Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems

Kowalczyk, Piotr and Glendinning, Paul (2011) Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems. [MIMS Preprint]

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Abstract

In the paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems). We consider phase space topology of systems with 1-dimensional slow dynamics and 1-dimensional fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and the di®erentiability is lost across the switching surface. In the full system the slow manifold is no longer continuous, and there is an O(") discontinuity across the switching manifold, but the discontinuity cannot qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncov- ered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and the presence of a boundary-equilibrium bifurcation of a fold type is shown.

Item Type:	MIMS Preprint
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:	Ms Lucy van Russelt
Date Deposited:	06 Mar 2011
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1590

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