# NONLINEAR DYNAMICAL SYSTEMS 2006-2007

Kursledare: Mario Natiello tel 046/2220919

Förkunskaper: Nödvändiga förkunskaper är matematikkurserna till och med Linjär analys. Det är en fördel att ha tränat analytiska bevis, t ex i fortättningskursen Analysens grunder.

Tid: Lp 1 och 2 enligt schema för F4, E4, D4, I4.

Omfattning: Föreläsningar 28 timmar. Övningar 14 timmar. Tre inlämningsuppgifter och en tentamen.

Språk: English (vid behov)

Poäng: 4p poäng i civilingenjörsexamen. Kursen kan kompletteras med ett projekt och ger då 6p.

Litteratur:

• Solari, Natiello, Mindlin: Nonlinear dynamics, IOP, 1996.
• S. Spanne, Föreläsningar i Olineära dynamiska system, LTH, 1995
• K. G. Andersson, L-Ch. Böiers, Ordinära differentialekvationer, Studentlitteratur

Anmälan: Via kursanmälningssystemet.

More than 100 years ago it was discovered that processes occurring in nature may exhibit a very complicated and even chaotic behavior. These systems are highly unpredictable by means of standard methods. They are of a completely different nature than linear systems. This observation initiated a new and very powerful field in mathematics and natural sciences: the theory of nonlinear dynamical systems.

The mathematical theory of dynamical systems investigates those general structures which are the basis of evolutionary processes. This kind of processes is met in most scientific fields, from population Dynamics, to Laser Physics or Physical Chemistry. Most derived properties are applicable to many different evolutionary models no matter whether they occur in mathematics, physics, biology, economy or other fields. The aim of the theory of dynamical systems is to give qualitative assertions about such processes. Because one meets evolutionary processes in so many different fields their general nature is very complex and rich. Nevertheless the dynamical systems theory substantially helps to understand, predict or even control chaotic behaviour, i.e. one can ''order the chaos''.

On one hand the theory of nonlinear dynamical systems is a field by its own while on the other hand it developed many powerful tools which can be used all over mathematics and natural sciences. Vice versa the theory of nonlinear systems uses methods from nearly all parts of mathematics and is strongly influenced by other sciences.

The aim of the lectures is to develop an insight into this modern theory. A special emphasis is given to the problems and questions which occur from physics to automatic control theory.

## Contents

Continuous and discrete time systems
ODE's as examples of dynamical systems (existence and uniqueness of solutions)
Phase space analysis (local structures) and geometric methods
Local and structural stability (Grobman--Hartman theorem, Lyapunov theory)
Chaotic and strange attractors (dynamic, combinatorial description)
Bifurcation theory and transition to chaos

Lars Vretare