We demonstrate a method for tracking the onset of nonlinear oscillations (Hopf bifurcation) in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system but instead relies on feedback controllability. This makes the approach potentially applicable in an experiment. The main advantage of our method is that it allows one to vary parameters directly along the stability boundary. In other words, there is no need to observe the transient oscillations of the dynamical system for a long time to determine their decay or growth. Moreover, the procedure automatically tracks the change of the critical frequency along the boundary and is able to continue the Hopf bifurcation curve into parameter regions where other modes are unstable. We illustrate the basic ideas with a numerical realization of the classical autonomous dry friction oscillator.
|Publication status||Published - 10 Apr 2006|
Additional information: This is the preprint version of a contribution to the 6th International
Conference on Modern Practice in Stress and Vibration Analysis, Bath (United
Sponsorship: The research of Jan Sieber is supported by EPSRC grant GR/R72020/01.
- unstable nonlinear oscillation
- Hopf bifurcation