Georgia Institute of Technology College of Sciences

This concerns general gradient-like dynamical systems in Banach space with the property that there is a manifold along which solutions move slowly compared to attraction in the transverse direction. Conditions are given on the energy (or, more generally, Lyapunov functional) that ensure solutions starting near the manifold stay near for a long time or even forever. Applications are given with the vector Allen-Cahn and Cahn-Morral equations. This is joint work with Giorgio Fusco and Georgia Karali.

Consider a $T$-preserving probability measure $m$ on a dynamical system $T:X\to X$. The occupation time of a measurable function is the sequence $\ell_n(A,x)$ ($A\subset \mathbb R, x\in X$) defined as the number of $j\le n$ for which the partial sums $S_jf(x)\in A$. The talk will discuss conditions which ensure that this sequence, properly normed, converges weakly to some limit distribution. It turns out that this distribution is Mittag-Leffler and in particular the result covers the case when $f\circ T^j$ is a fractal Gaussian noise of Hurst parameter $>3/4$.

The restricted three body problem models the motion of a body of zero mass under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which describe Keplerian orbits. In 1922, Chazy conjectured that this model had oscillatory motions, that is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. Its existence was not proven until 1960 by Sitnikov in a extremely symmetric and carefully chosen configuration. In 1973, Moser related oscillatory motions to the existence of chaotic orbits given by a horseshoe and thus associated to certain transversal homoclinic points. Since then, there has been many atempts to generalize their result to more general settings in the restricted three body problem.In 1980, J. Llibre and C. Sim\'o, using Moser ideas, proved the existence of oscillatory motions for the restricted planar circular three body problem provided that the ratio between the masses of the two primaries was arbitrarily small. In this talk I will explain how to generalize their result to any value of the mass ratio. I will also explain how to generalize the result to the restricted planar elliptic three body problem. This is based on joint works with P. Martin, T. M. Seara. and L. Sabbagh.

The so-called Hopf-zero singularity consists in a vector field in $\mathbf{R}^3$ having the origin as a critical point, with a zero eigenvalue and a pair of conjugate purely imaginary eigenvalues. Depending of the sign in the second order Taylor coefficients of the singularity, the dynamics of its unfoldings is not completely understood. If one considers conservative (i.e. one-parameter) unfoldings of such singularity, one can see that the truncation of the normal form at any order possesses two saddle-focus critical points with a one- and a two-dimensional heteroclinic connection. The same happens for non-conservative (i.e. two-parameter) unfoldings when the parameters lie in a certain curve (see for instance [GH]).However, when one considers the whole vector field, one expects these heteroclinic connections to be destroyed. This fact can lead to the birth of a homoclinic connection to one of the critical points, producing thus a Shilnikov bifurcation. For the case of $\mathcal{C}^\infty$ unfoldings, this has been proved before (see [BV]), but for analytic unfoldings it is still an open problem.Our study concerns the splittings of the one and two-dimensional heteroclinic connections (see [BCS] for the one-dimensional case). Of course, these cannot be detected in the truncation of the normal form at any order, and hence they are expected to be exponentially small with respect to one of the perturbation parameters. In [DIKS] it has been seen that a complete understanding of how the heteroclinic connections are broken is the last step to prove the existence of Shilnikov bifurcations for analytic unfoldings of the Hopf-zero singularity. Our results [BCSa, BCSb] and [DIKS] give the existence of Shilnikov bifurcations for analytic unfoldings. [GH] Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York (1983), 376--396. [BV] Broer, H. W. and Vegter, G., Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension. Ergodic Theory Dynam. Systems, 4 (1984), 509--525. [BSC] Baldoma;, I., Castejon, O. and Seara, T. M., Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity. Journal of Dynamics and Differential Equations, 25(2) (2013), 335--392. [DIKS] Dumortier, F., Ibanez, S., Kokubu, H. and Simo, C., About the unfolding of a Hopf-zero singularity. Discrete Contin. Dyn. Syst., 33(10) (2013), 4435--€“4471. [BSCa] Baldoma, I., Castejon, O. and Seara, T. M., Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (I). Preprint: https://arxiv.org/abs/1608.01115 [BSCb] Baldoma, I., Castejon, O. and Seara, T. M., Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II). The generic case. Preprint: https://arxiv.org/abs/1608.01116