Synchronization of limit cycle oscillations in diffusively-coupled systems

We present analytical and numerical conditions to verify whether limit cycle oscillations synchronize in diffusively coupled systems. We consider both compartmental ODE models, where each compartment represents a spatial domain of components interconnected through diffusion terms with like components in different compartments, and reaction-diffusion PDEs with Neumann boundary conditions. In both the discrete and continuous spatial domains, we assume the uncoupled dynamics are determined by a nonlinear system which admits an asymptotically stable limit cycle. The main contribution of the paper is a method to certify when the stable oscillatory trajectories of a diffusively coupled system are robust to diffusion, and to highlight cases where diffusion in fact leads to loss of spatial synchrony. We illustrate our results with a relaxation oscillator example.