Grazing-sliding bifurcations, the border collision normal form, and the curse of dimensionality for nonsmooth bifurcation theory

Mike R Jeffrey, P Glendinning

Research output: Contribution to journalArticle (Academic Journal)peer-review

117 Downloads (Pure)

Abstract

In this paper we show that the border collision normal form of continuous but nondifferentiable discrete time maps is affected by a curse of dimensionality: it is impossible
to reduce the study of the general case to low dimensions, since in every dimension the bifurcation produces fundamentally different attractors (contrary to the case of smooth systems).
In particular we show that the n-dimensional border collision normal form can have invariant
sets of dimension k for integer k from 0 to n. We also show that the border collision normal
form is related to grazing-sliding bifurcations of switching dynamical systems. This implies
that the dynamics of these two apparently distinct bifurcations (one for discrete time dynamics, the other for continuous time dynamics) are closely related and hence that a similar
curse of dimensionality holds for this bifurcation.
Original languageEnglish
JournalNonlinearity
Publication statusIn preparation - 2012

Fingerprint Dive into the research topics of 'Grazing-sliding bifurcations, the border collision normal form, and the curse of dimensionality for nonsmooth bifurcation theory'. Together they form a unique fingerprint.

Cite this