Tobia Dondè


Ph.D Cycle
XXXI
Supervisor
Fabio Zanolin
Institutional Office Address
DMIF, Room NS2, Phone: +39 0432 558401
Research Project Title
Topological methods for dynamical systems and chaos
Research Project Description
My research project is three-folded.
The first fold is devoted to chaos detecting. Starting from the Stretching Along the Paths (SAP) method introduced by Papini and Zanolin (2002) for ODEs, the aim is to fit SAP into the infinite-dimensional framework of PDEs. There are two main paths to follow: the first one is to “spatially perturb” some ODEs on which SAP works well, e.g. the Duffing equation; the second one is to generalize the Melnikov method for chaos detecting in PDEs, enhancing it through SAP (Joint work with Fabio Zanolin).
The second fold’s topic is the transversal notion of persistence in ecological models. Persistence in many-species models can be roughly described as the non-extinction condition: it is akin to the existence of a compact attractor and is strongly related to the flow’s behaviour on the boundary. The theoretical goal is to give some useful criteria for persistence and to prove existence of periodic solutions once persistence is given. As for applications, persistence can be achieved in some interesting ecological systems, from the SIR model to a predator-prey model with a diseased predator which shows some rich dynamics.
The third fold, more recent, is about celestial mechanics and focuses on the N-centre problem. Here the aim is to describe the dynamics of parabolic orbits in the plane in a symbolic way, either in the case of scattering solutions or in the case of semi-bounded and bounded (chaotic) solutions (Joint work with Duccio Papini).