 
Summary: The Center for Control, Dynamical Systems, and Computation
University of California at Santa Barbara
Fall 2009 Seminar Series
Presents
Arnol'd Tongues Arising from a GrazingSliding Bifurcation
Hinke Osinga
Department of Engineering Mathematics
University of Bristol
Friday, November 13, 2009 3:00  4:00 PM ESB 2001
Abstract:
We investigate a system with a sliding surface that has an unstable periodic orbit of focus type. This
scenario is typical for a forced friction oscillator, which we use as the leading example. Sliding corre
sponds to sticking in this model. The instability of the periodic orbit is induced by the friction coefficient,
which decreases with increased relative speed between the contacting surfaces. We derive a normal
form return map near the onset of this instability and show that attracting invariant polygons arise in the
system. We are able to construct a fractallike bifurcation diagram that shows how the number of sides
of the polygon varies as a function of the parameters. The polygons can be viewed as the analogon
of an invariant torus for piecewisesmooth systems. Indeed, the dynamics on the polygons is phase
locked in large regions of the parameter space that form socalled Arnol'd tongues. However, not all is
equivalent to smooth systems! The Arnol'd tongues look more like sausages and there is more to the
