Multiple attractors in grazing-sliding bifurcations in Filippov type ows

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Abstract

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.

Bibliographical metadata

Original languageEnglish
JournalIMA Journal of Applied Mathematics
StateAccepted/In press - 26 Feb 2016