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Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Jul;92(1):012904. Epub 2015 Jul 6.

Explosive synchronization is discontinuous.

Author information

Institute of Physics, Potsdam University, 14469 Potsdam, Germany.
Department of Physics, East China Normal University, Shanghai 200062, China.
Department of Mathematics, Imperial College London, London SW72AZ, United Kingdom.
Institute of Mathematical and Computer Sciences, Universidade de São Paulo, São Paulo, Brazil.


Spontaneous explosive is an abrupt transition to collective behavior taking place in heterogeneous networks when the frequencies of the nodes are positively correlated with the node degree. This explosive transition was conjectured to be discontinuous. Indeed, numerical investigations reveal a hysteresis behavior associated with the transition. Here, we analyze explosive synchronization in star graphs. We show that in the thermodynamic limit the transition to (and out of) collective behavior is indeed discontinuous. The discontinuous nature of the transition is related to the nonlinear behavior of the order parameter, which in the thermodynamic limit exhibits multiple fixed points. Moreover, we unravel the hysteresis behavior in terms of the graph parameters. Our numerical results show that finite-size graphs are well described by our predictions.


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