Guiding Simulations and Experiments using Continuation

When applying continuation of periodic solutions to high-dimensional finite element models one might face a dilemma. The mesh resolution and thus the dimension N of the model are typically chosen such that a given computer system can store the information necessary to perform one integration step for dimension N, but not for larger dimensions.

In other words, a model is usually implemented as a carefully derived implicit integration scheme tailored for numerically stable simulations with the highest spacial resolution admitted by the computational power available. On the other hand, stable numerical methods for periodic solutions, for example, multiple shooting or collocation, typically require the simultaneous storage and manipulation of information for K>1 states, which would imply that periodic solutions cannot be computed without a significant reduction of the model's resolution.

The recently developed method of control based continuation allows the continuation of periodic solutions without a reduction of the model resolution, and even directly in physical experiments. Moreover, both a simulation as well as an experiment can run asynchronously from the actual continuation method, which communicates with the simulation or experiment by setting a control target and by taking measurements.

The key ideas of this approach are (1) to introduce a control scheme that locally stabilizes periodic solutions without perturbing them, and (2) to use continuation to guide the simulation or experiment around folds and through bifurcation points. In this talk we will present a Matlab toolbox for control based continuation and illustrate its application with a lab experiment of an impact oscillator that exhibits a large hysteresis loop.

We will indicate current challenges with this method and how we intend to tackle them.