Dia: Dimecres 12 de setembre del 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

(*)Resum:
The kinematic framework of the Newtonian n-body problem consists of distinctly heterogeneous elements: time, translational and rotational degrees of freedom, an overall scale, and shape degrees of freedom. In his famous and influential paper on the three-body problem, Lagrange eliminated the translations and rotations but retained time and scale. I shall discuss two issues: 1) What happens if Lagrange's reduction is taken to its logical end point, so that Newtonian n-body dynamics is described by timeless curves in the space of shapes of the instantaneous configurations (Shape Space)? I shall argue that systematic investigation of this question is desirable and is likely to cast much interesting light on the structure of Newtonian dynamics, in particular its failure to be fully scale invariant. 2) Is it possible to formulate a non-Newtonian dynamics of point particles as a geodesic principle on Shape Space? Does there exist a dynamics of pure shape? This second question will be answered in the affirmative, and a scale-invariant generalization of Newtonian celestial mechanics will be presented. This new theory shows how the dynamics of the universe could be scale invariant even though locally observed physics is not.

Dia: Dimecres 19 de setembre del 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

(*)Resum:
We develop a simple model or cartoon of the large scale under lying dynamics of the estuarine flow. Such flows are time periodic and hence can be reduced to the study of a 3D Poincare map. As a result of nonslip boundary conditions then the boundaries to the flow are 2 D invariant manifolds, on which the dynamics needs not preserve area. We study the dynamics, via a Poincare map, on one of these 2D invariant manifolds. We relate the presence of structures and the gemometry of the tangle of unstable and stable manifolds of fixed and periodic orbits to the transport, discharge and patchiness of pollution released into such flows.
Ref: Stirling, J.R. Physica D 144 (2000), 169-193.

Dia: Dimecres 3 d'octubre del 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

(*)Treball conjunt amb H. Broer i R. Vitolo.

Resum:
A low dimensional model of general circulation of the atmosphere is investigated. The differential equations are subject to periodic forcing, where the period is one year. A three dimensional Poincaré mapping $\mathscr{P}$ depends on three control parameters $F,$ $G,$ and $\eps$, the latter being the relative amplitude of the oscillating part of the forcing. This paper provides a coherent inventory of the phenomenology of $\mathscr{P}_{F,G,\eps}$. For $\eps$ small, a Hopf-saddle-node bifurcation $\mathcal{HSN}$ of fixed points and quasi-periodic Hopf bifurcations of invariant circles occur, persisting from the autonomous case $\eps=0$. For $\eps=0.5$, the above bifurcations have disappeared. Different types ofstrange attractors are found in four regions (chaotic ranges) in $\{F,G\}$ and the related routes to chaos are discussed.

Dia: Dimecres 17 d'octubre del 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

(*)Treball conjunt amb H. Broer, I. Hoveijn, M. van Noort i G. Vegter.

Resum:

his paper is concerned with the global coherent (i.e., non-chaotic) dynamics of the parametrically forced pendulum. The system is studied in a one and one half degree of freedom Hamiltonian setting with two parameters, where a spatio-temporal symmetry is taken into account. Our explorations are restricted to sufficiently large regions of coherent dynamics in phase space and parameter plane. At any given parameter point we restrict to a bounded subset of phase space, using KAM theory to exclude an infinitely large region with trivial dynamics.

In the absence of forcing the system is integrable. Analytical and numerical methods are used to study the dynamics in a parameter region away from integrability, where the results of a perturbation analysis of the nearly integrable case are used as a starting point. We organize the dynamics by dividing the parameter plane in fundamental domains, guided by the linearized system at the upper and lower equilibria.

Away from integrability some features of the nearly integrable coherent dynamics persist, while new bifurcations arise. On the other hand, the chaotic region increases.

Dia: Dimecres 14 de novembre del 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

Resum:

In this talk we will revisit the Taylor method for the numerical integration of initial value problems of Ordinary Differential Equations (ODEs). The main issue is to present a computer program that, given a set of ODEs, produces the corresponding Taylor numerical integrator. The step size control adaptively selects both order and step size to achieve a prescribed error, and trying to minimize the global number of operations. The package provides support for several extended precision arithmetics, including user-defined types.

We will also discuss the performance of the resulting integrator in some examples. As it can select the order of the approximation used, it has a very good behaviour for high accuracy computations. In fact, if we are interested in a very accurate computation in extended precision arithmetic, it becomes the best choice by far. The main drawback is that the Taylor method is an explicit method, so it has all the limitations of these kind of schemes. For instance, it is not suitable for stiff systems.

Dia: Dimecres 21 de novembre del 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

Resums:

(*)
We give a completely solution of the problem of the construction of the field of force capable of generating the given orbits.
We propose a generalization of the Dainelli, Bertrand and Joukovski problems. A new approach to solve the Suslov problem is obtained.

(**) Treball conjunt amb Claudia Valls
We consider the classical Arnold's example of diffusion with two equal parameters. Such system has two-dimensional partially hyperbolic invariant tori. We mainly focus on the tori whose ratio of frequencies is the golden mean. We present formal approximations of the three-dimensional invariant manifolds associated to this torus and numerical globalization of these manifolds. This allows to obtain the splitting (of separatrices) vector and to compute its Fourier components. It is apparent that the Melnikov vector provides the dominant order of the splitting provided the contribution of each harmonic is computed after a suitable number of averaging steps, depending on the harmonic. We carry out the first order analysis of the splitting based on that approach, mainly looking for bifurcations of the zero level curves of the components of the splitting vector and of the homoclinic points.

Dia: Dimecres 28 de novembre del 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

Resums:

(*)
We describe the flow in an estuary using a 3 dimensional time periodic vector field, in which the dynamics in the vertical plane can be separated from that in the horizontal. We then reduce the dynamics in the vertical plane to the study of 2d Hamiltonian Poincare map and make predictions as to the time taken for pollution to leave the estuary and the regions of different mixing and hence the presence of patchiness within the clouds of pollution. We finish with suggestions for optimal strategies for the release of pollution into such flows.

(**) Treball conjunt amb Claudia Valls.

Dia: Dimecres 5 de desembre de 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

Resum:

The symmetric collinear four-body problem is a special case of the general Newtonian four-body problem in which the bodies are distributed symmetrically about the center of masses on a fixed common line. This problem represents a special class of N-body systems with two degrees of freedom.

Both analytical and numerical studies of SC4BP have been conducted in the case of negative energy. Especially, symbol sequences are directly obtained by numerical simulations.

It is found that some kinds of symbol words are unrealizable, contrary to the existence of other kinds of periodic sequences. The period coincides with the "winding number" of the invariant manifold associated with the critical point on the total collision manifold.

Moreover, escape criteria are constructed analytically. The distribution of initial points on the surface of section for escape is numerically confirmed.

Dia: Dimecres 12 de desembre de 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

Dia: Dimecres 20 de desembre de 2001.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

Dia: Dimecres 13 de febrer de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

Dia: Dimecres 20 de febrer de 2002.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

Dia: Dimecres 27 de febrer de 2002.

Lloc: Aula 1, Facultat de Matemàtiques, UB.

Dia: Dimecres 13 de març de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

Dia: Dimecres 17 d'abril de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

Dia: Dimecres 24 d'abril de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

Dia: Dimecres 8 de maig de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

Dia: Dimecres 19 de juny de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

Dia: Dimecres 26 de juny de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

Dia: Dimecres 3 de juliol de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

Dia: Dimecres 17 de juliol de 2002.

Lloc: Aula 5, Facultat de Matemàtiques, UB.

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Last updated: Wed Jul 17 15:41:23 MEST 2002