26 Sep 2002

Ida Friestad Pedersen "Investigation of parametric resonance in systems with many degrees of freedom".

I will start by describing the phenomenon of parametric resonance (without mixing in other kinds of processes) and specifying some necessary conditions on a system in order for this to occur. Then I will find a normal form and draw the bifurcation diagrams and phase portraits for a general one-dimensional system that fulfills these conditions. 
The next step will be to extend this to a general system with n degrees of freedom by using a Lagrangian formulation, and show that many results from the one-dimensional case still hold. In particular, we will see that the relatively simple normal form and bifurcation diagrams obtained in the  one-dimensional case can be viewed as typical for the case of parametric resonance in a system with n degrees of freedom. However, we will also see that in certain exceptional and transitional cases of internal resonance between two of the modes, the response of the system can be very complicated (even chaotic).