Resonance Zones in Two-Parameter Families of Circle Homeomorphisms

Abstract

We consider a two-parameter family of diffeomorphisms of the circle where one of the parameters controls the amount of rigid rotation while the second controls the nonlinearity. In particular, we show that the regions in the parameter plane for which the map has a periodic orbit of a particular rotation number (resonance zones) increase in size linearly as the second parameter is increased from zero. This is a discretization of the phenomenon known as “phase locking” for ordinary differential equations. Using this, we obtain some results on the smoothness of the curves between the resonance zones.