In this work we investigate the stability of synchronized states for the Kuramoto model on scale-free and random networks in the presence of white noise forcing. We show that for a fixed coupling constant, the robustness of the globally synchronized state against the noise is dependent on the noise intensity on both kinds of networks. At low noise intensities the random networks are more robust against losing the coherency but upon increasing the noise, at a specific noise strength the synchronization among the population vanishes suddenly. In contrast, on scale-free networks the global synchronization disappears continuously at a much larger critical noise intensity respect to the random networks.
We investigate synchronization in the Kuramoto model with noise on a star graph. By revising the case of a complete graph, we propose a closed form of self-consistency equation for the conventional order parameter and generalize it for a star graph. Using the obtained self-consistency equation, we demonstrate that there is a crossover between the abrupt synchronization at small noise and the continuous phase transition for quite large noise. We probe this crossover numerically and analytically.
A small-world network (SW) of similar phase oscillators, interacting according to the Kuramoto model is studied numerically. It is shown that deterministic Kuramoto dynamics on the SW networks has various stable stationary states. This can be attributed to the "defect patterns" in a SW network which is inherited to it from deformation of "helical patterns" in its parent regular one. Turning on an uncorrelated random force, causes the vanishing of the defect patterns, hence increasing the synchronization among oscillators for intermediate noise intensities. This phenomenon which is called "stoc…
Reflecting boundary conditions cause two one-dimensional random walks to synchronize if a common direction is chosen in each step. The mean synchronization time and its standard deviation are calculated analytically. Both quantities are found to increase proportional to the square of the system size. Additionally, the probability of synchronization in a given step is analyzed, which converges to a geometric distribution for long synchronization times. From this asymptotic behavior the number of steps required to synchronize an ensemble of independent random walk pairs is deduced. Here the sync…
We present a linear stability analysis of the incoherent state in a system of globally coupled, identical phase oscillators subject to colored noise. In that we succeed to bridge the extreme time scales between the formerly studied and analytically solvable cases of white noise and quenched random frequencies.
We consider one typical system of oscillators coupled through disordered link configurations in networks, i.e., a finite population of coupled phase oscillators with distributed intrinsic frequencies on a random network. We investigate collective synchronization behavior, paying particular attention to link-disorder fluctuation effects on the synchronization transition and its finite-size scaling (FSS). Extensive numerical simulations as well as the mean-field analysis have been performed. We find that link-disorder fluctuations effectively induce {\em uncorrelated random} fluctuations in freq…
Synchronization is a widespread phenomenon in science and technology. We here study noise-induced synchronization in a quantum spin chain subjected to local Gaussian white noise. We demonstrate stable (anti)synchronization between the endpoint magnetizations of a quantum $XY$ model with transverse field of arbitrary length. Remarkably, we show that noise applied to a single spin suffices to reach stable (anti)synchronization, and demonstrate that the two synchronized end spins are entangled. We additionally determine the optimal noise amplitude that leads to the fastest synchronization along t…
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.
Relationships between inter-cluster synchronization phenomena and external noise are studied on the basis of noise level-free analysis. We consider a mean-field model of ensembles of coupled limit cycle oscillators with two natural frequencies, which are subjected to external white Gaussian noise. Using a nonlinear Fokker-Planck equation approach, we analytically derive the order parameter equations associated with nonequilibrium phase transitions in the thermodynamic limit. Solving them numerically, we systematically investigate the model parameter dependences of the appearance and disappeara…
We analyze the noise induced synchronization between a collective variable characterizing a complex system with a finite number of interacting bistable units and time periodic driving forces. A random phase process associated to the collective stochastic variable is defined. Its average phase frequency and average phase diffusion are used to characterize the phenomenon. Our analysis is based on numerical solutions of the corresponding set of Langevin equations.
We report the experimental evidence of noise-induced phase synchronization in a vertical cavity laser. The polarized laser emission is entrained with the input periodic pump modulation when an optimal amount of white, gaussian noise is applied. We characterize the phenomenon, evaluating the average frequency of the output signal and the diffusion coefficient of the phase difference variable. Their values are roughly independent on different waveforms of periodic input, provided that a simple condition for the amplitudes is satisfied. The experimental results are compared with numerical simulat…
A solvable model of noise effects on globally coupled limit cycle oscillators is proposed. The oscillators are under the influence of independent and additive white Gaussian noise. The averaged motion equation of the system with infinitely coupled oscillators is derived without any approximation through an analysis based on the nonlinear Fokker--Planck equation. Chaotic synchronization associated with the appearance of macroscopic chaotic behavior is shown by investigating the changes in averaged motion with increasing noise intensity.
In this article, we present a systematic study of the critical behavior of phase oscillators with multiplicative noise from a thermodynamic equilibrium approach. We have already presented the thermodynamics of phase noise oscillators and mapped out in detail the behavior of free energy, entropy, and specific heat in a previous work [P. D. Pinto, F.A. Oliveira, A.L.A. Penna, Phys. Rev. E 93, 052220 (2016)], in which we also introduced the concept of synchronization field. This proved to be important in order to understand the effect of multiplicative noise in the synchronization of the system. …
We consider a system of phase oscillators with random intrinsic frequencies coupled through sparse random networks, and investigate how the connectivity disorder affects the nature of collective synchronization transitions. Various distribution types of intrinsic frequencies are considered: uniform, unimodal, and bimodal distribution. We employ a heterogeneous mean-field approximation based on the annealed networks and also perform numerical simulations on the quenched Erdos-Renyi networks. We find that the connectivity disorder drastically changes the nature of the synchronization transitions…
We study the noise effects in a driven system of globally coupled oscillators, with particular attention to the interplay between driving and noise. The self-consistency equation for the order parameter, which measures the collective synchronization of the system, is derived; it is found that the total order parameter decreases monotonically with noise, indicating overall suppression of synchronization. Still, for large coupling strengths, there exists an optimal noise level at which the periodic (ac) component of the order parameter reaches its maximum. The response of the phase velocity is a…
In this article, we present an approach for the thermodynamics of phase oscillators induced by an internal multiplicative noise. We analytically derive the free energy, entropy, internal energy, and specific heat. In this framework, the formulation of the first law of thermodynamics requires the definition of a synchronization field acting on the phase oscillators. By introducing the synchronization field, we have consistently obtained the susceptibility and analyzed its behavior. This allows us to characterize distinct phases in the system, which we have denoted as synchronized and parasynchr…
We elucidate how the presence of noise may significantly interact with the synchronization mechanism of systems exhibiting frequency-locking. The response of these systems exhibits a rich variety of behaviors, such as resonances and anti-resonances which can be controlled by the intensity of noise. The transition between different locked regimes provokes the development of a multiple enhancement of the effective diffusion. This diffusion behavior is accompanied by a crest-like peak-splitting cascade when the distribution of the lockings is self-similar, as it occurs in periodic systems that ar…
Many natural and social phenomena are characterized by synchronization. The Kuramoto model, taking into account the basic ingredients for observing synchronized states, allows to study mathematically synchronization in a simplified but nontrivial picture. Here we study how a noise that is correlated on a finite time-scale $τ$ impacts the ability of the Kuramoto model to achieve synchronization. We develop an approximated theory that allows to compute the critical coupling constant $k_c$ as a function of the correlation time $τ$. We obtain that that $k_c(τ)$ decreases as $τ$ increases indicatin…
We study synchronization of random one-dimensional linear maps for which the Lyapunov exponent can be calculated exactly. Certain aspects of the dynamics of these maps are explained using their relation with a random walk. We confirm that the Lyapunov exponent changes sign at the complete synchronization transition. We also consider partial synchronization of nonidentical systems. It turns out that the way partial synchronization manifests depends on the type of differences (in Lyapunov exponent or in contraction points) between the systems. The crossover from partial synchronization to comple…
We investigate the role of noise in the phenomenon of stochastic synchronization of switching events in a rocked, overdamped bistable potential driven by white Gaussian noise, the archetype description of Stochastic Resonance. We present a new approach to the stochastic counting process of noise-induced switching events: starting from the Markovian dynamics of the nonstationary, continuous particle dynamics one finds upon contraction onto two states a non-Markovian renewal dynamics. The output frequency is determined as the velocity of the underlying discrete phase dynamics. The phenomenon of …
We analyse the effect of synchronization between noise and periodic signal in a two-state spatially extended system analytically. Resonance features are demonstrated. To have the maximum cooperation between signal and noise, it is shown that noise strength at resonance should increase linearly with the frequency of the signal. The time scale of the process at resonance is also shown to increase linearly with the period of the signal.
The problem of a spatially correlated noise affecting a complex system is studied in this paper. We present a comprehensive analysis of a 2D model polymer chain, driven by the spatially correlated Gaussian noise, for which we have varied the amplitude and the correlation length. The chain model is based on a bead-spring approach, enriched with a global Lennard-Jones potential and angular interactions. We show that spatial correlations in the noise inhibit the chain geometry dynamics, enhancing the preservation of the polymer shape. This is supported by the analysis of correlation functions of …
We have extended the study of the Kuramoto model with additive Gaussian noise running on the KKI-18 large human connectome graph. We determined the dynamical behavior of this model by solving it numerically in an assumed homeostatic state, below the synchronization crossover point we determined previously. The de-synchronization duration distributions exhibit power-law tails, characterized by the exponent in the range $1.1 < τ_t < 2$, overlapping the in vivo human brain activity experiments by Palva et al. We show that these scaling results remain valid, by a transformation of the ultra-slow e…
We study the influence of telegraph noise on synchrony of limit cycle oscillators. Adopting the phase description for these oscillators, we derive the explicit expression for the Lyapunov exponent. We show that either for weak noise or frequent switching the Lyapunov exponent is negative, and the phase model gives adequate analytical results. In some systems moderate noise can desynchronize oscillations, and we demonstrate this for the Van der Pol--Duffing system.
We study effects of independent white noise on synchronization phenomena in ensembles of coupled limit cycle oscillators with different native frequencies. We consider a simple model where the ensemble consists of two inter-connected clusters with own native frequencies and mean-field couplings are introduced between intra- and inter-clusters. Taking advantage of nonlinear mean-field coupling concept together with the law of large numbers valid in the thermodynamic limit, we employ a nonlinear Fokker-Planck equation approach that turns out to be noise level-free analysis, to analytically deriv…
Synchronization is shown to occur in spatially extended systems under the effect of additive spatio-temporal noise. In analogy to low dimensional systems, synchronized states are observable only if the maximum Lyapunov exponent $Λ$ is negative. However, a sufficiently high noise level can lead, in map with finite domain of definition, to nonlinear propagation of information, even in non chaotic systems. In this latter case the transition to synchronization is ruled by a new ingredient : the propagation velocity of information $V_F$. As a general statement, we can affirm that if $V_F$ is finite…
We consider the celebrated Kuramoto model with nearest neighbour interactions, arranged on a two-dimensional square lattice in presence of two kinds of noise - annealed and quenched. We focus on both the steady state and relaxation dynamics of the model. The bare model with annealed noise on finite $2D$ lattice, in the stationary state, exhibits a crossover from synchronization to desynchronization as noise strength varies. Finite-size scaling (FSS) analysis reveals that, in the thermodynamic limit, this crossover becomes a true phase transition, which is Kosterlitz-Thouless ($KT$)-type analog…
In this paper we study synchronization of random clustered networks consisting of Kuramoto oscillators. More specifically, by developing a mean-field analysis, we find that the presence of cycles of order three does not play an important role on network synchronization, showing that the synchronization of random clustered networks can be described by tree-based theories, even for high values of clustering. In order to support our findings, we provide numerical simulations considering clustered and non-clustered networks, which are in good agreement with our theoretical results.
We consider the Kuramoto model of globally coupled phase oscillators with time-delayed interactions, that is subject to the Ornstein-Uhlenbeck (Gaussian) colored or the non-Gaussian colored noise. We investigate numerically the interplay between the influences of the finite correlation time of noise $τ$ and the time delay $τ_{d}$ on the onset of the synchronization process. Both cases for identical and nonidentical oscillators had been considered. Among the obtained results for identical oscillators is a large increase of the synchronization threshold as a function of time delay for the colore…
We study collective behavior of locally coupled limit-cycle oscillators with random intrinsic frequencies, spatially extended over $d$-dimensional hypercubic lattices. Phase synchronization as well as frequency entrainment are explored analytically in the linear (strong-coupling) regime and numerically in the nonlinear (weak-coupling) regime. Our analysis shows that the oscillator phases are always desynchronized up to $d=4$, which implies the lower critical dimension $d_{l}^{P}=4$ for phase synchronization. On the other hand, the oscillators behave collectively in frequency (phase velocity) e…
We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of 1-d maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon ana…
We study the effect of common noise on coupled active rotators. While such a noise always facilitates synchrony, coupling may be attractive or repulsing. We develop an analytical approach based on a transformation to approximate angle-action variables and averaging over fast rotations. For identical rotators, we describe a transition from full to partial synchrony at a critical value of repulsive coupling. For nonidentical rotators, the most nontrivial effect occurs at moderate repulsive coupling, where a juxtaposition of phase locking with frequency repulsion (anti-entrainment) is observed. W…
We study a generic model of globally coupled rotors that includes the effects of noise, phase shift in the coupling, and distributions of moments of inertia and natural frequencies of oscillation. As particular cases, the setup includes previously studied Sakaguchi-Kuramoto, Hamiltonian and Brownian mean-field, and Tanaka-Lichtenberg-Oishi and Acebrón-Bonilla-Spigler models. We derive an exact solution of the self-consistent equations for the order parameter in the stationary state, valid for arbitrary parameters in the dynamics, and demonstrate nontrivial phase transitions to synchrony that i…
The Vicsek model for the self-propelled particles is investigated with the respect to the introduction of the stochastic perturbation of the dynamics. It is shown that such a dependence can be thought in terms of the isomorphism of the Vicsek model with the Kuramoto model of spontaneous synchronization. They are isomorphic at least within the mean-field approach. The isomorphism between two models allows to state the dependence of the type of the transition in Vicsek model on the noise perturbation. Two types of noise the scalar and the vector ones lead to qualitatively different behavior with…
Spatially extended dynamical systems, namely coupled map lattices, driven by additive spatio-temporal noise are shown to exhibit stochastic synchronization. In analogy with low-dymensional systems, synchronization can be achieved only if the maximum Lyapunov exponent becomes negative for sufficiently large noise amplitude. Moreover, noise can suppress also the non-linear mechanism of information propagation, that may be present in the spatially extended system. A first example of phase transition is observed when both the linear and the non-linear mechanisms of information production disappear…
We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered. The derivation is performed within the framework of the phase reduction, which fully accounts for the amplitude degrees of freedom. Firstly, we consider the case of identical oscillators subject to intrinsic noise, obtain the synchronization condition, and find that the distribution of phase deviations always possesses lower-law heavy tails. Secondly, we cons…
A geometric approach is introduced for understanding the phenomenon of phase synchronization in coupled nonlinear systems in the presence of additive noise. We show that the emergence of cooperative behaviour through a change of stability via a Hopf bifurcation entails the spontaneous appearance of a gauge structure in the system, arising from the evolution of the slow dynamics, but induced by the fast variables. The conditions for the oscillators to be synchronised in phase are obtained. The role of weak noise appears to be to drive the system towards a more synchronized behaviour. Our analys…
