Adaptive detection of instabilities: an experimental feasibility study

We implement a practical protocol for the active, on-line detection of bifurcations in experimental systems, based on real-time identification and feedback control ideas. Current experimental practice for the detection of bifurcations typically requires long observation times in the vicinity of marginally stable solutions, as well as frequent re-settings of the experiment for the detection of turning point or subcritical bifurcations. The approach exemplified here addresses these issues drawing from numerical bifurcation detection procedures. The main idea is to create an augmented experiment, using the experimental bifurcation parameter(s) as additional state variables. We implement deterministic laws for the evolution of these new variables by coupling the experiment with an on-line, computer-assisted identification/feedback protocol. The “augmented” experiment (the closed-loop system) thus actively converges to what, for the original experiment (the open-loop system), is a bifurcation point. We apply this method to the real-time, computer-assisted detection of period-doubling bifurcations in an electronic circuit. The method succeeds in actively driving the circuit to the bifurcation points, even in the presence of modest experimental uncertainties, noise, and limited resolution. The active experimental tracing of a codimension-1 bifurcation boundary in two-parameter space is also demonstrated.