Multiple attractors in grazing-sliding bifurcations in an explicit example of a Filippov type flow

Glendinning, Paul and Kowalczyk, Piotr and Nordmark, Arne (2011) Multiple attractors in grazing-sliding bifurcations in an explicit example of a Filippov type flow. [MIMS Preprint]

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Abstract

We present a constructed explicit example of a three-dimensional Filippov type flow where we show the birth of multiple attractors in grazing-sliding bifurcations. To the best of our knowledge, it is the first such an example of a Filippov type flow where grazing-sliding bifurcation is shown to trigger birth of multiple attractors, reported in the literature. Three qualitatively different scenarios are shown; namely, birth of period-two and period-three stable orbits with one sliding segment, chaotic attractor coexisting with stable period-three orbit characterised by a segment of sliding motion, and a coexistence of a period-three sliding orbit with two sliding segments and a limit cycle with no sliding segments. Our work reveals an important feature of the normal form map used to construct the Filippov flow that would produce the desired dynamics. Namely, due the fact that the normal form that we use is valid only locally around the grazing-sliding bifurcations, the scale of the variation of the bifurcation parameter past the grazing-sliding had to be carefully chosen to see the dynamics predicted by the map. In other words, sufficiently small neighbourhood where the normal form is valid, in the context of nonsmooth bifurcations, seems to mean different order of magnitude in the range of the bifurcation parameter variation than in the context of smooth bifurcations.

Item Type:	MIMS Preprint
Additional Information:	CICADA
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 34 Ordinary differential equations MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
Depositing User:	Professor Paul Glendinning
Date Deposited:	29 Jul 2011
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1658

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