Abstract:
The detection of chaotic behaviours in dynamical models is supported, in most cases, by numerical arguments. Occasionally, it is possible to prove the existence of strange attractors in the neighbourhood of certain global configurations. Nevertheless, proofs are hard and the location, in a given system, of the appropriate global organizer is an involved task. The evidence for the genesis of chaos from singularities is much more recent.
Since the first studies on the bifurcation diagram of Hopf-zero singularities (three-dimensional vector fields with an equilibrium point at which the linear part has a zero and a pair of pure imaginary eigenvalues), it was well known that there was a specific type which could unfold chaotic behaviors. The role of these singularities as seeds of chaos has been widely accepted from that time. However, only recently it has been proved that, indeed, there are generic unfoldings of Hopf-zero singularities displaying strange attractors [I. Baldomá, S. Ibáñez, T.M. Seara, Hopf-Zero singularities truly unfold chaos, Commun. Nonlinear Sci. Numer. Simulat. 84 (2020) 105162]. The goal is to discuss these results and to set out future directions.
Video of the talk