Jan Sieber,University of Bristol

jan.sieber@bristol.ac.uk

<H3>Dynamics of delayed relay systems</H3>

Relay systems are systems of differential equations that follow 
two different smooth vectorfields in two different regions of their 
physical space. We consider systems where the relay switch between the two 
vector fields is subject to a time delay, giving rise to an 
infinite-dimensional phase space. Typically, this time delay induces 
periodic orbits that switch back and forth between the two vector fields. 
Generically, the return maps of these periodic orbits are smooth on their 
finite-dimensional image, which makes the dynamics near these orbits easy 
to study.  We motivate the study of this type of equations using a relevant 
application, the stabilization of an unstable equilibrium by feedback 
control in the presence of delay in the feedback loop.  Whereas 
stabilization of the equilibrium by linear state feedback becomes 
impossible if the delay in the feedback loop is larger than a critical 
value, this limit does not apply for relay systems. We construct simple 
switching manifolds that permit stable periodic orbits even with 
arbitrarily large delays in the control loop.  We also study the dynamics 
near periodic orbits that graze the switching manifold. These grazing 
events are of codimension one and cause local return maps that are no 
longer smooth but locally piecewise linear or square-root like. The 
bifurcation scenarios induced by these grazing events are different from 
standard bifurcations for smooth maps and have interesting consequences for 
the small-delay limit of delayed relay systems.