Synchronization of Coupled Oscillators in a Local One-Dimensional Kuramoto Model
J. Ochab, P. F. Góra
备注：
10 pages, 5 figures, presented at the Summer Solstice 2009 International Conference on Discrete Models of Complex Systems, submitted to the conference proceedings in Acta Phys. Pol. B
发表信息：
Acta Phys. Pol. B [Proceedings Supplement 3] (2010), 453
A modified Kuramoto model of synchronization in a finite discrete system of locally coupled oscillators is studied. The model consists of N oscillators with random natural frequencies arranged on a ring. It is shown analytically and numerically that finite-size systems may have many different synchronized stable solutions which are characterised by different values of the winding number. The lower bound for the critical coupling is given, as well as an algorithm for its exact calculation. It is shown that in general phase-locking does not lead to phase coherence in 1D.
We study the synchronization of oscillators with inertias and phase shifts, namely the second-order Kuramoto-Sakaguchi model. Using the self-consistent method, we find that the effect of inertia is the introduction of effective phase shifts. The discontinuous synchronization transition of the Kuramoto-Sakaguchi model changes to a continuous one when the value of inertia is small. In addition, we find a new synchronization process, in which with increasing coupling strength the system reaches an oscillating state instead of complete synchronization due to the cross-effect of phase shifts and in…
Optimal synchronization of Kuramoto oscillators: a dimensional reduction approach
最近提出的降维方法研究同步在仓本模型中建立最佳的网络拓扑结构，以促进或抑制同步。该方法的基础是在一个集体坐标的锁相振荡器的时间演化的介绍，在的精神，安东森anquiry。我们证明了仓本网络的最优同步要求二次函数$ω^T L ω$的最大化，其中$ω$代表振荡器的固有频率的向量，和$L$网络的拉普拉斯矩阵。许多最近得到的数值结果可以重新获得解析和更简单的方式从我们的最大化条件。提出了一种计算效率高的{爬山}重布线算法来生成具有最佳同步特性的网络。我们的方法可以很容易地适应的情况下，仓本模型的吸引力和排斥力的相互作用，并再次许多最近的数值结果可以重新推导在一个更简单，更清晰的分析方式。
We study two groups of nonidentical Kuramoto oscillators with differing frequency distributions. Coupling between the groups is repulsive, while coupling between oscillators of the same group is attractive. This asymmetry of interactions leads to an interesting synchronization behavior. For small coupling strength, the mean-fields of both groups resemble a more weakly coupled Kuramoto model. After increasing the coupling strength beyond a threshold, they reach the Bellerophon state of multiple clusters of averagely entrained oscillators. A further increase in the coupling strength then asympto…
We propose Moebius maps as a tool to model synchronization phenomena in coupled phase oscillators. Not only does the map provide fast computation of phase synchronization, it also reflects the underlying group structure of the sinusoidally coupled continuous phase dynamics. We study map versions of various known continuous-time collective dynamics, such as the synchronization transition in the Kuramoto-Sakaguchi model of non-identical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and dif…
An incorporation of higher-order interactions is known to lead an abrupt first-order transition to synchronization in otherwise smooth second-order one for pair-wise coupled systems. Here, we show that adaptation in higher-order coupling strength may yield completely different phenomena, notably, second-order transition to synchronization and tiered synchronization. Using the Ott-Antonsen approach, we perform rigorous theoretical calculations illustrating the origin of these emerging phenomena along with a complete description of all (un)stable states. Numerical simulations for limited-size ne…
We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles of identical oscillators with the global coupling. In such a setup, with an additional assumption that the initial distribution of oscillators is uniform on the sphere, one can derive equations for order parameters in closed form. The rate of synchronization in a real Kuramoto model depends on the dimension of the sphere. Specifically, synchronization is faster on higher-dimensional spheres. On the other side, real order parameter in complex Kuramoto model…
When higher-order interactions enhance synchronization: the case of the Kuramoto model
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（RIKEN跨学科理论与数学科学中心（iTHEMS））；Department of Systems and Control Engineering, Institute of Science Tokyo（东京科学大学系统与控制工程系）；Research Center for Autonomous Systems Materialogy, Institute of Science Tokyo（东京科学大学自主系统材料学研究中心）；Scuola Superiore Meridionale（南方高等学院）
A fundamental understanding of synchronized behavior in multi-agent systems can be acquired by studying analytically tractable Kuramoto models. However, such models typically diverge from many real systems whose dynamics evolve under non-negligible resource constraints. Here we construct a system of coupled Kuramoto oscillators that consume or produce resources as a function of their oscillation frequency. At high coupling, we observe strongly synchronized dynamics, whereas at low coupling we observe independent oscillator dynamics, as expected from standard Kuramoto models. For intermediate c…
Cortical regions without direct neuronal connections have been observed to exhibit synchronized dynamics. A recent empirical study has further revealed that such regions that share more common neighbors are more likely to behave coherently. To analytically investigate the underlying mechanisms, we consider that a set of n oscillators, which have no direct connections, are linked through m intermediate oscillators (called mediators), forming a complete bipartite network structure. Modeling the oscillators by the Kuramoto-Sakaguchi model, we rigorously prove that mediated remote synchronization,…
We show that an introduction of a phase parameter ($α$), with $0 \le α\le π/2$, in the interlayer coupling terms of multiplex networks of Kuramoto oscillators can induce explosive synchronization (ES) in the multiplexed layers. Along with the α values, the hysteresis width is determined by the interlayer coupling strength and the frequency mismatch between the mirror (inter-connected) nodes. A mean-field analysis is performed to support the numerical results. Similar to the earlier works, we find that the suppression of synchronization is accountable for the origin of ES. The robustness of ES …
Complete Synchronization and its Transition in Higher Harmonic Sakaguchi--Kuramoto Oscillators
CSIR-Central Mechanical Engineering Research Institute（印度中央机械工程技术研究所）；Academy of Scientific and Innovative Research (AcSIR)（科学创新研究学院）；Department of Mathematics, National Institute of Technology（国家技术学院数学系）；Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research（匈牙利科学院能源研究中心技术物理与材料科学研究所）；School of Computer Science, UPES（尤帕迪大学计算机科学学院）；Dhirubhai Ambani University（迪鲁拜·安巴尼大学）
Diversity of dynamical behaviors due to initial conditions: exact results with extended Ott--Antonsen ansatz for identical Kuramoto--Sakaguchi phase oscillators
Synchronization transition in Sakaguchi-Kuramoto model on complex networks with partial degree-frequency correlation
本文研究了复杂网络中坂口仓本模型向同步的转变。假设网络中某个百分比（$f$）的高度数节点的固有频率与其度数相关，其余节点的固有频率则取自某个标准分布，即Lorenz分布。详细研究了相位阻挫参数$α$和$f$的变化对系统向同步过渡的影响。在热力学极限下，导出了包含临界耦合强度（$λ_c$）和群角速度（$Ω_c$）的自洽方程.为了进行详细的研究，我们考虑了无标度网络上的SK模型以及Erdens-Rényi（ER）网络。有趣的是，爆炸性同步（ES）已被观察到在两个网络的$α$和$f$的值的不同范围。对于无标度网络，由于$f$的值被设置在$10\% \leq f \leq 70\%$内，因此与完全度-频率相关的情况相比，ES存在的$α$的值的范围被大大增强。另一方面，对于随机网络，当$f$的值取在$30\% \leq f \leq 50\%$内时，在$α$的窄窗口中观察到ES。在所有的情况下，临界耦合强度过渡到同步计算的解析推导的自洽方程显示出非常好的协议与数值结果。
Populations of oscillators are present throughout nature. Very often synchronization is observed in such populations if they are allowed to interact. A paradigmatic model for the study of such phenomena has been the Kuramoto model. However, considering real oscillations are rarely isochronous as a function of energy, it is natural to extend the model by allowing the natural frequencies to vary as a function of some dynamical resource supply. Beyond just accounting for a dynamical supply of resources, however, competition over a \emph{shared} resource supply is important in a variety of biologi…
What happens when the paradigmatic Kuramoto model involving interacting oscillators of distributed natural frequencies and showing spontaneous collective synchronization in the stationary state is subject to random and repeated interruptions of its dynamics with a reset to the initial condition? While resetting to a synchronized state, it may happen between two successive resets that the system desynchronizes, which depends on the duration of the random time interval between the two resets. Here, we unveil how such a protocol of stochastic resetting dramatically modifies the phase diagram of t…
