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**1 - 1**of**1**### UNIQUENESS OF RELAXATION OSCILLATIONS: A CLASSICAL APPROACH

"... Abstract. In a recent paper in this Journal, ..."

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by I Kosiuk, P Szmolyan

Venue: | Siam J. Appl. Dyn. Sys. |

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S P Hastings , J B Mcleod

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...es. All the proofs of uniqueness which we have seen for systems in this generality have used “energy” functions of some kind, perhaps in complicated ways, which means that considerable ingenuity may be necessary to use these methods for more general systems. In this paper we will consider the case of “relaxation oscillations”, which are seen in (1.1) when ε is small. Such systems have a parameter, say ε, such that if ε = 0 then one of the equations is degenerate in some way. We will not attempt to give a general definition of relaxation oscillations. The motivation for this paper was the work [2], where a system which to our knowledge has not been put in Lienard form was studied using the methods of geometric perturbation theory. This system was considered earlier, for example in [7], as a model of glycolytic oscillations. The geometry of the phase plane for this model is similar to that of (1.1). This means that it is easy to give a geometric argument to show the existence of a 1 2 S. P. HASTINGS. AND J. B. MCLEOD periodic solution, assuming only that the equilibrium point is unstable. However proving uniqueness and stability of this solution is more challenging. Our first result, Th...

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