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2012.66: Grazing-sliding bifurcations, the border collision normal form, and the curse of dimensionality for nonsmooth bifurcation theory

2012.66: Paul Glendinning and Mike Jeffrey (2012) Grazing-sliding bifurcations, the border collision normal form, and the curse of dimensionality for nonsmooth bifurcation theory.

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Abstract

In this paper we show that the border collision normal form of continuous but non-differentiable 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.

Item Type:MIMS Preprint
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Subjects:PACS 2003 > 05 Statistical physics, thermodynamics, and nonlinear dynamical systems
MIMS number:2012.66
Deposited By:Professor Paul Glendinning
Deposited On:03 July 2012

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