Collective organization of networked phase oscillators: from explosive synchronization to Bellerophon states

Stefano Boccaletti (CNR - Institute for Complex Systems, Italy)

I will discuss the spontaneous emergence of some collective dynamics in networked phase oscillators.
As a first step, I will discuss how synchronization may emerge in a graph. The vast majority of transitions to synchronization are of the second-order type, continuous and reversible. However, as soon as networked units with complex architectures of interactions are taken into consideration, abrupt and irreversible phenomena may emerge, namely explosive synchronization, which rather remind first-order like transitions.
In the second part of my talk, I will concentrate on a high order coherent state, the Bellerophon state, which is generically observed in the proximity of explosive synchronization at intermediate values of the coupling strength.
Bellerophon states are multi-clustered states emerging in symmetric pairs. In these states, oscillators belonging to a given cluster are not locked in their instantaneous phases or frequencies, rather they display the same long-time average frequency (a sort of effective global frequency). Moreover, Bellerophon states feature quantum traits, in that such average frequencies are all odd multiples of a fundamental frequency.