Applied Maths Seminar: Ian Melbourne -- Corkscrews and boomerang-like motions

I will describe recent work by PhD student David Chan on dynamical systems with symmetry
group E(3) -- rotations and translations in 3-D space.  This is the 3-D counterpart to
Barkley’s work on the transition from meandering to linear drifting spirals in planar
systems.  

The simplest dynamics (relative equilibrium) is a rigid corkscrew motion.  Hopf
bifurcation (or periodic forcing) typically induces a periodic oscillation of the
underlying corkscrew.  

The resonant case is much more interesting and quite surprising.  This occurs when the
frequency of rotation around the original corkscrew is an integer multiple of the
Hopf/forcing frequency.  Chan shows that resonant Hopf bifurcation (a codimension two
phenomenon) leads to a boomerang-like motion.