Control-based exploration of non-linear systems.

Control-based continuation (CBC) is an experimental method used to explore the bifurcation structure of non-linear systems. Feedback control and continuation techniques are used to discover and stabilize a system’s steady states non-invasively, thus allowing them to be observed. To date CBC has been applied to simple mechanical systems such as oscillating pendulums, springs and bending beams. We apply CBC in a spatially extended system for the first time. The system investigated is an air bubble confined in a Hele-Shaw channel filled with silicone oil. The bubble is placed in the centre of a straining flow and is unstable to both translation and deformation. Real-time feedback control is used to achieve bubble steady-states by injecting/withdrawing fluid from the channel, based on the bubble’s position and shape. A bifurcation diagram is constructed mapping the bubble’s deformation to the injection flow rate of the straining flow. Two unstable solution branches are found which are not observable otherwise.