Transitions from spatiotemporal chaos to cluster and complete synchronization states in a shift-invariant set of coupled nonlinear oscillators

We study the spatiotemporal dynamics of a ring of diffusely coupled single-well Duffing oscillators. The transitions from spatiotemporal chaos to cluster and complete synchronization states are particularly investigated, as well as the Hopf bifurcations to instability. It is found that the underlying mechanism of these transitions relies on the motion of the representative points corresponding to the system's nondegenerated spatial transverse Fourier modes in the parametric Strutt diagram. A scaling law is used to demonstrate that the compact interval of the scalar coupling parameter values leading to cluster synchronization broadens in a square-power-like fashion as the number of oscillators is increased. The analytical approach is confirmed by numerical simulations.