Control based bifurcation analysis for experiments

We introduce a method for tracking nonlinear oscillations and their bifurcations in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system nor the ability to set its initial conditions. Instead it relies on feedback stabilizability, which makes the approach applicable in an experiment. This is demonstrated with a proofof-concept computer experiment of the classical autonomous dry-friction oscillator, where we use a fixed time step simulation and include noise to mimic experimental limitations. For this system we track in one parameter a family of unstable nonlinear oscillations that forms the boundary between the basins of attraction of a stable equilibrium and a stable stick-slip oscillation. Furthermore, we track in two parameters the curves of Hopf bifurcation and grazing-sliding bifurcation that form the boundary of the bistability region. The research of J.S. was supported by EPSRC grant GR/R72020/01, and that of B.K. by an EPSRC Advanced Research Fellowship. J. Sieber ( ) Department of Engineering, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK e-mail: [email protected] B. Krauskopf Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, Queen’s Building, University of Bristol, Bristol BS8 1TR, UK