1. Center manifold reduction for infinite dimensional systems and normal form theory. Application to elementary bifurcations of reversible systems in small dimensions.
2. Traveling waves in the water wave problem. Spatial dynamics formulation.
3. Traveling waves in infinite lattices. Spatial dynamics formulation.
4. Analytic systems - Analytic center manifold up to exponentially small term. Remarks on "limiting" cases, where the reduction method does not apply.
   

Suppose you have data from a physical system that is behaving chaotically. What do you do? How do you analyze these data? What should you look for? What is the mechanism that generates chaos? For a large class of systems an algorithm now exists for addressing each of these questions successively and successfully. We will go through the steps of this algorithm, showing how each works using experimental data and pointing out the connection with topology. In the process we will develop a classification scheme for strange attractors.

Inside our cells, essential biological functions rely on complex interaction networks where molecular actors regulate each other so as to generate appropriate biochemical signals. Real-time monitoring in living cells has revealed that these networks are dynamic, spatially organized and highly nonlinear. In particular, typical nonlinear behavior such as bistability and oscillations is harnessed to multiplex cellular information, build clocks, memories or decision circuits. In this course, we will discuss a few case studies representative of systems biology. We will illustrate how nonlinear dynamics is essential to understand the design principles of most cellular regulatory networks.

Reservoir computing is a recent paradigm of employing dynamical systems for purposeful computation by means of machine learning. While originating in artificial neural networks, it is applicable to a broad range of dynamical systems. The course will contain a quick introduction to the basic concepts of machine learning and artificial neural networks, the origins of reservoir computing, different learning methods and perspectives on it, examples of computation with different types of dynamical systems, applications, some practical aspects, and trends of current research.