(494b) Oscillator Synchronization: Cluster Dynamics of a Complex Phase Transition

Occurring in many natural and engineered systems, the collective synchronization of oscillators has attracted much research attention to understand the behavior. Populations of coupled limit-cycle oscillators tend to synchronize by altering their frequencies and phase angles. Collective synchronization of this type has been observed in numerous biological and engineering systems. The process has similarities to a condensation phase transition, such as crystallization, with monomers attaching to clusters. Following this the well-known similarity of the synchronization to condensation, we explore how a distribution kinetics model for cluster growth can be adapted to describe synchronization dynamics. We find that oscillators that synchronize, or cluster, according to a reversible association-dissociation mechanism demonstrate behavior like the conventional stability analysis. Oscillators proceed through exponential time dependence before taking on power law behavior at intermediate times, as reported for model computations. The power varies for synchronization and depends on the rate coefficient but is constant for desynchronization. In terms of the coupling constant K and its critical value Kc, the coherence increases as (1 - Kc/K)^â, where â = 1, the generic value, rather than 1/2, the special case for sinusoidal coupling. The long-time limit is a stationary equilibrium state. We hypothesize that oscillator synchronization is the reversible addition of individual oscillators to a single synchronized cluster. The oscillators begin in an incoherent state and freeze into coherence by synchronization. Synchronization of the oscillator phase angle is governed by nonlinear dynamics that can be mathematically described by so-called phase models, for example, the Daido and the Kuramoto models. The models, developed for coupled oscillators, such as groups of chorusing crickets, flashing fireflies, and cardiac pacemaker cells, exhibit a spontaneous transition from incoherence to collective synchronization beyond a threshold in coupling strength. The models have been further analyzed and extended and linked to several physical problems, including Landau damping in plasmas, the dynamics of Josephson junction arrays, bubbly fluids, and coupled Brownian ratchets. The collective synchronization of an initially incoherent oscillator population with different natural frequencies is determined by nonlinear differential equations. When the coupling parameter is less than a critical value, the oscillators remain in a completely desynchronized incoherent state. Above this critical value, some oscillators with similar frequencies alter their phase angles and become synchronized. Oscillators with frequencies further away from the mean, i.e., in the tail of the distribution, remain desynchronized. As the coupling parameter is increased, more oscillators are recruited from the distribution and are synchronized. The coupling parameter, by marking the final division between synchronized (clustered) and desynchronized (unclustered) oscillators thus determines their equilibrium ratio. The model involves the cluster dynamics of synchronization, or condensation, where a cluster is defined as the sub-population of oscillators that become locked in phase at one frequency. The number of oscillators in the cluster defines the coherence. The micro-reversibility for individual oscillator synchronization allows macroscopic irreversible evolution to equilibrium (as observed in the phase models), which is well-known and understood in nonequilibrium statistical mechanics. Both synchronization and desynchronization are of interest, and the proposed model handles both without difficulty. Condensation into multiple clusters, rather than a single cluster of one frequency, is also a possible extension of the model. It is most interesting that the Daido model bifurcation behavior is identical to ours when the rate constants are appropriately defined in terms of the coupling constants. The hypothesis that synchronisation is comparable to condensation phenomenon and thus should be expected to follow the mathematics of the proposed clustering kinetics is therefore corroborated by Daido's result. In conclusion, it is perhaps not surprising in view of the universal character of phase transitions that very different processes, including crystallization and oscillator synchronization, display dynamic similarities. The reversible growth-dissociation of monomers on clusters is generic to condensation phase transitions, whether first- or second-order. Here, the rate coefficient for cluster growth replaces the coupling constant in the conventional linear stability approach. Several aspects of the conventional approach are reproduced: synchronization and desynchronization processes evolve to equilibrium states, the time evolution begins with exponential behavior and transitions to power law time dependence before arriving at equilibrium, a Fokker-Planck equation governs the cluster size distribution, and the long-time coherence (number of synchronized oscillators) varies with the rate coefficient in the same way the coherence varies with the coupling constant.