Abstract. Numerical techniques for dynamical systems are well developed today, at least for the computation of the most basic elements, and are implemented in packages of widespread use AUTO. However, some of these techniques are not viable when the systems have large dimension, like those typically arising from the discretization of partial differential equations. We will review the basic techniques for the computation and stability analysis of branches of equilibria and periodic orbits, focusing on the changes needed to deal with large-dimensional systems.

Remark. For this course, it is advisable to carry a personal laptop with the software MATLAB or OCTAVE already installed, in order to be able to practice on the case studies to be proposed.

Abstract. Dynamical systems is a very active field in pure and applied mathematics, that involves tools and techniques from many other areas such as analyses, geometry and number theory.

A dynamical system can be obtained by iterating a function or letting evolve in time the solution of a differential equation. Even if the rule of evolution is simple, the long term behavior of the system is often chaotic, meaning that the system depends on initial data. The most significative example of such system is the one given by the Lorenz equations: it has three equations of two degree polynomial on three variables.

The course will provide an introduction to Dynamical Systems, exploring the dynamics of some of the basic examples of this theory.

Driven by those examples we will introduce some of the phenomena and main concepts which one is interested in the study of this subject.

The course will cover:

Circle rotations; expanding maps of the circle and the shift map, the quadradict family, Smale horseshoe, Markov partitions and the CAT map and toral automorphisms, Limit inverse and the solenoide, Geometric Lorenz Flows.

Along the presentation of the examples, we will develop the basic notions as dynamical systems, orbits, fixed points, source, saddle, sink hyperbolic fixed points. After, we introduce a basic topological dynamics as transitivity, mixing, topological conjugacy and deduce which of the examples given above satisfy such notions.

References:

• Dynamical Systems, Stability, Symbolic Dynamics, and Chaos, C. Robinson.
• Geometric Theory of Dynamical Systems, W. De Melo and J. Palis.
• Three Dimensional Flows, V. Araujo and M. J. Pacifico.

Abstract. This course will start by reviewing key concepts of time-varying (i.e. nonautonomous) differential equations, including unique solvability, linearization along solutions, linear theory and stability theory, so that those with little knowledge in the area can appreciate the nonautonomous point of view, in contrast to well-know autonomous theory which is taught in most basic courses on differential equations. After describing those basic notions and some of the mathematical tools (primarily transformations and Taylor series expansion). I will discuss the central notion of exponential dichotomy which is the nonautonomous version of hyperbolicity. We will relate it to linear algebra (eigenvalues and eigenspaces) in the autonomous case. I will then introduce dichotomy spectrum and prove a spectral theorem. All this is part of linear theory and we will then apply the spectral theorem to block-diagonalize linear systems. Once these results of linear theory are discussed we are prepared to develop the nonlinear theory and prove the invariant manifold theorem. By then we have collected the main tools of the geometric theory of nonautonomous differential equations. We now go one step further and study nonautonomous differential equations which are motivated by modern applications e.g. in meteorology, oceanography and fluid dynamics which allow the time only to be in a bounded interval (called finite-time dynamics). None of the above developed techniques apply in this case and we have to start over asking the same questions (linear theory, hyperbolicity, etc.) as before but developing new mathematical tools. We will present recent work, prove a spectral theorem for finite-time dynamics and the course will end with open research questions.

1. Existence and uniqueness
2. Boundary behavior of maximal solutions
3. Linear differential equations
4. Dichotomy spectrum
5. Block diagonalization
6. Invariant manifolds
7. Finite-time dynamics

The course describes classical and important work on nonautonomous dynamical systems, motivates with applications why those results sometimes do not apply and builds on the classical theory to develop modern research results and formulate open problems.

Literature

1. H. Amann, Ordinary Differential Equations: Introduction to Nonlinear Analysis, De Gruyter Studies in Mathematics, 1990.
2. W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, Springer, 1998.
3. S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations 14 (2002), 243-258.
4. S. Siegmund, Reducibility of nonautonomous linear differential equations, J. London Math. Soc. 65 (2002), 397-410.
5. T.S. Doan, D. Karrasch, T.Y. Nguyen, S. Siegmund, A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents, J. Differential Equations 252 (2012), 5535-5554.