Olinjära dynamiska system

**Valid from:** Autumn 2014**Decided by:** FN1/Anders Gustafsson**Date of establishment:** 2015-03-15

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** FMAN15**Teaching language:** English

To give knowledge of and familiarity with concepts and methods from the theory of dynamical systems which are important in applications within almost all subjects in science and technology.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- independently be able to explain different methods to describe, qualitatively or quantitatively, the solution sets of ordinary differential and difference equations.
- be able to explain basic bifurcation theory and its relevance in technological contexts.
- be able to explain the mathematical meaning of the concept chaotic behaviour and its relevance in technological contexts.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to choose and use methods appropriate to describe, qualitatively and quantitatively, the solution sets of ordinary differential and difference equations.
- be able to use bifurcation theory to qualitatively describe how dynamical systems taken from applications in science and technology depend on a parameter.
- be able independently to identify and describe chaotic behaviour in examples taken from the applications.
- be able to write Matlab and Maple programs to solve mathematical problems within the framework of the course.
- in writing and orally, with clear logic and proper terminology, be able to explain the solution to a mathematical problem within the course.

Dynamical systems in discrete and continuous time. The fixed point theorem and Picard's theorem on the existence and uniqueness of solutions to ordinary differential equations. Phase space analysis and Poincaré's geometrical methods. Local stability theory (Liapunov's method and Hartman-Grobman's theorem). The central manifold theorem. Basic local bifurcation theory. Global bifurcations and transition to chaos. Chaotic and strange attractors (dynamics, combinatorial description).

- Natiello, M. & Schmeling, J.: Lecture notes in Nonlinear Dynamics. Matematikcentrum, 2013.
- Nonlinear dynamics, A two-way trip from physics to mathematics. ISBN 0750303808.

**Types of instruction:** Lectures, exercises

**Examination formats:** Written exam, oral exam, written assignments.
Written and/or oral test, to be decided by the examiner. Some written assignments to be completed before the exam.**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** FMAF05 Systems and Transforms, or similar

**Course coordinators:** **Web page:** http://www.maths.lth.se/matematiklth/personal/joerg/dynsys/huvud.html