Applied Maths Seminar: Krauskopf -- Experimental Continuation of Periodic Orbits through a Fold

Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol 

When a mathematical model is available, for example, in the form of a system of ordinary
differential equations, then it is possible to find and follow equilibria, periodic
solutions and their bifurcations in system parameters.  Numerical continuation is today
a well-established tool that is implemented in software packages such as AUTO, DsTool
and Content.  

However, in many situations it is impractical or even intractable to derive a
mathematical model of the system under consideration.  A particular example are hybrid
engineering tests, where a test specimen of interest (for example, a bridge cable) is
tested in the laboratory as if it were part of the entire structure (the bridge).  To
this end, the tested part is coupled dynamically via sensors and actuators to a computer
simulation of the remainder of the structure (such as the bridge deck).  

We present a continuation method that enables one to continue branches of solutions,
including periodic orbits, directly in an experiment.  A control-based setup in
combination with Newton iterations ensures that the periodic orbit can be continued even
when it is unstable.  Our method is demonstrated with the continuation of initially
stable rotations of a vertically forced pendulum experiment through a fold bifurcation
to find the unstable part of the branch.  

This is joint work with Jan Sieber, University of Portsmouth, and Alicia
Gonzalez-Buelga, Simon Neild and David Wagg, University of Bristol.

Location: New Law School Seminar 030 (Building F10)!