Summary

1. The course will start with a quick introduction to the N-body problem: symmetries and first integrals, reduction, central configurations and homographic motions.
2. After reduction, a relative equilibrium becomes an equilibrium from which Lyapunov families may bifurcate. The classical theorems will be discussed, in particular those of Lyapunov and Weinstein-Moser. Two examples will serve as illustration: Conley's approach of Hill's orbits in the restricted 3-body problem and Chenciner & Féjoz study of the flow in the neighborhood of the equilateral relative equilibrium of the equal mass 3-body problem.
3. With the aim of understanding the corresponding Lyapunov families, the Vertical Variational Equation of a (horizontal) relative equilibrium solution of the N-body problem will be introduced and the symmetries of its solutions studied in the case of the regular N-gon for N equal masses.
4. The minimization of the action integral under symmetry constraints will be introduced as a tool for the continuation of the local Lyapunov families. Already known to Poincaré, the collision problem, due to the weakness of the Newton potential, will be discussed and Marchal's theorem explained.
5. The course will end with a picture of what is known, theoretically or numerically on the global Lyapunov families (choreographies and Hip-Hops) which bifurcate from the regular N-gon relative equilibrium. It will show in particular the limitations of action minimization due to the existence of isomorphisms between the symmetries of different families.

References

• A. Chenciner, J. Féjoz, "Unchained polygons and the N-body problem", in Regular and chaotic dynamics Vol. 14, N°1, 64-115 (2009).
• A. Chenciner, "4 lectures on the N-body problem", in Hamiltonian Dynamical Systems and Applications, W. Craig ed. 21-52, Springer (2008).
• A. Chenciner, J. Féjoz, "The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium" in Discrete and Continuous Dynamical Systems, Series B, Vol.10, Number 2-3,421-438 (2008).

Summary

1. Maximal attractors and their Hausdorff dimension.
2. Hyperbolic attractors. Solenoid.
3. Different definitions of attractors. SRB measure. Palis paradigm.
4. Attractors with intermingled basins. Fubini nightmare.
5. Invisible parts of attractors. Fubini regained.
6. New type of attracting sets: bony attractors.
7. Limit cycles of quadratic vector fields.
8. Nonaccumulation of limit cycles to hyperbolic polycycles.
9. Infinitesimal Hilbert 16th problem.
10. Complex foliations.

Summary

1. First the classical notions of asymptotic expansions, Gevrey asymptotics and Borel summability will be presented, especially the basic theorem of Ramis-Sibuya on Gevrey-asymptotics.
2. For solutions of analytic singularly perturbed ODE, the existence of solutions having Gevrey asymptotics and the existence of summable solutions will be studied. Applications to the problems of overstability and the existence of canard-solutions are given.
3. For doubly singular equations, a new notion of monomial summability will be introduced. Near turning points, uniform asymptotics in the form of combined expansions and their relation to the classical "matching" will be discussed. If time permits, an example of parametric resurgence will also be presented.