On linearly coupled relaxation oscillations

Bélair, Jacques; Holmes, Philip

doi:10.1090/qam/745099

Authors: Jacques Bélair and Philip Holmes
Journal: Quart. Appl. Math. 42 (1984), 193-219
MSC: Primary 58F10; Secondary 34C15, 34E15, 70K05
DOI: https://doi.org/10.1090/qam/745099
MathSciNet review: 745099
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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 $1:\omega$. We give additional details on nonperiodic motions for the special case of $\omega = 2$.

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