Resumen de Study of invariant manifolds in two different problems: the Hopf-zero singularity and neural synchrony
Oriol Castejón Company
The main object of study of this thesis are invariant manifolds in the field of dynamical systems. We deal with two different and independent topics, namely, the study of exponentially small splitting of invariant manifolds in analytic unfoldings of the Hopf-zero singularity (in Part I) and the applications of dynamical systems in problems inspired by neuroscience (in Part II). In general, this thesis studies both theoretical and applied problems in dynamical systems, using analytical as well as computational tools. In Part I, we consider a certain class of generic unfoldings of the so-called Hopf-zero singularity. One can see that the truncation of the normal form at any finite order of such unfoldings possesses two saddle-focus critical points and, when the parameters lie on a certain curve, they are connected by a one- and a two-dimensional heteroclinic manifolds. However, considering 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 $C^\infty$ unfoldings, this was proved by Broer and Vegter during the 80's, but for analytic unfoldings it has remained an open problem. Recently, under some assumptions on the size of the splitting of the heteroclinic connections, Dumortier, Ibáñez, Kokubu and Simó proved the existence of Shilnikov bifurcations in the analytic case. Our study concerns the splitting of the one- and two-dimensional heteroclinic connections. These cannot be detected in the truncation of the normal form at any order, and hence they are exponentially small with respect to one of the perturbation parameters. We give asymptotic formulas of these splittings and, in particular, we prove that under generic conditions the main assumptions made by Dumortier, Ibáñez, Kokubu and Simó hold. In Part II, we deal with tools to provide an accurate prediction of phase variations in an oscillator subject to external stimuli. We construct a method based on the concepts of isochrons, Phase Response Functions (PRF) and Amplitude Response Functions (ARF). In particular, the method can be applied to neurons in a state of repetitive firing. In the special case of a pulse-train periodic stimulus, the application of this theoretical frame leads to a 2D map, one variable controlling phase jumps and the other controlling amplitude jumps. We compare these maps to the classical 1D maps obtained via Phase Response Curves (PRC) and we identify circumstances in which the 2D maps give a more accurate prediction of the synchronization. Moreover, we implement some numerical methods to compute the invariant curves of the 2D maps as well as the dynamics inside these curves. Finally, we compute Arnold tongues of these maps, which allow to determine regions in the parameter space for which the neuron is synchronized to the external input.
