Dynamical Systems

Dynamical system and representation theory

In this talk, a practical method is described for computing the classical normal form of vector fields near the bifurcation point. Some necessary formulas are derived and applied to the anharmonic oscillator, the Bogdanov-Takens bifurcation, the 3D nilpotent problem, and elastic pipe conveying fluid,  to demonstrate the applicability of the theoretical results.
 Then, a review will be given of the developments in the last decade concerning the classification of unique normal forms in 3D nilpotent problems.

From time-average replicator to best-response dynamics, and back

When a game is played over time, the players can change their actions throughout the game. The choice of action can follow different learning mechanisms leading to different types of dynamics.    In this talk I shall look at the relation between the time-average of replicator dynamics (RD) and best-response dynamics (BRD), and its corresponding fictitious play (FP).  It is known that the time-average of RD converges to an invariant set under BRD but not whether given an invariant set BRD there always exists a corresponding RD orbit.

Parabolic Flows Renormalised by Partially Hyperbolic Maps

We will discuss 3-dimensional parabolic flows which are renormalised by circle extensions of Anosov diffeormorphisms (this includes nilflows on the Heisenberg nilmanifold). We use the spectral information of the transfer operators associated to these partially hyperbolic maps to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow. (Joint work with Lucia Simonelli.)