Bernd Krauskopf
Department of Engineering Mathematics, University of Bristol
Experimental Continuation of Periodic Orbits through a Fold
Wednesday 10th March 14:05-14:55pm,
New Law School Seminar 030 (Building F10).
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.