Bifurcations which destroy the out-of-phase mode in a pair of linearly coupled relaxation oscillators

This paper presents two mechanisms, the Hopf bifurcation and the infinite period bifurcation, which lead to non-existence of the out-of-phase mode of oscillation in a pair of identical van der Pol relaxation oscillators with diffusive coupling in the displacements and velocities. We show that the out-of-phase mode is associated with a periodic solution in a two dimensional invariant out-of-phase manifold. Phase plane techniques are used to find the region of parameter space where this periodic solution exists and how it is destroyed at the boundaries of this region.