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Register a new userTraveling Speed of Clusters in the Kuramoto-Sakaguchi Model
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We study a variant of Kuramoto-Sakaguchi model in which oscillators are divided into two groups, each characterized by its coupling constant and phase lag. Specifically, we consider the case that one coupling constant is positive and the other negative, and calculate numerically the traveling speed of two clusters emerging in the system and average separation between them as well as the order parameters for positive and negative oscillators, as the two coupling constants, phase lags, and the fraction of positive oscillators are varied. An expression explaining the dependence of the traveling speed on these parameters is obtained and observed to fit well the numerical data. With the help of this, we describe the conditions for the traveling state to appear in the system.
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The Kuramoto-Sakaguchi model for coupled phase oscillators with phase-frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the collective dynamics of finite size Kuramoto-Sakaguchi models. We find that the inclusion of the effects of rogue oscillators is essential to obtain an accurate description, in contrast to the original Kuramoto model where we show that their effects can be ignored. We further introduce a more accurate ansatz function to describe the shape of synchronized oscillators. Our results from this extended collective coordinate approach reduce in the thermodynamic limit to the well-known mean-field consistency relations. For finite networks we show that our model reduction describes the collective behavior accurately, reproducing the order parameter, the mean frequency of the synchronized cluster, and the size of the cluster at given coupling strength, as well as the critical coupling strength for partial and for global synchronization.
We study the global bifurcations of frequency weighted Kuramoto model in low-dimension for network of fully connected oscillators. To study the effect of non-zero-centered frequency distribution, we consider two symmetric Lorentzians as an example. We derive the stability diagram of the system and show that the infinite-dimensional problem reduces to a flow in four dimensions. Using the system symmetries, it can be further reduced to two dimensions. Using this analytic framework, we obtain bifurcation boundaries of the system, which is compatible with our numeric simulations. We show that the system has three types of transitions to synchronized state for different parameters of the frequency distribution: (1) a two-step transition, representative of standing waves, (2) a continuous transition, as in the classical Kuramoto model, and (3) a first-order transition with hysteresis. Numerical simulations are also conducted to confirm analytic results.
The Kuramoto model is a canonical model for understanding phase-locking phenomenon. It is well-understood that, in the usual mean-field scaling, full phase-locking is unlikely and that it is partially phase-locked states that are important in applications. Despite this, while there has been much attention given to the existence and stability of fully phase-locked states in the finite N Kuramoto model, the partially phase-locked states have received much less attention. In this paper, we present two related results. Firstly, we derive an analytical criterion that, for sufficiently strong coupling, guarantees the existence of a partially phase-locked state by proving the existence of an attracting ball around a fixed point of a subset of the oscillators. We also derive a larger invariant ball such that any point in it will asymptotically converge to the attracting ball. Secondly, we consider the large N (thermodynamic) limit for the Kuramoto system with randomly distributed frequencies. Using some results of De Smet and Aeyels on partial entrainment, we derive a deterministic condition giving almost sure existence of a partially entrained state for sufficiently strong coupling when the natural frequencies of the individual oscillators are independent identically distributed random variables, as well as upper and lower bounds on the size of the largest cluster of partially entrained oscillators. Interestingly in a series on numerical experiments we find that the observed size of the largest entrained cluster is predicted extremely well by the upper bound.
Globally coupled ensembles of phase oscillators serve as useful tools for modeling synchronization and collective behavior in a variety of applications. As interest in the effects of simplicial interactions (i.e., non-additive, higher-order interactions between three or more units) continues to grow we study an extension of the Kuramoto model where oscillators are coupled via three-way interactions that exhibits novel dynamical properties including clustering, multistability, and abrupt desynchronization transitions. Here we provide a rigorous description of the stability of various multicluster states by studying their spectral properties in the thermodynamic limit. Not unlike the classical Kuramoto model, a natural frequency distribution with infinite support yields a population of drifting oscillators, which in turn guarantees that a portion of the spectrum is located on the imaginary axes, resulting in neutrally stable or unstable solutions. On the other hand, a natural frequency distribution with finite support allows for a fully phase-locked state, whose spectrum is real and may be linearly stable or unstable.
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.
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