Using central manifold theorem in the analysis of master-slave synchronization networks

This work presents a stability analysis of the synchronous state for one-way master-slave time distribution networks with single star topology. Using bifurcation theory, the dynamical behavior of second-order phase-locked loops employed to extract the synchronous state in each node is analyzed in function of the constitutive parameters. Two usual inputs, the step and the ramp phase perturbations, are supposed to appear in the master node and, in each case, the existence and the stability of the synchronous state are studied. For parameter combinations resulting in non-hyperbolic synchronous states the linear approximation does not provide any information, even about the local behavior of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behavior of the original system in a local neighborhood of these points. Thus, the local stability can be determined.