On linearly coupled relaxation oscillations

Author:

Bélair Jacques,Holmes Philip

Abstract

We study the dynamical behavior of a pair of linearly coupled relaxation oscillators. In such systems vastly different time scales play a crucial rôle, and solutions may be viewed as consisting of portions of slow drift linked by rapid jumps. This feature enables us to reduce the analysis from four dimensional phase space to that of a two dimensional system with discontinuous but well determined behavior at certain points on the phase plane. We determine the existence and stability of periodic motions for identical oscillators and oscillators with an uncoupled frequency ratio of 1 : ω 1:\omega . We give additional details on nonperiodic motions for the special case of ω = 2 \omega = 2 .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics

Reference28 articles.

1. Bélair [1983a], Phase locking in linearly coupled relaxation oscillators, Ph. D. Thesis, Cornell Univ.

2. Bélair [1983b], Une application de l’analyse nonstandard dans l’etude d’oscillateurs de relaxation, preprint

3. Benoit, J.—L. Callot, F. Diener and M. Diener [1980], Chasse au canard, IRMA

4. Lecture Notes in Mathematics, Vol. 470;Bowen, Rufus,1975

5. On non-linear differential equations of the second order. I. The equation 𝑦̈-𝑘(1-𝑦²)𝑦+𝑦=𝑏𝜆𝑘𝑐𝑜𝑠(𝜆𝑡+𝑎),𝑘 large;Cartwright, M. L.;J. London Math. Soc.,1945

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