In this thesis we study the dynamics of the in-phase mode for a system of three coupled van der Pol oscillators. Two of the oscillators are taken to be identical and coupled indirectly via a third oscillator whose natural frequency may vary. We use the singular perturbation method known as two-variable expansion to obtain a slow-flow, which is then analyzed using the computer algebra system MACSYMA. We find analytical representations for saddle-node and Hopf bifurcation curves in the parameter space and explore the dynamics found in the in-phase phase space.

The motivation for this work comes from the presence of circadian melatonin rhythms in the eyes of Japanese quail. Recent experiments showing the rhythms in the eyes to be tightly coupled and in-phase with each other have strengthened the hypothesis that the eyes are the location of the central pacemaker for Japanese quail. The melatonin rhythm in each eye is modeled as a van der Pol limit-cycle oscillator. Furthermore, the eyes cannot directly communicate to each other, but do so via a connection to an extra-ocular circadian system, here represented by the third oscillator.