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Coupled phase oscillators are used widely in describing cooperative
phenomena in physics, biology, chemistry, engineering, and so on.
We study the models of Winfree and Kuramoto of synchronization of phase
oscillators. Both models show that for typical distribution of
natural frequencies a synchronous behaviour emerges when coupling
intensity between oscillators exceeds certain value. A coherence is
suitably described with an order parameter. The order parameter
attains non-zero value for couplings stronger than the critical,
thus making the onset of synchronization a phase transition.
For the Kuramoto model the phase transition is of first or second order depending on the type of distribution function of the natural frequencies of the oscillators. The transition is of first order when the distribution has a plateau where the seed of the synchronized cluster is formed. The exponents characterizing the dependence of the order parameter on the coupling strength are derived analytically for both first- and second-order phase transitions. We also consider an analytically solvable version of the Winfree model of synchronization of phase oscillators. It is obtained that the transition from incoherence to partial death state is characterized by third or even higher order phase transitions according to Ehrenfest classification. The order depends on the type of distribution function of natural frequencies of the oscillators. The corresponding critical exponents are found analytically and confirmed numerically. The transition to partial death is considered also in more general setting when the interaction intensity depends on the Kuramoto order parameter r as Krz-1, where z is an additional parameter. If z is smaller than some particular value zc dependent on the distribution of natural frequencies of the oscillators, the critical exponents remain unchanged. For z>zc there is a first order transition with hysteresis. |
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