## Abstract

Sudden changes in a dynamical system can be modelled by mixtures of slow and fast

timescales, or by combining smooth change with sudden switching. In sets of ordinary

diﬀerential equations, the former are modelled using singular perturbations, the latter using

discontinuities. The relation between the two is not well understood, and here we develop

a method called pinching, which approximates a singularly perturbed dynamical system by

a discontinuous one, by making a discontinuous change of variables. We study pinching in

the context of the canard phenomenon at a folded node. The folded node is a singularity

associated with loss of normal hyperbolicity in slow-fast systems with (at least) two slow

variables, and canards are special solutions that characterize the local dynamics. Pinching

yields an approximation in terms of the two-fold singularity of discontinuous (Filippov)

systems, which arises generically in three or more dimensions, and remains a subject of

interest in its own right.

timescales, or by combining smooth change with sudden switching. In sets of ordinary

diﬀerential equations, the former are modelled using singular perturbations, the latter using

discontinuities. The relation between the two is not well understood, and here we develop

a method called pinching, which approximates a singularly perturbed dynamical system by

a discontinuous one, by making a discontinuous change of variables. We study pinching in

the context of the canard phenomenon at a folded node. The folded node is a singularity

associated with loss of normal hyperbolicity in slow-fast systems with (at least) two slow

variables, and canards are special solutions that characterize the local dynamics. Pinching

yields an approximation in terms of the two-fold singularity of discontinuous (Filippov)

systems, which arises generically in three or more dimensions, and remains a subject of

interest in its own right.

Original language | English |
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Publication status | In preparation - 2013 |