Arnol'd Tongues Arising from a Grazing-Sliding Bifurcation

The Neimark-Sacker bifurcation, or Hopf bifurcation for maps, is a well-known bifurcation for smooth dynamical systems. At this bifurcation a periodic orbit loses stability, and, except at certain "strong" resonances, an invariant torus is born. The dynamics on the torus is organized by Arnold tongues in parameter space; inside the Arnold tongues phase-locked periodic orbits exist that disappear in saddle-node bifurcations on the tongue boundaries, and outside the tongues the dynamics on the torus is quasi-periodic. In this paper we investigate whether a piecewise-smooth system with sliding regions may exhibit an equivalent of the Neimark-Sacker bifurcation. The vector field defining such a system changes from one region in phase space to the next, and the dividing (or switching) surface contains a sliding region if the vector fields on both sides point toward the switching surface. We consider the grazing-sliding bifurcation at which a periodic orbit becomes tangent to the sliding region and provide conditions under which it can be thought of as a Neimark-Sacker bifurcation. We find that the normal form of the Poincare map derived at the grazing-sliding bifurcation is, in fact, noninvertible. The resonances are again organized in Arnold tongues, but the associated periodic orbits typically bifurcate in border-collision bifurcations that can lead to more complicated dynamics than simple quasi-periodic motion. Interestingly, the Arnold 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 nonsmoothness 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 Neimark-Sacker bifurcation, lie on a smooth torus. The deformation of the Arnold tongues is more complicated; in contrast to the standard scenario, we find several coexisting pairs of periodic orbits near the points where the Arnold tongues close in the piecewise-smooth system. Nevertheless, the unfolding near the double border-collision points is also predicted as a typical scenario for nondegenerate smooth systems.