Dynamical systems describe the evolution of the possible states of the system (forming the state space) as time varies. In practical examples these systems depend on parameters: for some coefficients the values are only approximately known and other parameters enter from the outset as values to be controled and adjusted. Bifurcation theory studies how the behaviour of dynamical systems changes under variation of parameters, especially where a quantitatively small change of a parameter value leads to a qualitative change in the dynamics. This concerns both discrete and continuous dynamical systems.
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Bifurcations of equilibria: saddle-node and Hopf bifurcation. Bifurcations of fixed points, bifurcations of periodic orbits. Simplifying co-ordinates, Lie brackets, normal form theory, normally hyperbolic invariant manifolds. Bifurcation diagrams of systems depending on two parameters. Dynamical systems preserving an extra structure. Families of conditionally periodic tori, quasi-periodic bifurcations. Delay equations and their bifurcations.
Teachers: Heinz Hanßmann and Yuri Kuznetsov
Learning goals: After completion of the course, the student
1) is able to rework a given text into a coherent and understandable presentation;
2) has a good understanding of the mathematics in the field of the seminar;
3) can formulate relevant and challenging exercises.
The grade of the course is determined by one or more presentations (together 80%) and the homework assingments accompanying the other presentations (20%).
Toetsmatrix:
| |
Presentation(s) |
Homework |
Total |
| Goal 1 |
40 |
0 |
40 |
| Goal 2 |
20 |
20 |
40 |
| Goal 3 |
20 |
0 |
20 |
| Total |
80 |
20 |
100 |
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