Tere M-Seara (Universitat Politècnica de Catalunya) and Jaime Paradela Diaz (University of Maryland)

Since the works of Johannes Kepler (1571-1630), and later the works of Isaac Newton (1642-1727) about the universal laws of gravitation, it is well known that the possible motion of two bodies moving under their mutual Newtonian gravitational forces can be on ellipses, parabolas or hyperbolas. In this talk we will recall Kepler results about the motion of two bodies and we will talk about the possible motions when one considers the more realistic model of three bodies. This problem, already considered by Henri Poincaré (1854-1912), is still far for being understood, and one can found many works proving the existence of different type of solutions for the three body problem like periodic orbits, asymptotic orbits to them, among others. In this talk we will give some light about the possible motions that three bodies can have if we wait enough time and we will try to answer questions like:

• Will the orbits be bounded or can the bodies fall apart (and therefore going to “infinity”)?
• Can one find chaotic motion in the three body problem and therefore unpredictability?

These and related questions will be the main goals of this talk, where we will see how the mathematical tools coming from Dynamical Systems can be used to deal with this complex problem. In particular, we will prove the existence of of oscillatory motions, that is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region.

Consider a one-parameter family of smooth surface diffeomorphisms unfolding a quadratic homoclinic tangency to a hyperbolic fixed point. It is well known that in this unfolding there exists a Newhouse domain, i.e. an open set of parameters for which the corresponding diffeomorphisms exhibit persistent homoclinic tangencies. Moreover, quite a number of surprisingly exotic dynamical phenomena have been shown to exist for residual sets of parameters contained in the Newhouse domain.

The analysis of the unfolding is certainly more subtle when the homoclinic tangency is associated to a persistent degenerate saddle, i.e. a parabolic fixed point which exists for all values of the parameter and for which the topological picture of the local dynamics resembles that of a hyperbolic fixed point (existence of invariant manifolds and C^0 Lambda lemma). These degenerate saddles appear naturally in several models in Celestial Mechanics, in particular, in the so-called restricted 4 Body Problem. We prove that, in a particular configuration of the latter model which can be reduced to an area preserving map, there exists a degenerate saddle with a quadratic homoclinic tangency that unfolds generically (as we move the masses of the bodies). Moreover, and despite the fact that the C^1 Lambda lemma does not hold for this degenerate saddle, we show that the dynamics at the unfolding of the tangency can be renormalized, with the critical Hénon map showing up in the limit process. This implies the existence of a Newhouse domain in the parameter space (the masses of the bodies) and a residual subset of parameters for which there exist hyperbolic sets of large Hausdorff dimension which are accumulated by elliptic islands. This is joint work with J.M. Garrido and P. Martín.