NON-LINEAR DYNAMICAL SYSTEMS | FMA140 |

**Aim**

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. In addition, the course should develop the student's ability in general to assimilate and communicate mathematical theory, to express problems in science and technology in mathematical terms and to solve problems using the theory of dynamical systems.

*Knowledge and understanding*

For a passing grade the student must

be able to describe 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.

*Skills and abilities*

For a passing grade the student must

be able to use bifurcation theory in order to describe qualitatively 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.

with access to the resources of a library be able independently to assimilate and sum up the contents of a text in technology in which methods and results from the theory of dynamical systems are used.

**Contents**

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 Poincare'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 change to chaos. Chaotic and strange attractors (dynamics, combinatorial description).

**Literature**

Spanne, S: Föreläsningar i Olineära dynamiska system. KF-Sigma 1997.

Andersson, K-G & Böiers, L-C: Ordinära differentialekvationer. Studentlitteratur 1992. ISBN 91-44-29952-4.

Some supplementaty material.

Additional text: Solari, H.G., Natiello, M.A. & Mindlin, G.B: Nonlinear dynamics: A two-way trip from physics to mathematics. Taylor & Francis, 1996. ISBN 0750303808.