Noise effects on synchronization in systems of coupled oscillators

We study the dynamics of a system of coupled oscillators of distributed natural frequencies, by including the features of both thermal noise, parametrized by a temperature, and inertial terms, parametrized by a moment of inertia. For a general unimodal frequency distribution, we report here the complete phase diagram of the model in the space of dimensionless moment of inertia, temperature, and width of the frequency distribution. We demonstrate that the system undergoes a nonequilibrium first-order phase transition from a synchronized phase at low parameter values to an incoherent phase at high values. We provide strong numerical evidence for the existence of both the synchronized and the incoherent phase, treating the latter analytically to obtain the corresponding linear stability threshold that bounds the first-order transition point from below. In the limit of zero noise and inertia, when the dynamics reduces to the one of the Kuramoto model, we recover the associated known continuous transition. At finite noise and inertia but in the absence of natural frequencies, the dynamics becomes that of a well-studied model of long-range interactions, the Hamiltonian mean-field model. Close to the first-order phase transition, we show that the escape time out of metastable states scales exponentially with the number of oscillators, which we explain to be stemming from the long-range nature of the interaction between the oscillators.

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“…Similar disappearance of the hysteresis loop with increase of T was reported in Ref. [14]. equilibrium free energy [21].…”

Section: Phase Diagramsupporting

confidence: 88%

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

2014

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“…These characteristic features as K is varied for given T agree well with the results of Ref. [7]. Figure 1 (b) shows that the critical coupling strength K c , beyond which synchronization sets in, increases monotonically with the noise level T , demonstrating the suppression of synchronization by noise.…”

Section: Phase Synchronizationsupporting

confidence: 87%

Phase synchronization and noise-induced resonance in systems of coupled oscillators

Hong

2000

Phys. Rev. E

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We study synchronization and noise-induced resonance phenomena in systems of globally coupled oscillators, each possessing finite inertia. The behavior of the order parameter, which measures collective synchronization of the system, is investigated as the noise level and the coupling strength are varied, and hysteretic behavior is manifested. The power spectrum of the phase velocity is also examined and the quality factor as well as the response function is obtained to reveal noise-induced resonance behavior.

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“…On the other hand, it is expected that fluctuations tend to destabilize various quantization states, reducing the stability interval and the emergence of multistability. This is consistent with smoothing out of the step structure [4] and suppression of hysteresis [22], observed in the presence of noise. In summary, we have constructed a topological argument for subharmonic mode locking in a driven system of coupled oscillators with inertia.…”

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confidence: 88%

Topological interpretation of subharmonic mode locking in coupled oscillators with inertia

Choi

Thouless

2001

Phys. Rev. B

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A topological argument is constructed and applied to explain subharmonic mode locking in a system of coupled oscillators with inertia. Via a series of transformations, the system is shown to be described by a classical XY model with periodic bond angles, which is in turn mapped onto a tightbinding particle in a periodic gauge field. It is then revealed that subharmonic quantization of the average phase velocity follows as a manifestation of topological invariance. Ubiquity of multistability and associated hysteresis are also pointed out.PACS numbers: 05.45. Xt, 74.50.+r In a nonlinear oscillator system driven periodically, the competition between the natural frequency and the driving frequency in general leads to either an almost periodic motion or a periodic one, depending on the parameter range [1]. The latter, called mode locking, is characterized by the quantization at rational values of the average phase velocity. In particular subharmonic mode locking, appearing in the presence of the inertia term, results in the devil's staircase structure. One of the well-known examples is the Josephson junction driven by combined direct and alternating currents, with the capacitance playing the role of inertia [2]. Governed by the same equation of motion as a driven pendulum, it displays dc voltage plateaus in the current-voltage characteristics, known as Shapiro steps [3]. Similar voltage quantization has also been observed in arrays of Josephson junctions, yielding integer giant Shapiro steps [4,5] and subharmonic steps [6] according to the absence/presence of capacitive terms. Unfortunately, in spite of the deceptively simple equation of motion, even the single-junction problem has resisted complete analytical solutions, especially, in the presence of the capacitive term, except for the results mainly based on the circle map [7] and on the approximate analysis by means of expansion and averaging [8,9]. Accordingly, such mode locking phenomena in arrays have been demonstrated mostly by numerical simulations. On the other hand, the topological argument, proposed for the system without the capacitive term [10], reveals topological invariance of the system as the physical origin of quantization [11]. As in the case of the quantum Hall effect [12], the topological argument does not provide quantitative information, e.g., on the locking structure. Nevertheless it not only clarifies the nature of quantization but also provides a link between dynamics and statics by interpreting (dynamical) mode locking in terms of (static) topological invariance.In this work, we construct a topological argument for the system with inertia, and apply the idea to Josephsonjunction arrays or systems of coupled oscillators, with attention to the resulting subharmonic locking. For this purpose, we consider an appropriate canonical transformation of the dynamic equations of motion and the corresponding Fokker-Planck equation, the stationary solution of which gives the effective Hamiltonian in the form of a classical XY model with periodic bond...

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Noise effects on synchronization in systems of coupled oscillators

Cited by 26 publications

References 23 publications

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

Phase synchronization and noise-induced resonance in systems of coupled oscillators

Topological interpretation of subharmonic mode locking in coupled oscillators with inertia

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