Ensemble order parameter equations in star network (original) (raw)
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Order parameter analysis of synchronization transitions on star networks
2017
Collective behaviors of populations of coupled oscillators have attracted much attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynam- ical mechanism of collective synchronizations by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe-Strogatz transformation, Ott-Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to diverse col- lective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions are revealed in the star-network model by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.
arXiv: Chaotic Dynamics, 2018
We study the Kuramoto-Sakaguchi (KS) model composed by any N identical phase oscillators symmetrically coupled. Ranging from local (one-to-one, R = 1) to global (all-to-all, R = N/2) couplings, we derive the general solution that describes the network dynamics next to an equilibrium. Therewith we build stability diagrams according to N and R bringing to the light a rich scenery of attractors, repellers, saddles, and non-hyperbolic equilibriums. Our result also uncovers the obscure repulsive regime of the KS model through bifurcation analysis. Moreover, we present numerical evolutions of the network showing the great accordance with our analytical one. The exact knowledge of the behavior close to equilibriums is a fundamental step to investigate phenomena about synchronization in networks. As an example, at the end we discuss the dynamics behind chimera states from the point of view of our results.
Synchronization in starlike networks of phase oscillators
Physical Review E
We fully describe the mechanisms underlying synchronization in starlike networks of phase oscillators. In particular, the routes to synchronization and the critical points for the associated phase transitions are determined analytically. In contrast to the classical Kuramoto theory, we unveil that relaxation rates to each equilibrium state indeed exist and remain invariant under three levels of descriptions corresponding to different geometric implications. The special symmetry in the coupling determines a quasi-Hamiltonian property, which is further unveiled on the basis of singular perturbation theory. Since starlike coupling configurations constitute the building blocks of technological and biological real world networks, our paper paves the way towards the understanding of the functioning of such real world systems in many practical situations.
Low-dimensional behavior of Kuramoto model with inertia in complex networks
Scientific Reports, 2014
Low-dimensional behavior of large systems of globally coupled oscillators has been intensively investigated since the introduction of the Ott-Antonsen ansatz. In this report, we generalize the Ott-Antonsen ansatz to second-order Kuramoto models in complex networks. With an additional inertia term, we find a low-dimensional behavior similar to the first-order Kuramoto model, derive a self-consistent equation and seek the time-dependent derivation of the order parameter. Numerical simulations are also conducted to verify our analytical results.
Equilibria in Kuramoto Oscillator Networks: An Algebraic Approach
SIAM Journal on Applied Dynamical Systems
Kuramoto networks constitute a paradigmatic model for the investigation of collective behavior in networked systems. Despite many advances in recent years, many open questions remain on the solutions for systems composed of coupled Kuramoto oscillators on complex networks. In this article, we describe an algebraic method to find equilibrium points for this kind of system without using standard approximations in the limit of infinite system size or the continuum limit. To do this, we use a recently introduced algebraic approach to the Kuramoto dynamics, which results in an explicitly solvable complex-valued equation that captures the dynamics of the original Kuramoto model. Using this new approach, we obtain equilibria for both the nonlinear original Kuramoto and complex-valued systems. We then completely classify all equilibria in the case of complete graphs originally studied by Kuramoto. Finally, we go on to study equilibria in networks of coupled oscillators with phase lag, in generalized circulant networks, multi-layer networks, and also random networks. Contents
Kuramoto model with frequency-degree correlations on complex networks
Physical Review E, 2013
We study the Kuramoto model on complex networks, in which natural frequencies of phase oscillators and the vertex degrees are correlated. Using the annealed network approximation and numerical simulations we explore a special case in which the natural frequencies of the oscillators and the vertex degrees are linearly coupled. We find that in uncorrelated scale-free networks with the degree distribution exponent 2 < γ < 3, the model undergoes a first-order phase transition, while the transition becomes of the second order at γ > 3. If γ = 3, the phase synchronization emerges as a result of a hybrid phase transition that combines an abrupt emergence of synchronization, as in first-order phase transitions, and a critical singularity, as in second-order phase transitions. The critical fluctuations manifest themselves as avalanches in synchronization process. Comparing our analytical calculations with numerical simulations for Erdős-Rényi and scale-free networks, we demonstrate that the annealed network approach is accurate if the the mean degree and size of the network are sufficiently large. We also study analytically and numerically the Kuramoto model on star graphs and find that if the natural frequency of the central oscillator is sufficiently large in comparison to the average frequency of its neighbors, then synchronization emerges as a result of a first-order phase transition. This shows that oscillators sitting at hubs in a network may generate a discontinuous synchronization transition.
Chaos (Woodbury, N.Y.), 2015
We analyze star-type networks of phase oscillators by virtue of two methods. For identical oscillators we adopt the Watanabe-Strogatz approach, which gives full analytical description of states, rotating with constant frequency. For nonidentical oscillators, such states can be obtained by virtue of the self-consistent approach in a parametric form. In this case stability analysis cannot be performed, however with the help of direct numerical simulations we show which solutions are stable and which not. We consider this system as a model for a drum orchestra, where we assume that the drummers follow the signal of the leader without listening to each other and the coupling parameters are determined by a geometrical organization of the orchestra.
Spontaneous collective synchronization in the Kuramoto model with additional non-local interactions
Journal of Physics A: Mathematical and Theoretical, 2017
In the context of the celebrated Kuramoto model of globally-coupled phase oscillators of distributed natural frequencies, which serves as a paradigm to investigate spontaneous collective synchronization in many-body interacting systems, we report on a very rich phase diagram in presence of thermal noise and an additional non-local interaction on a one-dimensional periodic lattice. Remarkably, the phase diagram involves both equilibrium and non-equilibrium phase transitions. In two contrasting limits of the dynamics, we obtain exact analytical results for the phase transitions. These two limits correspond to (i) the absence of thermal noise, when the dynamics reduces to that of a non-linear dynamical system, and (ii) the oscillators having the same natural frequency, when the dynamics becomes that of a statistical system in contact with a heat bath and relaxing to a statistical equilibrium state. In the former case, our exact analysis is based on the use of the so-called Ott-Antonsen ansatz to derive a reduced set of nonlinear partial differential equations for the macroscopic evolution of the system. Our results for the case of statistical equilibrium are on the other hand obtained by extending the well-known transfer matrix approach for nearestneighbor Ising model to consider non-local interactions. The work offers a case study of exact analysis in many-body interacting systems. The results obtained underline the crucial role of additional non-local interactions in either destroying or enhancing the possibility of observing synchrony in mean-field systems exhibiting spontaneous synchronization.
Explosive first-order transition to synchrony in networked chaotic oscillators
2012
Critical phenomena in complex networks, and the emergence of dynamical abrupt transitions in the macroscopic state of the system are currently a subject of the outmost interest. We report evidence of an explosive phase synchronization in networks of chaotic units. Namely, by means of both extensive simulations of networks made up of chaotic units, and validation with an experiment of electronic circuits in a star configuration, we demonstrate the existence of a first order transition towards synchronization of the phases of the networked units. Our findings constitute the first prove of this kind of synchronization in practice, thus opening the path to its use in real-world applications. PACS:89.75.Hc, 89.75.Kd, 05.45.Xt The understanding of the spontaneous emergence of collective behavior in ensembles of networked dynamical units constitutes a fascinating challenge in science. Despite the fact that critical phenomena in networks have been intensively studied, the physics literature [1] almost exclusively reports continuous phase transitions. However, it has been recognized that, although very few in number , there are physical processes which might lead to sharp, discontinuous transitions of a global order parameter. The last several years have also witnessed an ever-increasing interest in studying networked systems composed of nonlinear dynamical units , and in particular, in the emergence of synchronization phenomena . Within this latter context, some advances have been made for the case of non-equilibrium synchronization transitions of chaotic systems , being, however, all the reported cases examples of second order phase transitions.