| Speaker: José Pablo Mujica Affiliation: Universidad Técnica Federico Santa María Time: 2 pm Thursday, 2 May, 2019 Location: 303-257 |
| In many areas of application, one encounters models whose behavior evolves in slow and fast episodes; well-known examples come from chemical reactions (non-harmonic oscillations), neuron systems (spiking and bursting) and climate models (sudden changes in the atmosphere’s pressure). Their mathematical description leads to vector-field models, called slow-fast systems, that have state variables separated into groups that evolve on different time scales. In classical dynamical systems, the ‘skeleton’ of phase space is given by equilibrium points, periodic orbits, and their invariant manifolds. For slow-fast dynamical systems, one also adds manifolds along which the flow is very slow compared to the rest of the dynamics; these are known as slow manifolds. In this talk, we will discuss some of the ideas from the theory of slow-fast dynamical systems and describe some of the consequences for the overall dynamics when invariant manifolds interact with slow manifolds. As a leading example, we consider a three-dimensional model with two slow and one fast variables, in which the manifolds of interest are two-dimensional surfaces. In particular, we will discuss the creation of recurrent dynamics, its connection with a Shilnikov homoclinic bifurcation, a chaotic scenario and further possible structures. |