The Advanced course on Central Configurations, Periodic Orbits and
Beyond in Celestial Mechanics is a joint activity of the DANCE
(Dinámica, Atractores y
Nolinealidad: Caos y Estabilidad) Spanish network with with the CRM in
the framework of the research programme Central Configurations, periodic orbits and Beyond
It is the 11th winter school in Dynamical Systems of the DANCE network.
This series of winter schools aims at training their participants both
theoretically and in applications in the field of the nonlinear science;
with the aim that theory and applications enforce each other.
This will be done in an atmosphere of informal discussion, interchange
of ideas and critical discussion of results. Attention will be paid to
the numerical and computational issues.
These winter schools should help the basic training of young
researchers, whilst opening new fields for senior ones. As a byproduct,
the courses are planned to receive official recognition in some
doctorate programs. There will be a number of partial and full
accommodation grants for young participants.
Lluís Alsedà (Universitat Autònoma de Barcelona)
Montserrat Corbera (Universitat de Vic)
Josep Maria Cors (Universitat Politècnica de Catalunya)
Jaume Llibre (Universitat Autònoma de Barcelona)
Enrique Ponce (Universidad de Sevilla)
Title: Central Configurations of the
Newtonian N-Body Problem
A central configuration is a special
arrangement of N point masses such that the gravitational acceleration vector
of each mass due to the other masses is proportional to its position vector
relative to the center of mass. The first examples of central configurations
date to the 18th century investigations of Euler and Lagrange. Since then, they
have played an important role in the qualitative study of the dynamics. In
addition to providing simple periodic solutions, they are relevant for studies
of collision, scattering to infinity and bifurcation of the integral manifolds.
This course will cover some of these
applications and then focus on methods for studying central configurations for
their own sake. We will focus on questions of existence and enumeration of
various types of central configurations, including algebraic-geometrical
approaches to Smale’s 6th problem: Is the number of central configurations
Title: Dynamical properties in Hamiltonian
Objective: To describe the main mechanisms
leading to a fairly global description of the dynamics in conservative systems.
1. Basic symplectic 2D maps. Return maps,
standard, separatrix and H\'enon maps. Invariant curves, invariant manifolds, splitting of
separatrices, invariant Cantor sets, creation
of chaos, Lyapunov exponents. Measure problems. Basic theoretical results and
2. Some key theoretical results: averaging
theory under periodic and quasi-periodic excitation; normal forms, convergence problems and
Gevrey properties; KAM theory; Nekhorosev theory, stable/unstable/centre manifolds.
3. Symbolic and numeric tools for the
computation of periodic and quasi-periodic solutions and invariant manifolds.
4. Applications: Global behavior of orbits
near the libration points in the RTBP; some problems and subproblems of the general 3-body
system. Escape, capture and diffusion.
Title: Periodic solutions via averaging
The goal of these lectures is to study the
periodic solutions of autonomous differential systems in R^n via the averaging
This theory is based essentially in two
theorems, we shall present them at any order and for arbitrary dimension.
We shall apply this theory to the study of
the periodic solutions of the van der Pol differential equation, Lienard
differential systems, the Rossler differential system, and to some Hamiltonian
Please check the following link: http://dance-net.org/rtns2014/
Registration includes: attendance to the lectures, documentation package, copy of the course notes, lunch tickets, social dinner, and coffee breaks.
Early bird registration fee: 300 €
Early bird Registration: before November 20, 2013
Deadline for registration and payment: January 7th, 2014
IMPORTANT: To register click on one of the Register buttons at the beginning or at the end of the page.
If you wish to apply for a grant, please fill out the registration form of the RTNS2014
A group reservation has been made in Hotel Campus, in the UAB campus, where the CRM is located. For further information click here
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If you have any questions please contact:
Neus Portet (email@example.com)