The Neimark-Sacker bifurcation, or Hopf bifurcation for maps, is a well-known bifurcation for smooth dynamical systems. At a Neimark-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 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 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 Poincare 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 Neimark-Sacker bifurcation. We give a normal form of the Poincare 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 Neimark-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 piecewise-smooth system. Nevertheless, the unfolding near the double border-collision points is also predicted as a typical scenario for nondegenerate smooth systems.
|Published - 2009
Bibliographical noteAdditional information: With five accompanying animations (GIF format).
Sponsorship: RS was supported by grant EP/C544048/1 from the Engineering and Physical Sciences Research
Council (EPSRC) and HMO by an EPSRC Advanced Research Fellowship and an IGERT grant.
- Engineering Mathematics Research Group