Self-organization dynamics and statistical physics of synchronization in complex systems

Order-parameter theory of self-organized synchronization in complex systems is proposed． Low-dimensional dynamical equations of order parameters are derived in terms of dimension reduction schemes such as Ott-Antonsen ansatz． The mechanisms, forms, and manifestations of the emergence of synchronization are explored at both microscopic and macroscopic levels． Order parameter dynamics of synchronization in heterogeneously coupled oscillators are investigated． It is found that coupling heterogeneity may induce the Bellerophon state． Explosive desynchronization transitions are revealed in coupled oscillators with multiplex network topologies, and these transitions are irreversible． Synchronization dynamics is studied for oscillators with nonlinear couplings． Various transitions and scaling relations are investigated at the onset of synchronization． Three types of transitions (explosive, continuous and hybrid synchronizations) are identified． These studies shed light on both theoretical understandings and potential applications of collective behaviors in complex systems．