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We discuss the appearance of oscillatory and standing wave states insecond-order oscillator networks showing that it is a special case of a moregeneral mechanism involving secondary synchronized clusters induced by inertia.Using a time-periodic mean-field ansatz, we find a bistable mechanism involvinga stable fixed point and an invariant curve of an appropriate Poincar\'e map.The bistability and the devil's staircase associated to the rotation number onthe invariant curve provide an explanation for the appearance of the secondarysynchronized clusters. The effect of inertias in the self-organization processis analyzed through a simplified model. This shows that the effect of giantsynchronized clusters on the other oscillators is weakened by inertias, thusleading to secondary synchronized clusters during the transition process tosynchronization.

Oscillatory states, standing waves, and additional synchronized clusters in networks of second-order oscillators: uncovering the role of inertia