Two dimensional coupled oscillators array with rhombus structure and its application in active antenna array

Beam scanning and forming can be achieved by coupled oscillators array without phase shifter. Active antenna array based on coupled oscillators array has the virtue of low cost, high integration, and high efficiency. Traditional two dimensional coupled oscillators array has been arranged on rectangular lattices, and phase difference of adjacent elements is limited to [-90°, 90°]. Therefore, the beam scanning range is limited to [-30°, 30°] from normal for half wavelength element spacing. A new two dimensional coupled oscillators array with rhombus structure is presented. Phase control method and phase error of the array are also provided. Stability of the array is analyzed, and stable condition is given. When this coupled oscillators array with rhombus structure is used in active antenna array, theoretical results show that phase difference of adjacent elements reach the limit of [-180°, 180°] along the horizontal and vertical directions. Therefore, it has wider beam scanning range than that of a rectangular lattice structure.     

Pinning Cluster Synchronization of Coupled Nonidentical Harmonic Oscillators Under Directed Topology

This paper addresses cluster synchronization of coupled harmonic oscillators over directed fixed and switching topologies, where uncoupled harmonic oscillators in the same cluster have the identical node dynamics, while any pair of nodes in different clusters are essentially distinguishing according to their local dynamics. A pinning control protocol is proposed for such coupled harmonic oscillators system, and then some graphic topology conditions easy to verify are established for cluster synchronization. The main contributions of the present investigation include: (i) cluster synchronization problem of coupled nonidentical harmonic oscillators is addressed over directed topology; (ii) desynchronizing motion of harmonic oscillators from different clusters doesn't depend on negative coupling weights but leader of every cluster; (iii) this paper deals with what kind of harmonic oscillators ought to be pinned in order to reach cluster synchronization. Finally, numerical examples and simulations demonstrate the obtained theoretical results.     

Synchronization of coupled harmonic oscillators in a dynamic proximity network

In this paper, we revisit the synchronization problems for coupled harmonic oscillators in a dynamic proximity network. Unlike many existing algorithms for distributed control of complex dynamical networks that require explicit assumptions on the network connectivity, we show that the coupled harmonic oscillators can always be synchronized, without imposing any network connectivity assumption. Moreover, we also investigate the synchronization with a leader and show that all harmonic oscillators can asymptotically attain the position and velocity of the leader, again without any assumption on connectivity of the followers. Numerical simulation illustrates the theoretical results.     

Desynchronization of coupled limit-cycle oscillators through nonlinear output regulation

Synchronization is one of the most important phenomena in coupled oscillators. However, sometimes synchronization may be harmful and suppression of collective synchrony or desynchronization is desired. In this paper, we propose a control strategy for the desynchronization of an ensemble of all-to-all coupled Stuart–Landau oscillators. First, the desynchronization problem is redefined in the nonlinear output regulation framework. Then, we design an output regulator (stimulation) which forces Stuart–Landau oscillators (as a paradigm for limit-cycle oscillators) to track exogenous sinusoidal references with different phases. Finally, by considering some modifications, the initial version of the controller is improved to be more applicable in neuronal ensembles as an application of the desynchronization problem. The proposed stimulation is robust against variations of oscillators’ frequencies and can adapt itself with variations of the coupling strength. Mathematical analysis and simulation results reveal the efficiency of the proposed technique.     

Synchronization of Instantaneous Coupled Harmonic Oscillators With Communication and Input Delays

This paper investigates the synchronization issue of instantaneous coupled harmonic oscillators with communication and input delays. A distributed synchronization algorithm is proposed for such oscillator systems, and some general algebraic criteria on exponential convergence are established for the algorithm. The main contributions of the present investigations include: (i) the directed network topology is first considered in the case of instantaneous interaction; (ii) both the time‐varying communication and input delays are simultaneously addressed; (iii) the effects of communication and input delays on synchronization performance of coupled harmonic oscillators are discussed. It is shown the communication delays have a more important impact than the input delays on determining synchronization dynamics of coupled harmonic oscillators. Explicitly, in absence of communication delays, coupled harmonic oscillators can achieve synchronization oscillatory motion, whereas, so long as communication delays are nonzero at infinite multiple impulsive times, its synchronization (or consensus) state is zero. Finally, numerical examples demonstrate the obtained theoretical results.     

Sampled-data synchronisation of coupled harmonic oscillators with communication and input delays subject to controller failure

This paper considers the sampled-data synchronisation problems of coupled harmonic oscillators with communication and input delays subject to controller failure. A synchronisation protocol is proposed for such oscillator systems over directed network topology, and then some general algebraic criteria on exponential convergence for the proposed protocol are established. The main features of the present investigation include: (1) both the communication and input delays are simultaneously addressed, and the directed network topology is firstly considered and (2) the effects of time delays on synchronisation performance are theoretically and numerically investigated. It is shown that in the absence of communication delays, coupled harmonic oscillators can achieve synchronisation oscillatory motion. Whereas if communication delays are nonzero at infinite multiple sampled-data instants, its synchronisation (or consensus) state is zero. This conclusion can be used as an effective control strategy to stabilise coupled harmonic oscillators in practical applications. Furthermore, it is interesting to find that increasing either communication or input delays will enhance the synchronisation performance of coupled harmonic oscillators. Subsequently, numerical examples illustrate and visualise theoretical results.     

Global dynamics of two coupled parametrically excited van der Pol oscillators

Using a combination of analytical and numerical methods, the global bifurcations and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators are investigated in detail. With the aid of the method of multiple scales, the slow flow equations are obtained. Based on the slow flow equations, normal form theory and the techniques of choosing complementary space are applied to find the explicit expressions of the simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. By the simpler normal form, using the global perturbation method developed by Kovacic and Wiggins, the analysis of global bifurcation and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators is given. The results indicate that there exists a Silnikov type single-pulse homoclinic orbit for this class of system which implies the chaotic motions can occur. Numerical simulations are also given and verify the analytical predictions.     

两个耦合Van der Pol振子系统的一阶近似守恒量

用直接积分法计算两个耦合Van der Pol振子系统的一阶近似守恒量,将两个耦合Van der Pol振子系统看成是未受微扰系统与微扰项的迭加,先通过坐标变换将未受微扰系统解耦,并对解耦系统的3种可能状态进行讨论,得到未受微扰系统的13个精确守恒量,再考虑微扰项对精确守恒量的影响,运用一阶近似守恒量的性质,得到1个稳定的一阶近似守恒量.另外,由13个精确守恒量直接得到13个平凡的一阶近似守恒量.     

耦合调和振子网络系统的联合连通同步

论文分析了耦合调和振子网络系统在联合连通网络拓扑结构下的引导-跟随同步问题.假定每个网络拓扑结构图不连通,但它们在有限时间内能够联合连通,利用代数图论,李雅普诺夫稳定性理论和La Salle不变原理,证明了该系统的同步稳定性.最后,数值模拟进一步验证了所得理论结果的正确性和有效性.     

Synchronization of coupled harmonic oscillators with local interaction

This paper studies synchronization of coupled second-order linear harmonic oscillators with local interaction. We analyze convergence conditions over, respectively, directed fixed and switching network topologies by using tools from algebraic graph theory, matrix theory, and nonsmooth analysis. It is shown that the coupled harmonic oscillators can be synchronized under mild network connectivity conditions. Examples are given to validate the convergence conditions. The theoretical result is also applied to synchronized motion coordination of multi-agent systems as a proof of concept.     

Evaluation of beam steering in circular planar array of coupled microwave oscillators

A Green's function to describe the time behavior of the locking phases in a circular planar array of mutually coupled microwave oscillators is proposed. Using this Green's function the dynamic behavior of the array can be described for any arbitrary free running frequency of elements of the oscillator array. Beam steering is realized via detuning the edge oscillators of the array from a reference frequency and the beam direction is controlled by the amount of detuning for each oscillator at the edge. Some detuning functions such as sinusoidal, triangular and rectangular ones have been applied and, the resulting array patterns are compared with one another. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2011.     

基于拓扑马蹄理论的双耦合振子系统混沌现象研究

目前,耦合振子网络中的群体混沌现象已经成为混沌研究的新兴热点。因为群体混沌的发现历史较短,缺少成熟的研究理论和方法,主要的研究手段还是集中在诸如数值计算、功率谱和Lyapunov指数等较为粗糙的方法,难以描述群体混沌发生机制,缺乏严格数学意义下的判定。借助拓扑马蹄理论,对一双耦合振子构成的四维连续系统中的群体混沌现象进行了深入研究,在其庞加莱映射的相空间中找到了一维拉伸的拓扑马蹄,不仅严格判定了双耦合振子系统中群体混沌,而且揭示了群体混沌行为发生的动力学机制。     

Detecting synchronisation of biological oscillators by model checking

We define a subclass of timed automata, called oscillator timed automata, suitable to model biological oscillators. Coupled biological oscillators may synchronise, as emerging behaviour, after a period of time in which they interact through physical or chemical means. We introduce a parametric semantics for their interaction that is general enough to capture the behaviour of different types of oscillators. We instantiate it both to the Kuramoto model, a model of synchronisation based on smooth interaction, and to the Peskin model of pacemaker cells in the heart, a model of synchronisation based on pulse interaction. We also introduce a logic, Biological Oscillators Synchronisation Logic (BOSL), that is able to describe collective synchronisation properties of a population of coupled oscillators. A model checking algorithm is proposed for the defined logic and it is implemented in a model checker. The model checker can be used to detect synchronisation properties of a given population of oscillators. This tool might be the basic step towards the generation of suitable techniques to control and regulate the behaviour of coupled oscillators in order to ensure the reachability of synchronisation.     

Synchronization in complex networks of phase oscillators: A survey

The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.     

Beyond synchronization: String instability in coupled harmonic oscillator systems

In this paper, we consider the problem of disturbance response and error amplification for a simple system of coupled harmonic oscillators. We first suppose that identical oscillators are connected in a string in which each oscillator attempts to track its predecessor by using the same control law that depends on the relative position information from its immediate predecessor. Such an oscillator string is called a homogeneous oscillator string with predecessor‐following architecture. Motivated by terminology from the problem of vehicle platooning, we say that the synchronized oscillator system is string unstable if the effect of a disturbance to the lead oscillator is amplified as it propagates along the string. With the use of a new Bode‐like integral relation that must be satisfied by the complementary sensitivity function, we provide sufficient conditions for string instability. The sufficient conditions show that any string of oscillators that satisfies certain time domain performance specifications and bandwidth limitations must necessarily be string unstable. We further introduce a concept of time headway for the oscillator system and extend our analysis of string instability to consider the heterogeneous oscillator string and a more general communication range. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust semiglobal swarm tracking of coupled harmonic oscillators with input saturation and external disturbance

The robust semiglobal swarm tracking problem of N coupled harmonic oscillators and 1 actual leader with input saturation and external disturbance on a directed communication topology is considered, in which the N coupled harmonic oscillators are referred to followers. First, the low‐and‐high gain feedback technique is introduced to construct a relative state‐dependent control algorithm. Then, an observer‐based control algorithm is designed based on the low‐and‐high gain feedback technique and the high‐gain observer design methodology. Sufficient conditions are derived to guarantee robust semiglobal swarm tracking for state‐feedback control and output‐feedback control, respectively. Numerical simulations are finally provided to verify the theoretic results.     

Semi‐global and global containment control of multi‐agent systems with second‐order dynamics and input saturation

This paper considers both semi‐global and global containment control for a second‐order multi‐agent system that is composed by a network of identical harmonic oscillators or double integrators with multiple leaders and input saturation. A distributed low gain feedback algorithm is proposed to solve the semi‐global containment control problem for the network whose topology is directed and initial condition is taken from any a priori given bounded set. In particular, by using a parametric Lyapunov equation approach, M‐matrix properties and algebraic graph theory, an upper bound of the low gain parameter is estimated such that the low gain feedback matrix can be analytically determined without involving numerical computation. Furthermore, under the assumption that the induced subgraph formed by the followers is strongly connected and detail balanced, two linear feedback protocols are designed for coupled harmonic oscillators and coupled double integrators, respectively, to asymptotically achieve the global containment control of the network with any initial condition. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback

Motivated by the recent development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of mean-field feedback and analyze the robustness of its phase-locking with respect to general inputs. Under standard assumptions, this system can be reduced to a modified version of the Kuramoto model of coupled nonlinear oscillators. In the first part of the paper we present an analytical study on the existence of phase-locked solutions under generic interconnection and feedback configurations. In particular we show that, in general, no oscillating phase-locked solutions can co-exist with any non-zero proportional mean-field feedback. In the second part we prove some robustness properties of phase-locked solutions (namely total stability). This general result allows in particular to justify the persistence of practically phase-locked states if sufficiently small feedback gains are applied, and to give explicit necessary conditions on the intensity of a desynchronizing mean-field feedback. Furthermore, the Lyapunov function used in the analysis provides a new characterization of the robust phase-locked configurations in the Kuramoto system with symmetric interconnections.     

具有脉冲耦合复杂时滞动力网络的同步

考虑了一个具有脉冲耦合的复杂时滞动力网络的同步问题.基于脉冲时滞动力系统扩展的Halanay不等式,给出了网络同步的一个充分条件.所获结果表明,即使网络节点之间仅在一些离散时刻存在瞬时连接,网络仍然能够达到同步.进一步将所得结果应用于一个由混沌FHN神经元振子为动力节点所构成的一个无标度的动力网络,数值仿真结果表明了理...     

Finite‐time consensus tracking for harmonic oscillators using both state feedback control and output feedback control

This paper investigates the distributed finite‐time consensus‐tracking problem for coupled harmonic oscillators. The objective is to guarantee a team of followers modeled by harmonic oscillators to track a dynamic virtual leader in finite time. Only a subset of followers can access the information of the virtual leader, and the interactions between followers are assumed to be local. We consider two cases: (i) The followers can obtain the relative states between their neighbors and their own; and (ii) Only relative outputs between neighboring agents are available. In the former case, a distributed consensus protocol is adopted to achieve the finite‐time consensus tracking. In the latter case, we propose a novel observer‐based dynamic protocol to guarantee the consensus tracking in finite time. Simulation examples are finally presented to verify the theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.