Effective potential approach to hybrid synchronization transitions

Abstract

The Kuramoto model exhibits different types of synchronization transitions depending on the type of natural frequency distribution. To obtain these results, the Kuramoto self-consistency equation (SCE) approach has been used successfully. However, this approach affords only limited understanding of more detailed properties such as the stability and finite size effect. Here, we extend the SCE approach by introducing an effective potential, that is, an integral version of the SCE. We examine the landscape of this effective potential for second-order, first-order, and hybrid synchronization transitions in the thermodynamic limit. In particular, for the hybrid transition, we find that the minimum of effective potential displays a plateau across the region in which the order parameter jumps. This result suggests that the effective free energy can be used to determine a type of synchronization transition. For finite systems, the effective potential contains local minima at which the system can be trapped. Using numerical simulations, we determine the stability of the system as a function of system size and simulation time.

Hysteresis and criticality in hybrid percolation transitions

“…A hybrid phase transition is a discontinuous transition but with criticality. Such a transition occurs in several a) Electronic mail: bkahng@snu.ac.kr complex systems, including cascade failure of interdependent networks 5 , k-core [6][7][8] , cooperative epidemic spreading [9][10][11][12] and synchronization [13][14][15][16][17] . However, a possible hysteresis together with critical behavior in hybrid phase transitions has been rarely reported thus far, even though hysteresis in discontinuous transitions was investigated in explosive synchronization [18][19][20] and percolation [21][22][23][24][25][26][27][28][29] in complex systems.…”

“…Thus, the transition is hybrid. A recent study showed that an ad-hoc potential 17 , proposed analogously to the Landau free energy, exhibits a flat landscape as it appears in colloidal metamagnetic systems 3,4 . The order parameter is uniquely determined except at the critical point.…”