JORNADES D'INTRODUCCIÓ ALS SISTEMES DINÀMICS I A LES EDP'S (JISD2006)

Nonsmooth dynamical systems occur in a wide variety of applications including systems with impacts, stick-slip friction or saturation effects. The dynamics of these systems is very different from those of smooth systems. It will be shown how to classify all possible behaviours of nonsmooth linear maps, both in 1D and in 2D, both for continuous maps and for discontinuous maps. In addition, the behaviours present in nonsmooth nonlinear maps will also be considered.

The basic theory of codimension 1 nonsmooth bifurcations will be covered and examples will be given of codimension 2 bifurcations. Period adding bifurcations will be discussed as well as robust chaos. It is planned to show how nonsmooth bifurcations can be removed by the addition of smoothing effects or of noise.

Throughout the course, the theory will be illustrated by many examples taken from applications. In particular a worked example taken from engineering (high speed milling) will be used to illustrate many aspects of the theory. Also improvements to secure communication will be demonstrated by ann application of the basic theory and observed behaviour in DC/DC converters will be explained. Finally students will be introduced to the many unsolved problems and challenges that still dominate this subject.

This course will be concerned with the analysis of the structural stability of piecewise smooth dynamical systems. It will be shown that these systems can undergo unexpected transitions due to the presence of discontinuity boundaries partitioning the state space into different regions associated to different system functional forms. These transitions are unique to nonsmooth dynamical systems and cannot be observed in their smooth counterparts.

Grazing bifurcations will be discussed in detail and related to the occurrence of complex dynamics and the onset of instability in some representative examples in mechanical engineering, systems biology and power electronics. The structural stability of Filippov systems will be also discussed, highlighting the role played by sliding motion in organising a different class of events termed as sliding bifurcations.

A methodology to derive an analytical approximation of the system Poincare map close to each of these nonsmooth bifurcation events will be presented. These maps can be used to characterise analytically the structural stability of the systems of interest but also, as will be suggested in the talk, to design nonconventional switching controllers for smooth and nonsmooth dynamical systems.

The first aim of the course is to describe the main mathematical techniques and results used in the exploration of the semiclassical and quantum behavior of chaotic dynamical systems. The level of the course mainly addresses graduate students in mathematics and physics, even if also both undergraduate students and young researcher should benefit from it.

After a general introduction to the main classical results and definitions concerning the theory of classical chaotic dynamical systems, quantization and semi-classical analysis, the course will focus on recent mathematical and numerical results concerning the statistical properties of quantum eigenvalues/eigenfunctions and (long) time propagation of quantum states for quantized discrete dynamical systems.

Hyperbolic chaotic maps on the two-dimensional torus, together with certain billiard flows, will represent the main examples discussed through all the course. Finally, a particular emphasis will be put on the possible prospectives and relations with contiguous field, such as number theory and the relations between the theory of quantum maps described in the course and quantum computation.

The basic prerequisite for the course are basic and fundamental notions in: calculus, measure and ergodic theory, classical and quantum mechanics and number theory.

An idea of the mathematics and of the results we aim to present in this course, without some more recent advance, are contained in The Mathematical Aspects of Quantum Maps", M. Degli Esposti and S. Graffi Eds., Lecture Notes in Physics 618, Springer-Verlag (2003).

Reaction-diffusion equations arise in many fields in physics, chemistry and biology. One of the most interesting features in unbounded domains is the propagation of fronts. The course will be devoted to propagation phenomena in reaction-diffusion equations, with focus on well-known planar fronts, on curved fronts in homogeneous media, and on pulsating fronts in periodic excitable media. Such problems involve linear and nonlinear elliptic and parabolic equations.

It is of interest to study the effects of various phenomena such as diffusion, reaction, advection, as well as the geometry of the domain, on the speed of fronts. Some asymptotic regimes, like the limit of large drifts, and some more general notions could also be discussed.

This course is divided into two parts. In Part I, I will explain the basic theory of order-preserving dynamical systems that was developed mainly in 1980's and 1990's. In particular, we discuss the relation between symmetry and stability, and also discuss convergence of orbits near a manifold of equilibria. This theory will then be applied to the study of travelling waves.

In Part II, I will discuss blow-up in nonlinear heat equations. Topics include: classification of type I and type II blow-up, local and global asymptotics, critical exponents, heteroclinic connections via singular solutions. Emphasis will be placed on dynamical systems point of views.

In this set of lectures I will describe how one can use ideas of dynamical systems theory to give a quite complete picture of the long time asymptotics of solutions of the two-dimensional Navier-Stokes equation.

I will discuss the existence and properties of invariant manifolds for dynamical systems defined on Banach spaces and review the theory of Lyapunov functions, again concentrating on the aspects of the theory most relevant to infinite dimensional dynamics. I will then explain how one can apply both of these techniques to the two-dimensional Navier-Stokes equation to prove that any solution with integrable initial vorticity will will be asymptotic to a single, explicitly computable solution known as an Oseen vortex. If time permits I will describe certain extensions of this theory to the three-dimensional Navier-Stokes equations.