Bifurcation theory has undergone a big development with inspiration from dynamical systems theory, computational dynamics, etc. As a result, bifurcation analysis has become a very important tool to study the dynamics of the nonlinear dynamical systems. State-of-the-art tools for the bifurcation analysis are therefore important to improve our understanding of real systems coming from biological, chemical or physical sciences or engineering.

Our aim is to offer an approach to the main ideas on bifurcation analysis through a variety of topics explaining the most updated numerical techniques for dynamical systems described by either ODEs or PDEs.

The first two courses will deal with the bifurcation analysis of ODEs related to the biological sciences, showing phenomena such as bursting or multistability. Multistability enhances the flexibility of nervous systems and has far reaching implications for motor control, dynamic memory, information processing, and decision making of humans and animals.

The two other courses will deal with the bifurcation theory applied to PDEs such as the ones coming from atmospheric and oceanic models, the dynamics of fluid particles, its stability properties and bifurcations. Efficient algorithms for the continuation of periodic orbits of high-dimensional dissipative dynamical systems, and for analyzing their stability will be applied to thermal convection fluid dynamics problems.

The advanced course length will be one week. It will consist on 4 short courses (4-5 hours each course and some computer practicum, 20 hours in total) plus several 45 minutes or 20 minutes talks.