Finite-size scaling in globally coupled phase oscillators with a general coupling scheme

Abstract

We investigate a critical exponent related to synchronization transition in globally coupled nonidentical phase oscillators. The critical exponents of susceptibility, correlation time, and correlation size are significant quantities to characterize fluctuations in coupled oscillator systems of large but finite size and understand a universal property of synchronization. These exponents have been identified for the sinusoidal coupling but not fully studied for other coupling schemes. Herein, for a general coupling function including a negative second harmonic term in addition to the sinusoidal term, we numerically estimate the critical exponent of the correlation size, denoted by ν+, in a synchronized regime of the system by employing a non-conventional statistical quantity. First, we confirm that the estimated value of ν+ is approximately 5/2 for the sinusoidal coupling case, which is consistent with the well-known theoretical result. Second, we show that the value of ν+ increases with an increase in the strength of the second harmonic term. Our result implies that the critical exponent characterizing synchronization transition largely depends on the coupling function.

Nonstandard transitions in the Kuramoto model: a role of asymmetry in natural frequency distributions

“…We start from referring to two analogies. The Kuramoto model shares macroscopic aspects with a Hamiltonian system in the critical exponents concerning with the applied external force (see [37][38][39][40] for the Kuramoto model and [41][42][43][44][45] for the Hamiltonian system). Another analogy is used in the analysis on the Landau damping [46], which is originally studied in plasma physics, and is applied to the Kuramoto model [47].…”

Microscopic correlations in the finite-size Kuramoto model of coupled oscillators

“…The qualitative features, established in the thermodynamic limit, remain approximately valid for finite ensembles. Here, similar to finite-size effects in equilibrium phase transitions, the order parameter, i. e. the macroscopic mean field, fluctuates with an amplitude that depends on the ensemble size in a nontrivial way [6][7][8][9][10]. These fluctuations are most pronounced close to the criticality, and can be attributed to weak chaoticity of the finite population dynamics [11,12].…”

Nonstandard scaling law of fluctuations in finite-size systems of globally coupled oscillators

“…A synchronization transition in a system of N coupled oscillators takes place when the coupling strength surpasses a critical value at which an order parameter changes from zero to a non-zero value. Universality of synchronization phenomena can be partially confirmed by the fact that the scaling property of the order parameter in the vicinity of the critical synchronization transition point does not depend on the details of the system elements [2,[7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In the thermodynamic limit N → ∞, the study of the universal scaling law with respect to the coupling strength has been a topic of primary interest [7,8].…”