Numerical continuation is a computational tool for investigating the bifurcation structure of a dynamical system with respect to the system parameters. It is most often used to "carve up" parameter space into regions of qualitatively different behaviour by finding and tracking bifurcations (e.g., Hopf bifurcations) as the system parameters change. This talk will give a brief introduction to the theory behind numerical continuation and go on to discuss recent developments in the field.
When a differential equation model of a particular experiment is available it is (reasonably) straightforward to apply the standard techniques of numerical continuation to find and continue, for example, periodic orbits by means of collocation with orthogonal polynomials or with some other discretisation method. Similarly, if instead a black-box numerical integrator (e.g., a finite-element solver) is available, numerical continuation can be combined with a shooting-type method to find the orbits. This talk will focus on the case where neither an explicit model or black-box integrator are available. Instead, we focus on the application of numerical continuation techniques to physical lab experiments with all the associated constraints.
This talk will outline possible solutions to this problem using methods such as Pyragas-type time delay control and OGY control. In both these cases, the desire is to make the controller non-invasive, that is, the invariant sets of the underlying experiment (whether stable or unstable) are preserved in the controlled system. The key idea is that in the controlled system the unstable orbits become stable, thus enabling numerical continuation to be employed. Initial experimental work on several nonlinear devices will be presented and future directions on tracking bifurcations in physical experiments will be discussed.