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The Center for Control, Dynamical Systems, and Computation University of California at Santa Barbara
 

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 Grazing-Sliding 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 fractal-like 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 piecewise-smooth systems. Indeed, the dynamics on the polygons is phase-
locked in large regions of the parameter space that form so-called 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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics