Arnol'd Tongues Arising from a Grazing-Sliding Bifurcation

The Nĕımark-Sacker bifurcation, or Hopf bifurcation for maps, is a well-known bifurcation for smooth dynamical systems. At a Nĕımark-Sacker bifurcation a periodic orbit loses stability and, except for certain so-called strong resonances, an invariant torus is born; the dynamics on the torus can be either quasi-periodic or phase locked, which is organized by Arnol′d tongues in parameter space. Inside the Arnol′d tongues phase-locked periodic orbits exist that disappear in saddle-node bifurcations on the tongue boundaries. In this paper we investigate whether a piecewisesmooth system with sliding regions may exhibit an equivalent of the Nĕımark-Sacker bifurcation. The vector field defining such a system changes from one region in phase space to the next and the dividing so-called switching surface contains a sliding region if the vector fields on both sides point towards the switching surface. The existence of a sliding region has a superstabilizing effect on periodic orbits interacting with it. In particular, the associated Poincaré map is non-invertible. We consider the grazing-sliding bifurcation at which a periodic orbit becomes tangent to the sliding region. We provide conditions under which the grazing-sliding bifurcation can be thought of as a Nĕımark-Sacker bifurcation. We give a normal form of the Poincaré map derived at the grazing-sliding bifurcation and show that the resonances are again organized in Arnol′d tongues. The associated periodic orbits typically bifurcate in border-collision bifurcations that can lead to dynamics that is more complicated than simple quasi-periodic motion. Interestingly, the Arnol′d tongues of piecewise-smooth systems look like strings of connected sausages and the tongues close at double border-collision points. Since in most models of physical systems non-smoothness is a simplifying approximation, we relate our results to regularized systems. As one expects, the phase-locked solutions deform into smooth orbits that, in a neighborhood of the Nĕımark-Sacker bifurcation, lie on a smooth torus. The deformation of the Arnol′d tongues is more complicated; in contrast to the standard scenario, we find several coexisting pairs of periodic orbits near the points where the Arnol′d tongues close in the piecewisesmooth system. Nevertheless, the unfolding near the double border-collision points is also predicted as a typical scenario for nondegenerate smooth systems. AMS subject classifications. 70K30, 70K45, 70K50, 70K70, 37E05, 37G15