Co-dimension-Two Grazing Bifurcations in Single-Degree-of-Freedom Impact Oscillators

Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.