.
Títol:Arnold diffusion via variational methods in non isochronous Hamiltonian systems.
Resum: We consider nearly integrable, non-isochronous, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $T_d = O((1/\mu)\log(1/\mu))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d$ is optimal as a consequence of a general stability result proved via classical perturbation theory.
A càrrec de: Stefanella Boatto, IMCCE, Observatoire de Paris i Dept. de Mathematiques, Universite Paris 13.
Títol: Point vortex dynamics on the plane and on the sphere: the stability of relative equilibria configurations .
Resum: In 1883, while studying and modeling the atomic structure J.J. Thomson investigated the {\it linear} stability of corotating point vortices in the plane (see J.J. Thomson , ``A Treatise of the Motion of Vortex Rings'', Macmillan (1883), pag 94-108). In particular, his interest was in configurations of identical vortices equally spaced along the circumference of a circle, i.e., located at the vertices of a regular polygon. He proved that for six of fewer vortices the polygonal configurations are stable, while for seven vortices -- the Thomson heptagon -- he erroneously concluded that the configuration is slightly unstable (Morikawa \& Swenson (1971)). It took over slightly more than a century to make some progresses on this problem! In his PhD thesis (Princeton, 1985), D.G. Dritschel proved that the Thomson heptagon is neutrally stable and that for eight or more vortices the corresponding polygonal configurations are linearly unstable. Recently (1999) H.E. Cabral and D.S. Schmidt proved that for seven or fewer vortices the polygonal configurations are {\it non-linearly} stable in the plane.
For the spherical case the results are much more recent! In 1993 D.G. Dritschel and L.M. Polvani determined the ranges of linear stability -- in terms of the latitude-- of polygonal configurations. By a similar method to the one used by Dritschel in the planar case, Dritschel and Polvani showed that at the pole, for $N<7$ the configuration is stable, for $N=7$ it is neutrally stable and for $N>7$ it is unstable. In 1998 J.E. Marsden and S. Pekarsky proved that for $N=3$ the range of non linear stability is the whole sphere (they also had stability results for vortices with different vorticities $k_1$, $k_2$ and $k_3$). H.E. Cabral and myself (2001) determined the ranges of non-linear stability for $N<7$.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Alexei Vasiliev, Space Research Institute, Moscow.
Títol: Resonant phenomena in slowly perturbed rectangular billiards .
Resum: One considers a slowly rotating rectangular billiard with slowly moving borders. Methods of canonical perturbation theory are used to describe the dynamics of a billiard particle. In the process of slow evolution certain resonance conditions can be satisfied. The phenomena of scattering on a resonance and capture into a resonance are studied. These phenomena lead to destruction of adiabatic invariance in the system.
A càrrec de: Jose Tomas Lazaro, Dept. de Matematica Aplicada I, UPC .
Títol: Pseudo-normal forms around saddle-center or saddle-focus equilibria .
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Sergey Gonchenko, Dept. Diff. Eq., Univ. Nizhny Novgorod.
Títol: On two-dimensional diffeomorphisms with a "totally mixed" dynamics .
Resum: We consider two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. We suppose that this heteroclinic cycle contains two saddle periodic orbits such that the Jacobian at one point is less than one and the Jacobian at the other point is greater than one. Then, we prove that, near such a diffeomorphism, the so-called Newhouse regions exist in the space of dynamical systems, where diffeomorphisms having simultaneously infinitely many saddle, stable and completely unstable periodic orbits as well as infinitely many stable and unstable closed invariant curves, are dense. Moreover, this property is generic, when the closures of the sets of these orbits contain nontrivial hyperbolic sets.
A càrrec de: Vladimir Gonchenko, Dept. Diff. Eq., Univ. Nizhny Novgorod .
Títol: On local bifurcations and chaotic dynamics in reversible maps .
Resum: We study the problem on the breakdown of conservativity in two-dimensional reversible maps with chaotic dynamics. We show that there exist two different ways in which conservativity is lost. One way leads to the existence of attractor and repeller which are separated from each other. In the second scenario we have persistent intersection between the attractor and repeller.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de:Tere M Seara.
Títol: Existència de difusió en un problema amb "gaps" .
Resum: Introduim un mecanisme per a obtenir difusi\'o en sistemas a-priori inestables.
Esta basat en el fet que les reson\`ancies, que destruieixen els tors KAM primaris, creen tors secundaris i tors de dimensi\'o m\'es baixa. Argumentem que aquests tors poden ser incorporats a les cadenes de transici\'o ocupant el lloc del tors KAM primaris.
Establim rigurosament l'exist\` encia d'aquest mecanisme en un model simple ja estudiat previament.
Podeu baixar-vos preprint de:
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Rafael Ramírez Ros, Dept. de Matemàtica Aplicada I, UPC .
Títol: Persistencia de orbitas heteroclinicas en aplicaciones twist y en billares .
Resum: Consideramos la dinamica del billar dentro de un elipsoide $Q$ de dimension $n$. Suponemos que $Q$ tiene un unico diametro; es decir, suponemos que uno de sus ejes es mayor que el resto. El diametro es una trayectoria hiperbolica de periodo dos cuyas variedades invariantes estables e inestable coinciden. Por tanto, las orbitas homoclinicas forman un conjunto de dimension $n$ en el espacio de fases.
Se prueba que existe un entero $l=l(Q)>0$ con la siguiente propiedad: Persisten al menos $l(Q)$ orbitas homoclinicas primarias bajo cualquier perturbacion $C^3$-peque\~na del elipsoide $Q$. Ademas, la cantidad $l(Q)$ se mueve desde seis (cuando el elipsoide es de revolucion) hasta $8n$ (cuando el elipsoide es generico). La cota es optima en el caso generico.
La prueba se basa en que las aplicaciones que modelan la dinamica del billar son aplicaciones twist. De hecho, la cota $l(Q)$ se deduce de otra cota $l=l(f)>0$ mas general, valida para perturbaciones twist $C^2$-peque\~nas de aplicaciones twist $f$ que posean un par de conjuntos periodicos hiperbolicos cuyas variedades invariantes estable e inestable tengan una interseccion limpia a lo largo de una subvariedad que verifique unas condiciones de compacidad y finitud.
En la practica, la propiedad mas dificil de establecer es el caracter limpio de las intersecciones.
(Este trabajo se esta desarrollando con Sergey Bolotin y Amadeu Delshams.)
A càrrec de: Alex Haro, Dept. de Matemàtica Aplicada i Anàlisi, UB .
Títol: Varietats invariants en sistemes quasiperiodics: teoria, computacio i aplicacions .
Resum:Expliquem el metode de parametritzacio per demostrar l'existencia de tors normalment hiperbolics i les seves varietats invariants en sistemes dinamics forçats quasiperiodicament.
Primer desenvoluparem la teoria lineal (teoria espectral d'operadors de tranferencia sobre rotacions), per despres aplicar-la a l'analisi de les equacions d'invariança.
Una de les avantatges d'aquestes demostracions es que son constructives i es podem implementar com algorismes numerics. L'ultima part del seminari la dedicarem a descriure alguns exemples concrets.
Últim dia de seminari abans de les vacances de Nadal
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Laurent Niederman, U. Paris-Sud (Orsay) .
Títol: Hamiltonian stability and subanalytic geometry .
A càrrec de: Marian Gidea, Northeastern Illinois University .
Títol: Topologically Crossing Heteroclinic Connections to Invariant Tori .
A càrrec de: Carles Simó, Dept. Matemàtica Aplicada i Anàlisi, UB .
Títol: Henon-like strange attractors in a family of maps of the solid torus (*) .
Occurrence of Henon-like strange attractors holds for a set of parameter values with positive measure.
Main characteristic of Henon-like strange attractors is that they coincide with the closure of the 1D unstable manifold of a saddle periodic orbit.
Under suitable hypotheses, we also show that the closure of the unstable manifold of an invariant circle of saddle type has a basin of attraction with nonempty interior.
(Joint work with H.W.Broer and R.Vitolo)
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Joaquim Puig, Dept. Matemàtica Aplicada i Anàlisi, UB .
Títol: Espectre de Cantor per a l'operador Almost Mathieu (*) .
H=-x_{n+1}-x_{n-1} + b cos(2\pi\omega n + \phi) x_n
a $l^2(Z)$. Introduirem resultats de reductibilitat i localització desenvolupats recentment, i veurem com aquests permeten demostrar que si $\omega$ és diofàntic i $b$ és diferent de zero, llavors l'espectre d'aquest operador és un conjunt de Cantor.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: James Stirling, Centre de Recerca Matemàtica .
Títol: Mathematical modelling in sport (*) .
(*)Resum:
We give a examples of how mathematical modeling is used to understand
processes in sport. We start by giving a general overview of areas
currently employing mathematical tools to analyise the processes involved.
We then go onto look in more detail at the use of mathematics to model the
physiological and biomechanical response to exercise. It should also be
noted that these two areas are not only fundamental to the science of
technique and training methodology in sport but are also fundamental areas
of medicine. As a result improvements in the mathematical annalysis of
such problems will have far reaching benefits.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Mercè Ollé, Dept. de Matemàtica Aplicada I, UPC .
Títol: Numerical exploration of the Hamiltonian-Hopf bifurcation(*) .
(*)Resum:
We give a numerical description of the neighbourhood of a fixed point
of a symplectic map undergoing a transition from linear stability to
complex instability, i.e., the so called Hamiltonian-Hopf bifurcation.
We have considered both the direct and inverse cases.
The study is based on the numerical computation of the Lyapunov families of invariant curves near the fixed point. We show how these families, jointly with their invariant manifolds and the invariant manifolds of the fixed point organise the phase space around the bifurcation.
(Joint work with A. Jorba)
A càrrec de: Pau Martín, Dept. de Matemàtica Aplicada IV, UPC .
Títol: Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems(**) .
(**)Resum:
We show that if a set of Lyapunov exponents of a regular orbit is such
that sums of these exponents do not give a Lyapunov exponent outside this
set, and all of them are negative, then there is an invariant set of
smooth manifolds along the orbit associated to this class of exponents.
In particular, we can obtain the usual strong stable manifolds.
We stablish the same results for orbits whose uniform hyperbolicity rates satisfy some non-resonant conditions.
It should be noted that, in general, these systems of invariant manifolds are not a foliation.
(Joint work with E. Fontich i R. de la Llave)
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Alex Haro, Dept. de Matemàtica Aplicada, UB .
Títol: Varietats invariants en sistemes quasiperiodics: teoria, computacio i aplicacions .
Resum:
Expliquem el metode de parametritzacio per demostrar l'existencia
de tors normalment hiperbolics i les seves varietats invariants en
sistemes dinamics forçats quasiperiodicament.
Primer desenvoluparem la teoria lineal (teoria espectral d'operadors de tranferencia sobre rotacions), per despres aplicar-la a l'analisi de les equacions d'invariança.
Una de les avantatges d'aquestes demostracions es que son constructives i es podem implementar com algoritmes numerics. L'ultima part del seminari la dedicarem a descriure alguns exemples concrets.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: David Sauzin, IMCCE, ASD, Observatoire de Paris .
Títol:Construction of unstable Hamiltonian systems arbitrarily close to integrable .
Resum:
This is a joint work with Jean-Pierre Marco which started as a collaboration
with Michael Herman (``Stability and instability for Gevrey
near-integrable Hamiltonian systems'', to appear in {\em Publ. Math.
I.H.E.S.}).
When a completely integrable Hamiltonian system~$h$, which is written in action-angle coordinates, is perturbed, the action variables remain stable over exponentially long time intervals.
The hypotheses are 1) the quasi-convexity of~$h$, 2) the Gevrey-$\alpha$ regularity of~$h$ and of the perturbation, with $\alpha\ge1$.
This is a generalization of the Nekhoroshev Theorem (1977), which corresponds to the analytic case ($\alpha=1$).
The stability time is governed by an exponent which can be chosen to be~$1/2n\alpha$ in general, but which can be improved in~$1/2(n-2)\alpha$ for the orbits passing close to a double resonance of~$h$.
Moreover, for $\alpha>1$, the existence of Gevrey-$\alpha$ functions with compact support allows us to prove the optimality of the previous result:
for three degrees of freedom or more, we construct systems exhibiting unstable orbits for which the speed of drift is optimal. We also discuss the relationship between these examples and Arnold's mechanism of instability.
In this seminar, I shall try to explain the construction of these examples
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Henk Broer, University of Groningen, Department of Mathematics .
Títol: Geometry of KAM tori for nearly integrable Hamiltonian systems (joint work with R.H. Cushman and F. Fassò) .
Resum:
We obtain a global version of the Hamiltonian KAM theorem for
invariant Lagrangean tori by glueing together local KAM conjugacies
with help of a partition of unity. In this way we find a global Whitney
smooth conjugacy between a nearly-integrable system and an integrable
one. This leads to preservation of geometry, which allows us to
define all the nontrivial geometric invariants like monodromy or
Chern classes of an integrable system also for nearly integrable systems.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Yuri Fedorov, Dept. Matemàtica Aplicada I, UPC .
Títol:About integrable discretizations of the Euler top and projective billiards .
Resum:
First, we briefly consider various approaches to discretization of the classical Euler top and its generalizations. Then we describe an integrable two-valued symplectic map B on the 4-dimensional extended Stiefel variety V(2,3). The map admits two different reductions, namely, to the Lie group SO(3) and to its algebra so(3).
The first reduction provides a new discretization of the motion of the classical Euler top in space and has a transparent geometric interpretation, which can be regarded as a discrete version of the celebrated Poinsot model of motion and which inherits some properties of another discrete system, the elliptic billiard.
On the other hand, the reduction of B to the algebra so(3) gives a new explicit discretization of the Euler top in the space of its angular momentum, which preserves first integrals of the continuous system.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Jean-Pierre Eckmann, Dept. Physique Théorique, Univ. Genève .
Títol: Liapunov Multipliers and Decay of Correlations in Dynamical Systems .
A càrrec de: Jean-Pierre Eckmann, Dept. Physique Théorique, Univ. Genève .
Títol: Non-equilibrium steady states .
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Joao Lopes Dias, Univ. Cambridge, Centre for Math. Sciences-DAMTP .
Títol: Multidimensional dynamic renormalisation and KAM theory .
Resum: We construct a renormalisation operator acting on the space of analytic vector fields of the d-torus, and show its convergence to a trivial limit set of integrable systems. This allows one to derive several KAM-type of results which will be discussed. The renormalisation scheme is essentially a multidimensional continued fractions expansion and a elimination of non-resonant terms corresponding to large-divisors.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Anna Litvak-Hinenzon .
Títol: On energy surfaces, bifurcations and resonances .
Resum: A geometrical scheme is developed to study near integrable Hamiltonian systems with n≥3 d.o.f. This scheme combines energy-momentum bifurcation diagrams (EMBD), Fomenko graphs and plots of the unperturbed energy surfaces in the frequency space to identify potential instabilities of the near integrable system. The proposed geometrical interpretation provides connection between bifurcations in the structure of the unperturbed energy surfaces, folds in the EMBD, resonant lower dimensional tori and instabilities of perturbed orbits. A priori stable, a priori unstable and bifurcating systems are compared and discussed.
Lloc: Aula 7, Facultat de Matemàtiques, UB.
A càrrec de: Philippe Robutel, IMCCE, Paris .
Títol: A preliminary study of the Trojans resonant structure based on frequency analysis .
Lloc: Aula 5, Facultat de Matemàtiques, UB.
A càrrec de: Victor Donnay, Department of Mathematics, Bryn Mawr College .
Títol: Geodesic flows on embedded surfaces: ergodic and non-ergodic examples .
Resum: In joint work with C. Pugh, we show how to construct embedded surfaces with Anosov geodesic flow making use of a finite horizon construction. This construction generalizes earlier constructions of embedded surfaces whose geodesic flow is ergodic (but not Anosov).For these ergodic but not Anosov examples,we show that under a small perturbation, they become non-ergodic. Non-ergodicity is produced by creating elliptic islands by applying a perturbation to a homoclinic connection of a parabolic periodic point.
A càrrec de: K. Yagasaki, Gifu University .
Títol: The Method of Melnikov for Homoclinic and Heteroclinic Orbits in Hamiltonian Systems with Saddle-Centers .
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