We study the transient behavior in coupled dissipative dynamical systems based on the linear analysis around the steady state. We find that the transient time is minimized at a specific set of system parameters and show that at this parameter set, two eigenvalues and two eigenvectors of the Jacobian matrix coalesce at the same time; this degenerate point is called the exceptional point. For the case of coupled limit-cycle oscillators, we investigate the transient behavior into the amplitude death state, and clarify that the exceptional point is associated with a critical point of frequency locking, as well as the transition of the envelope oscillation

We propose a method of estimating inter-modular connectivity in a hierarchical modular network. The method is based on an analysis of inverse phase synchronization applied to the local field potentials on a hierarchical modular network of phase oscillators. For a strong-coupling strength, the inverse phase synchronization index of the local field potentials for two modules depends linearly on the corresponding inter-modular connectivity defined as the number of links connecting the modules. The method might enable us to estimate the inter-modular connectivity in various complex systems from the inverse phase synchronization index of the mesoscopic modular activities.