Synchronous/asynchronous patterns in three-cell networks with multiple time delays

Some patterns of synchrony/asynchrony in the dynamics of coupled cell systems can be predicted by symmetry. However, in the system without symmetry, the different patterns of periodic solutions may exist as well. We consider a general model including three cells with multiple time delays that connect in any possible manner. Our approach is based on the analytic construction by using a perturbation procedure together with the Fredholm alternative theory. Then we employ the PoincaréLindstedt series expansion to compute the Floquet exponents which determine the stability. Finally, we resort to numerical computation to get some insights about the leading term in the Floquet exponents, and the numerical simulations are given to confirm the theoretical results.