Synchronization of coupled harmonic oscillators is investigated. Coupling considered here is pairwise, unidirectional, and described by a nonlinear function (whose graph resides in the first and third quadrants) of some projection of the relative distance (between the states of the pair being coupled) vector. Under the assumption that the interconnection topology defines a connected graph, it is shown that the synchronization manifold is semiglobally practically asymptotically stable in the frequency of oscillations.
We investigate the synchronized collective behavior of the Kuramoto oscillators with inertia effect. Both the frequency synchronization for nonidentical case and the phase synchronization for identical case are in view. As an application of our general theory, we show the unconditional frequency synchronization for the three-nonidentical-oscillator case.
For original Kuramoto models with nonidentical oscillators, it is impossible to realize complete phase synchronization. However, this paper reveals that complete phase synchronization can be achieved for a large class of high-dimensional Kuramoto models with nonidentical oscillators. Under the topology of strongly connected digraphs, a necessary and sufficient condition for complete phase synchronization is obtained. Finally, some simulations are provided to validate the obtained theoretic results.
In this paper, we show that under a generic condition of the coefficient of a stochastic phase oscillator the Lyapunov exponent of the linearization along an arbitrary solution is always negative. Consequently, the generated random dynamical system exhibits a synchronization.
In networks of identical linear oscillators (e.g. pendulums undergoing small vibrations) coupled through both dissipative connectors (e.g. dampers) and restorative connectors (e.g. springs) the relation between asymptotic synchronization and coupling structure is studied. Conditions on the interconnection under which synchronization can be achieved for some selection of coupling strengths are established. How to strengthen those conditions so that synchronization is guaranteed for all admissible parameter values is also presented.
This paper addresses the amplitude and phase dynamics of a large system non-linear coupled, non-identical damped harmonic oscillators, which is based on recent research in coupled oscillation in optomechanics. Our goal is to investigate the existence and stability of collective behaviour which occurs due to a play-off between the distribution of individual oscillator frequency and the type of nonlinear coupling. We show that this system exhibits synchronisation, where all oscillators are rotating at the same rate, and that in the synchronised state the system has a regular structure related to…
Synchronization is studied in an array of identical linear oscillators of arbitrary order, coupled through a dynamic network comprising dissipative connectors (e.g., dampers) and restorative connectors (e.g., springs). The coupling network is allowed to contain interior nodes, i.e., those that are not directly connected to an oscillator. It is shown that the oscillators asymptotically synchronize if and only if the Schur complement (with respect to the boundary nodes) of the complex-valued Laplacian matrix representing the coupling has a single eigenvalue on the imaginary axis.
The role of restorative coupling on synchronization of coupled identical harmonic oscillators is studied. Necessary and sufficient conditions, under which the individual systems' solutions converge to a common trajectory, are presented. Through simple physical examples, the meaning and limitations of the theorems are expounded. Also, to demonstrate their versatility, the results are extended to cover LTI passive electrical networks. One of the extensions generalizes the well-known link between the asymptotic stability of the synchronization subspace and the second smallest eigenvalue of the La…
In this paper, new schemes to synchronize linearly or nonlinearly coupled chaotic systems with an adaptive coupling strength are proposed. Unlike other adaptive schemes, which synchronize coupled chaotic systems to a special trajectory (or an equilibrium point) of the uncoupled node by adding negative feedbacks adaptively; here, adaptive schemes for the coupling strength are used to synchronize coupled chaotic systems without knowing the synchronization trajectory. Moreover, in many applications, the state variables are not observable; instead, some functions of the states can be observed. How…
Coupled oscillator models where $N$ oscillators are identical and symmetrically coupled to all others with full permutation symmetry $S_N$ are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive - and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical s…
In this article we study synchronization of Kuramoto oscillators with heterogeneous frequencies, and where underlying topology is a graph of diameter two. When the coupling strengths between every two connected oscillators are the same, we find an analytic condition that guarantees an existence of a Positively Invariant Set (PIS) and demonstrate that existence of a PIS suffices for frequency synchronization. For graphs of diameter two, this synchronization condition is significantly better than existing general conditions for an arbitrary topology. If the coupling strengths can be different fo…
The stability of synchronization state in networks of oscillators are studied under the assumption that oscillators and their couplings have slightly mismatched parameters. A generalized master stability function is provided that takes the mismatches into account. Using this master stability function a lower bound on the probability of synchronization is derived for regular and random network models. The probability of stability of synchronization is then used to study the phase transition behavior of the networks. Numerical examples using van der Pol oscillators are used to illustrate the res…
We discuss the asymptotic frequency synchronization for the non-identical Kuramoto oscillators with time delayed interactions. We provide explicit lower bound on the coupling strength and upper bound on time delay in terms of initial configurations ensuring exponential synchronization. This generalizes not only the frequency synchronization estimate by Choi et al. [Physica D, 241, (2012), 735-754] for the non-identical Kuramoto oscillators without time delays but also improves previous result by Schmidt et al. [Automatica, 48, (2012), 3008-3017] in the case of homogeneous time delays where the…
We study synchronization of coupled Kuramoto oscillators with heterogeneous inherent frequencies and general underlying connectivity. We provide conditions on the coupling strength and the initial phases which guarantee the existence of a Positively Invariant Set (PIS) and lead to synchronization. Unlike previous works that focus only on analytical bounds, here we introduce an optimization approach to provide a computational-analytical bound that can further exploit the particular features of each individual system such as topology and frequency distribution. Examples are provided to illustrat…
We report erosion of synchronization in networks of coupled phase oscillators, a phenomenon where perfect phase synchronization is unattainable in steady-state, even in the limit of infinite coupling. An analysis reveals that the total erosion is separable into the product of terms characterizing coupling frustration and structural heterogeneity, both of which amplify erosion. The latter, however, can differ significantly from degree heterogeneity. Finally, we show that erosion is marked by the reorganization of oscillators according to their node degrees rather than their natural frequencies.
Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；Universität Hamburg（汉堡大学）；Saint Louis University（圣路易斯大学）；University of Oxford（牛津大学）；University of Exeter（埃克塞特大学）；Technische Universität München（慕尼黑工业大学）
The emergence of synchronization in a network of coupled oscillators is a pervasive topic in various scientific disciplines ranging from biology, physics, and chemistry to social networks and engineering applications. A coupled oscillator network is characterized by a population of heterogeneous oscillators and a graph describing the interaction among the oscillators. These two ingredients give rise to a rich dynamic behavior that keeps on fascinating the scientific community. In this article, we present a tutorial introduction to coupled oscillator networks, we review the vast literature on t…
We study synchronization of Kuramoto oscillators in strongly modular networks in which the structure of the network inside each community is averaged. We find that the dynamics of the interacting communities can be described as an ensemble of coupled planar oscillators. In the limit of a large number of communities, we find a low dimensional description of the level of synchronization between the communities. In this limit, we describe bifurcations between incoherence, local synchrony, and global synchrony. We compare the predictions of this simplified model with simulations of heterogeneous n…
