# Numerical Path Following and Bifurcation, TM

4 credit units

### Starting on Monday the 12th of March 2001, 1315, in MD1, Mathematical Center

Swedish title: Parameterberoende ekvationer och bifurkationer (4 poäng)

Homepage of Göran Starius
E-Mail:
`goran@math.chalmers.se`
Phone:
772 1097 (office)

### The general aim of the course

The main object of the course is to study methods for parameter dependent nonlinear systems of algebraic or transcendental equations. In applications such systems are often arrived at after a discretization of a parameter dependent continuous problem, most commonly a partial differential equation problem. Since we will study equilibria, stability considerations will play an important part and so will the underlying time dependent problem.
Because the systems are underdetermined the solution set will be a curve system, in the one parameter case. The determination of the structure of the solution set, including bifurcation and limit points, is an important and often extensive task.
There are several interesting engineering problem classes that lead to such nonlinear equations. One practically important class is buckling in structural mechanics. In fluid dynamics there are numerous bifurcation problems. To mention just a few, the fluttering of an airfoil, which will occur if the passing flow is fast enough, the vibrations of tubes depending on the speeds of the internal and outer flow, the Taylor vortex flow and driven cavity flow in hydrodynamics. In chemical kinetics the problems often have several possible steady states and can generally be modeled by nonlinear systems of algebraic equations.
The course is a mixture of mathematics, numerical analysis and scientific computing and is application oriented.

### The main themes of the course

Equilibrium points and stability for autonomous systems of ordinary differential equations. Singular points - limit points and bifurcation points and Hopf bifurcation. Path following using local parametrizations and the study of different predictor-corrector methods. Detection and calculation of singular points, the latter by using bifurcation equations. Discretization of certain differential equation problems in connection to the assignments mentioned below.

### Literature

R. Seydel: Practical Bifurcation and Stability Analysis: From equilibrium to Chaos. Springer-Verlag, 1994.
Can for e. g. be bought from the web-bookstore: http://www.blackwell.co.uk
or from the bookstore Cremona at Chalmers.

### Prerequisites

Only elementary calculus, linear algebra, numerical analysis and some familiarity with computing.

### Examination

Three completed assignments each consisting of a Matlab based part and a paper and pen part. Written examination.

### SCHEDULE

Week no 11-14 , Monday 1315-1500 , Wedesday 0800-0945 , room: 2306 (seminarierum 4)

Week no 17-20 , Monday 1315-1500 , room: 2306

Note that we will use the room MD1 the first time(March 12).

Last change: 2001-03-06 GS