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Syllabus for

Academic year
TMA146 - Numerical path following and bifurcations
 
Owner: TM
5,0 Credits (ECTS 7,5)
Grading: TH - Five, Four, Three, Not passed
Level: A
Department: 0702 - Matematik MV CTH/GU


Teaching language: Swedish

Course module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 No Sp
0101 Examination 5,0 c Grading: TH   5,0 c    

In programs

TM Teknisk matematik, Year 2 (elective)
EMMAS MSc PROGR IN ENGINEERING MATHEMATICS, Year 1 (elective)
TAUTA AUTOMATION AND MECHATRONICS ENGENEERING, Year 4 (elective)
TDATA COMPUTER SCIENCE AND ENGINEERING, Year 3 (elective)
TDATA COMPUTER SCIENCE AND ENGINEERING - Algorithms, Year 4 (elective)

Examiner:




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Eligibility:

For single subject courses within Chalmers programmes the same eligibility requirements apply, as to the programme(s) that the course is part of.

Course specific prerequisites

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

Aim

The main object of the course is to study methods for parameter dependent nonlinear systems of algebraic or transcendental equations. In applica-tions such systems are often arrived at after a discretization of a parameter dependent continuous problem, most commonly a partial differential equa-tion 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.

Content

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 parametriza-tions 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 prob-lems in connection to the assignments mentioned below.

Literature

R. Seydel: Practical Bifurcation and Stability Analysis: From equilibrium to Chaos. Springer-Verlag, 1994.

Examination

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


Page manager Published: Thu 03 Nov 2022.