Multiple attractors in grazing-sliding bifurcations in Filippov-type flows

We describe two examples of three-dimensional Filippov-type flows in which multiple attractors are created by grazing-sliding bifurcations. To the best of our knowledge these are the first examples to show multistability due to a grazing-sliding bifurcation in flows. In both examples, we identify the coefficients of the normal form map describing the bifurcation, and use this to find parameters with the desired behaviour. In the first example this can be done analytically, whilst the second is a dry-friction model and the identification is numerical. This explicit correspondence between the flows and a truncated normal form map reveals an important feature of the sensitivity of the predicted dynamics: the scale of the variation of the bifurcation parameter has to be very carefully chosen. Although no detailed analysis is given, we believe that this may indicate a much greater sensitivity to parameters than experience with smooth flows might suggest. We conjecture that the grazing-sliding bifurcations leading to multistability remained unreported in the literature due to this sensitivity to parameter variations.