My main research interests lie in the application of dynamical systems to investigate complex models of physical systems. I am interested in developing tools for the analysis and prediction of qualitative and quantitative behaviour of dynamical systems. I believe that a strong bridge between theory and applications is essential for the development of a vibrant mathematical community. My research focuses on constructing the building blocks of that bridge. This process has naturally led me into collaborations with a wide spectrum of scientists ranging from medical doctors, biologists, physicists, applied mathematicians to astrophysicists. However a sturdy bridge requires strong foundations. A significant component of my research involves the development of sophisticated mathematical tools and analysis techniques. My interests are essentially divided into two categories: spatially extended systems and low-dimensional chaos:
Spatially extended systems find an extremely wide range of applications in many branches of science. They originate from a complex interplay between local dynamics (e.g. reaction) and a global or localized interaction in space (e.g. coupling or diffusion). The interaction between space and time produces a plethora of rich phenomenological possibilities. These possibilities may be roughly divided into two categories: coherent structures and spatio-temporal chaos.
Within the class of coherent structures I am particularly interested in emergence mechanisms, dynamics, interactions and stability. Some of the applications I currently investigate include: nonlinear propagation (waves, solitons, pulses, fronts), interaction of solitons and pulses (soliton trains, metastability), finite time blow-up (scale invariance, self similarity), nonlinear localization (localized breathers, extended vibrations) and the accurate computation of numerical solutions using sophisticated adaptive mesh techniques.
Often the interaction between space and time does not produce coherent structures, but rather an irregular behaviour both in space and time: spatio-temporal chaos. I am interested in the characterization of spatio-temporal chaos both from a dynamics perspective and from a time series approach. In the former case one assumes complete knowledge of the equations of motion (e.g. equations of our model) and in the latter the only information provided about the dynamics is given by a collection (usually large) of discrete samples of some (or combination) of the system's variables at different spatial locations.
Equally important is the a priori determination of which models or class of models are prone to chaotic evolution. I am interested in a large class of Hamiltonian systems that exhibit a transition from regular to chaotic behaviour as their energy is increased. For some of these systems, it is indeed possible to estimate the energy threshold for this transition by analyzing the geometric features of the potential energy surface.