Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> A qualitative theory of the circuit dynamics of the Josephson-junction device using a simple circuit model is developed. By examining trajectories on a cylindrical phase space and on the surface of a torus, respectively, the main features of thel/V characteristics of the junction excited by D.C. and a.c. are explained, and their properties derived. These results lead to the interpretation of the steps of constant voltage in the a.c. characteristics as a manifestation of the synchronisation phenomenon <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> A new model for the study of power system stability via Lyapunov functions is proposed. The key feature of the model is an assumption of frequency-dependent load power, rather than the usual impedance loads which are subsequently absorbed into a reduced network. The original network topology is explicitly represented. This approach has the important advantage of rigorously accounting for real power loads in the Lyapunov functions. This compares favorably with existing methods involving approximations to allow for the significant transfer conductances in reduced network models. The preservation of network topology can be exploited in stability analysis, with the concepts of critical and vulnerable cutsets playing central roles in dynamic and transient stability evaluation respectively. Of fundamental importance is the feature that the Lyapunov functions give a true representation of the spatial distribution of stored energy in the system <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> A quasi-optical method for solid-state power combining with applications to high-power millimeter-wave generation is discussed. The approach uses two-dimensional planar arrays of weakly coupled oscillators. Limiting the strength of the coupling avoids multifrequency moding problems and simplifies the design. A radiating element is embedded in each oscillator so that the power combining is accomplished in free space. The concept which has been demonstrated with two prototype arrays, one using Gunn diodes and the other using MESFETs is discussed. A theoretical description of the coupled-oscillator arrays is presented for design purposes, and is used to investigate phasing problems and stability. Experiments are discussed which indicate that in-phase operation is facilitated by using a quasi-optical reflector element. which influences the operating frequency and coupling between the elements. Equivalent isotropic radiated powers of 22 W at 1% efficiency for a 16-element Gunn array and 10 W at 26% efficiency for a 16-element MESFET array which have been obtained at X-band are discussed. > <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Synchronization due to a weak global coupling with time delay in a semiconductor laser array is investigated both in the cw and self-pulsing regimes. A generalized form of the Kuramoto phase equations is derived and discussed analytically. The time delay is shown to induce in-phase synchronization in all dynamical regimes. Another form of synchronization is found which leads to local extinction of self-pulsing in the array. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> 1. Introduction. 2. Electromagnetic Transients. 3. Synchronous Machine Modeling. 4. Synchronous Machine Control Models. 5. Single-Machine Dynamic Models. 6. Multimachine Dynamic Models. 7. Multimachine Simulation. 8. Small-Signal Stability. 9. Energy Function Methods. Appendix A: Integral Manifolds for Model. Bibliography. Index. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> We investigate here possible neurocomputational features of networks of laser oscillators. Our approach is similar to classical optical neurocomputing where artificial neurons are lasers and connection matrices are holographic media. However, we consider oscillatory neurons communicating via phases rather than amplitudes. Memorized patterns correspond to synchronized states where the neurons oscillate with equal frequencies and with prescribed phase relations. The mechanism of recognition is related to phase locking. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> The Kuramoto model describes a large population of coupled limit-cycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory, bifurcation theory, and plasma physics. © 2000 Elsevier Science B.V. All rights reserved. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> We combine here two well-known and established concepts: microelectromechanical systems (MEMS) and neurocomputing. First, we consider MEMS oscillators having low amplitude activity and we derive a simple mathematical model that describes nonlinear phase-locking dynamics in them. Then, we investigate a theoretical possibility of using MEMS oscillators to build an oscillatory neurocomputer having autocorrelative associative memory. The neurocomputer stores and retrieves complex oscillatory patterns in the form of synchronized states with appropriate phase relations between the oscillators. Thus, we show that MEMS alone can be used to build a sophisticated information processing system (U.S. provisional patent 60/178,654). <s> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Abstract. The coordinated reset of neural subpopula-tions is introduced as an eﬀectively desynchronizingstimulation technique. For this, short sequences of high-frequency pulse trains are administered at diﬀerent sitesin a coordinated way. Desynchronization is easilymaintained by performing a coordinated reset withdemand-controlled timing or by periodically adminis-tering resetting high-frequency pulse trains of demand-controlled length. Unlike previously developed methods,this novel approach is robust against variations of modelparameters and does not require time-consuming cali-bration. The novel technique is suggested to be used fordemand-controlled deep brain stimulation in patientssuﬀering from Parkinson’s disease or essential tremor. Itmight even be applicable to diseases with intermittentlyemerging synchronized neural oscillations like epilepsy.1 IntroductionIn several neurological diseases like Parkinson’s disease(PD) or essential tremor, brain function is severelyimpaired by synchronization processes (Elble and Koller1990). Parkinsonian resting tremor appears to be causedby a population of neurons located in the thalamus andthe basal ganglia. These neurons ﬁre in a synchronizedand intrinsically periodical manner at a frequencysimilar to that of the tremor, regardless of any feedbacksignals (Llina´s and Jahnsen 1982; Pare et al. 1990; Lenzet al. 1994). In contrast, under physiological conditionsthese neurons ﬁre incoherently (Nini et al. 1995). Inpatients with PD, this cluster acts like a pacemaker andactivates premotor areas and the motor cortex (Albertset al. 1969; Volkmann et al. 1996), where the lattersynchronize their oscillatory activity (Tass et al. 1998).In patients with advanced PD or essential tremor whono longer respond to drug therapy, depth electrodes arechronically implanted in target areas like the thalamicventralis intermedius nucleus or the subthalamic nucleus(Benabid et al. 1991; Blond et al. 1992). Electrical deepbrain stimulation (DBS) is performed by administering apermanent high-frequency (HF) (> 100 Hz) periodicpulse train via the depth electrodes (Benabid et al. 1991;Blond et al. 1992). HF DBS has been developed empiri-cally, and its mechanism is not yet fully understood.Permanent HF stimulation basically mimics the eﬀect oftissue lesioning by suppressing neuronal ﬁring, which inturn suppresses the peripheral tremor (Wielepp et al.2001).HFDBSisreversibleandhasamuchlowerrateofside eﬀects than lesioning with thermocoagulation(Schuurman et al. 2000). Nevertheless, due to currentspread, for example, HF DBS may lead to severe sideeﬀects like dysarthria, dysesthesia, or cerebellar ataxia.For this reason stimulation techniques have beendeveloped that aim at desynchronizing the pacemaker’spathologically synchronized ﬁring in a demand-con-trolled way instead of by simply suppressing the neu-ronal ﬁring (Tass 2001a,b,c, 2002a,b,d). These methodsshare one particular feature: each stimulus consists oftwo qualitatively diﬀerent stimuli. The ﬁrst stimulus isstronger and resets (restarts) the cluster, whereas thesecond, weaker stimulus is a single pulse that is admin-istered after a constant time delay and desynchronizes byhitting the cluster in a vulnerable state. The goal of thereset is to control the dynamics of the cluster byrestarting the cluster in a stereotypical way. The resetmay be achieved by means of a strong single pulse (Tass2001a,c, 2002d), a HF pulse train (Tass 2001b), or a low-frequency pulse train (Tass 2002a,b).A desynchronizing single pulse is eﬀective only if ithits the population very precisely at a vulnerable phase(Tass 1999). Hence the composite stimulation techniquesonly work with an eﬀective reset that may require highstimulation intensities, and minor variations of criticalstimulation parameters (especially the delay betweenresetting stimulus and desynchronizing single pulse) mayabolish the desynchronizing eﬀect (Tass 2001c, 2002b).Thus these methods are not robust against larger vari-ations of the model parameters (frequencies of the <s> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Recently, many experiments and analyses with biped robots have been carried out. Steady walking of a biped robot implies a stable limit cycle in the state space of the robot. In the design of a locomotion control system, there are primarily three problems associated with achieving such a stable limit cycle: the design of the motion of each limb, interlimb coordination, and posture control. In addition to these problems, when environmental conditions change or disturbances are added to the robot, there is the added problem of obtaining robust walking against them. In this paper we attempt to solve these problems and propose a locomotion control system for a biped robot to achieve robust walking by the robot using nonlinear oscillators, each of which has a stable limit cycle. The nominal trajectories of each limb's joints are designed by the phases of the oscillators, and the interlimb coordination is designed by the phase relation between the oscillators. The phases of the oscillators are reset and the nominal trajectories are modified using sensory feedbacks that depend on the posture and motion of the robot to achieve stable and robust walking. We verify the effectiveness of the proposed locomotion control system, analyzing the dynamic properties of the walking motion by numerical simulations and hardware experiments. <s> BIB011 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> We present a system of coupled nonlinear oscillators to be used as programmable central pattern generators, and apply it to control the locomotion of a humanoid robot. Central pattern generators are biological neural networks that can produce coordinated multidimensional rhythmic signals, under the control of simple input signals. They are found both in vertebrate and invertebrate animals for the control of locomotion. In this article, we present a novel system composed of coupled adaptive nonlinear oscillators that can learn arbitrary rhythmic signals in a supervised learning framework. Using adaptive rules implemented as differential equations, parameters such as intrinsic frequencies, amplitudes, and coupling weights are automatically adjusted to replicate a teaching signal. Once the teaching signal is removed, the trajectories remain embedded as the limit cycle of the dynamical system. An interesting aspect of this approach is that the learning is completely embedded into the dynamical system, and does not require external optimization algorithms. We use our system to encapsulate rhythmic trajectories for biped locomotion with a simulated humanoid robot, and demonstrate how it can be used to do online trajectory generation. The system can modulate the speed of locomotion, and even allow the reversal of direction (i.e. walking backwards). The integration of sensory feedback allows the online modulation of trajectories such as to increase the basin of stability of the gaits, and therefore the range of speeds that can be produced <s> BIB012 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> In this paper, we propose a decentralized sensor network scheme capable to reach a globally optimum maximum-likelihood (ML) estimate through self-synchronization of nonlinearly coupled dynamical systems. Each node of the network is composed of a sensor and a first-order dynamical system initialized with the local measurements. Nearby nodes interact with each other exchanging their state value, and the final estimate is associated to the state derivative of each dynamical system. We derive the conditions on the coupling mechanism guaranteeing that, if the network observes one common phenomenon, each node converges to the globally optimal ML estimate. We prove that the synchronized state is globally asymptotically stable if the coupling strength exceeds a given threshold. Acting on a single parameter, the coupling strength, we show how, in the case of nonlinear coupling, the network behavior can switch from a global consensus system to a spatial clustering system. Finally, we show the effect of the network topology on the scalability properties of the network, and we validate our theoretical findings with simulation results. <s> BIB013 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering. © 2005 Elsevier B.V. All rights reserved. <s> BIB014 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> We show that the quadrature LC oscillator is best treated as two strongly coupled, nominally identical oscillators that are locked to the same frequency. Differential equations that extend Adler's description of locking to strong injection reveal the full dynamics of this circuit. With a simplifying insight, the analysis reveals all the modes of the oscillator, their stability, the effects of mismatch on quadrature phase accuracy, and through a novel use of the analysis, phase noise. <s> BIB015 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> This paper proposes a design methodology to stabilize isolated relative equilibria in a model of all-to-all coupled identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or to circular motion of all particles with fixed relative phases. The stabilizing feedbacks derive from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The results of the paper provide a low-order parametric family of stabilizable collectives that offer a set of primitives for the design of higher-level tasks at the group level <s> BIB016 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> In this paper we study the problem of stability for one of the most popular models of coupled phase oscillators, the Kuramoto model. The Kuramoto model is used to describe the phenomenon of collective synchronization, in which an enormous system of oscillators spontaneously locks to a common frequency although the oscillators have distinct natural frequencies. In the paper we consider the stability of the Kuramoto model of coupled oscillators with identical natural frequency and provide a stability analysis of phase difference equilibrium. The stability of the phase difference equilibrium make it possible to apply the Kuramoto model in pattern recognition. <s> BIB017 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview a wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks. We also discuss strong finite size effects in these systems and highlight open problems and perspectives. <s> BIB018 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> This paper proposes a design methodology to stabilize relative equilibria in a model of identical, steered particles moving in the plane at unit speed. Relative equilibria either correspond to parallel motion of all particles with fixed relative spacing or to circular motion of all particles around the same circle. Particles exchange relative information according to a communication graph that can be undirected or directed and time-invariant or time-varying. The emphasis of this paper is to show how previous results assuming all-to-all communication can be extended to a general communication framework. <s> BIB019 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Coordinated Control and Estimation for Multi-agent Systems: Theory and Practice Daniel J. Klein Chair of the Supervisory Committee: Professor Kristi A. Morgansen Aeronautics & Astronautics The problems studied in this thesis center on coordinated control and estimation tasks for multi-agent systems. The work begins technically, with a Kuramoto-inspired study of coordinated phasor centroid control for a group ofN homogeneous phase integrators. A main result is the development of a controller which guarantees asymptotic stability of the phasor centroid to a known reference vector. An extensions of the splay state to non-balanced group states is then explored. The work then moves on to a group of N heterogeneous agents wherein some agents act as leaders to the others, which obey a standard sinusoidal protocol. One interesting point here is that unlike in the related controlled linear agreement problem, symmetry of the communication topology does not determine uncontrollability. While these results are derived in continuous time, the next contributions shows that many of the desirable coordination properties can be preserved in discrete time. The work then moves to a more applied focus. A simulation study of coordinated target tracking is conducted, considering questions of control, estimation, and communication for distributed systems. Finally, the design and development of a computer vision system which enables a demonstration of some of the theory-oriented work on the University of Washington’s Underwater Fin-actuated Autonomous Vehicle testbed is discussed. <s> BIB020 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> This paper considers the integration of communication and control with respect to the task of coordinated heading control for a group of N vehicles. The heading control employed on each vehicle is a discretization of the well-known Kuramoto model of nonlinearly coupled oscillators. Stability for both all-to-all and random one-to-all broadcasts is shown to be dependent on the coupling strength, K, and the time discretization, DeltaT. For a desired system performance, DeltaT imposes a tight deadline by which the state information (M bits) must be propagated through the communication network. Routing optimization with respect to minimizing energy consumption is formulated considering the DeltaT deadline. Multi-hopping is optimal if time permits, but one-to-all broadcasting is more energy efficient when M/DeltaT is sufficiently large. <s> BIB021 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> This article has explored history, recent advances, and challenges in distributed synchronization for distributed wireless systems. It is focused on synchronization schemes based on exchange of signals at the physical layer and corresponding baseband processing, wherein analysis and design can be performed using known tools from signal processing. Emphasis has also been given on the synergy between distributed synchronization and distributed estimation/detection problems. Finally, we have touched upon synchronization of nonperiodic (chaotic) signals. Overall, we hope to have conveyed the relevance of the subject and to have provided insight on the open issues and available analytical tools that could inspire further research within the signal processing community. <s> BIB022 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> The problem of controlling locomotion is an area in which neuroscience and robotics can fruitfully interact. In this article, I will review research carried out on locomotor central pattern generators (CPGs), i.e. neural circuits capable of producing coordinated patterns of high-dimensional rhythmic output signals while receiving only simple, low-dimensional, input signals. The review will first cover neurobiological observations concerning locomotor CPGs and their numerical modelling, with a special focus on vertebrates. It will then cover how CPG models implemented as neural networks or systems of coupled oscillators can be used in robotics for controlling the locomotion of articulated robots. The review also presents how robots can be used as scientific tools to obtain a better understanding of the functioning of biological CPGs. Finally, various methods for designing CPGs to control specific modes of locomotion will be briefly reviewed. In this process, I will discuss different types of CPG models, the pros and cons of using CPGs with robots, and the pros and cons of using robots as scientific tools. Open research topics both in biology and in robotics will also be discussed. <s> BIB023 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Synchronization in complex networks Alex Arenas, 1, 2, 3 Albert D´ iaz-Guilera, 4, 2 Jurgen Kurths, 5 Yamir Moreno, 2, 6 and Changsong Zhou 7 Departament d’Enginyeria Inform` tica i Matem` tiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain a a Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50009, Spain Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Departament de F´ isica Fonamental, Universitat de Barcelona, 08028 Barcelona, Spain Institute of Physics, University of Potsdam PF 601553, 14415 Potsdam, Germany Department of Theoretical Physics, University of Zaragoza, Zaragoza 50009, Spain Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong (Dated: February 6, 2008) Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many eﬀorts de- voted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive nu- merical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to diﬀerent disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences. PACS numbers: 05.45.Xt,89.75.Fb,89.75.Hc Contents I. Introduction II. Complex networks in a nutshell III. Coupled phase oscillator models on complex networks A. Phase oscillators 1. The Kuramoto model 2. Kuramoto model on complex networks 3. Onset of synchronization in complex networks 4. Path towards synchronization in complex networks 5. Kuramoto model on structured or modular networks 6. Synchronization by pacemakers B. Pulse-coupled models C. Coupled maps IV. Stability of the synchronized state in complex networks A. Master Stability Function formalism 1. Linear Stability and Master Stability Function 2. Measures of synchronizability 3. Synchronizability of typical network models 4. Synchronizability and structural characteristics of networks 5. Graph theoretical bounds to synchronizability 6. Synchronizability of weighted networks 7. Universal parameters controlling the synchronizability B. Design of synchronizable networks 1. Weighted couplings for enhancing synchronizability 2. Topological modiﬁcation for enhancing synchronizability 3. Optimization of synchronizability C. Beyond the Master Stability Function formalism V. Applications A. Biological systems and neuroscience 1. Genetic networks 2. Circadian rhythms 3. Ecology 4. Neuronal networks 5. Cortical networks B. Computer science and engineering 1. Parallel/Distributed computation 2. Data mining 3. Consensus problems 4. Communication networks 5. Material and traﬃc ﬂow 6. Power-grids C. Social sciences and economy 1. Opinion formation 2. Finance 3. World Trade Web D. Perspectives VI. Conclusions Acknowledgments References I. INTRODUCTION Synchronization, as an emerging phenomenon of a pop- ulation of dynamically interacting units, has fascinated humans from ancestral times. No matter whether the phenomenon is spontaneous or induced, synchronization captivates our minds and becomes one of the most in- teresting scientiﬁc problems. Synchronization processes are ubiquitous in nature and play a very important role in many diﬀerent contexts as biology, ecology, climatol- ogy, sociology, technology, or even in arts ((Osipov et al., 2007; Pikovsky et al., 2001). It is known that synchrony <s> BIB024 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> The goal of the present paper is to highlight the fundamental differences of so-called synchronization or consensus algorithms when the agents to synchronize evolve on a compact homogeneous manifold (like the circle, sphere or the group of rotation matrices), instead of a vector space. For the benefit of understanding, the discussion is restricted to the circle. First, a fundamental consensus algorithm on R n is reviewed, from which continuous- and discretetime synchronization algorithms on the circle are deduced by analogy. It is shown how they are connected to Kuramoto and Vicsek models from the literature. Their convergence properties are similar to vector spaces only for specific graphs or if the agents are all located within a semicircle. Examples are proposed to illustrate several other possible behaviors. Finally, new algorithms are proposed to recover (almost-)global synchronization properties. <s> BIB025 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Motivated by recent interest for multi-agent systems and smart power grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with non-trivial transfer conductances. Our key insight is to exploit the relationship between the power network model and a first-order model of coupled oscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a non-uniform Kuramoto model. Here, non-uniform Kuramoto oscillators are characterized by multiple time constants, non-homogeneous coupling, and non-uniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols, we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditions reduce to and improve upon previously-available tests for the standard Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronization and transient stability of a power network to the underlying system parameters and initial conditions. <s> BIB026 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> This paper studies the problem of realizing a common software clock among a large set of nodes without an external time reference (i.e., internal clock synchronization), any centralized control, and where nodes can join and leave the distributed system at their will. The paper proposes an internal clock synchronization algorithm which combines the gossip-based paradigm with a nature-inspired approach, coming from the coupled oscillators phenomenon, to cope with scale and churn. The algorithm works on the top of an overlay network and uses a uniform peer sampling service to fulfill each node's local view. Therefore, differently from clock synchronization protocols for small scale and static distributed systems, here, each node synchronizes regularly with only the neighbors in its local view and not with the whole system. An evaluation of the convergence speed and the synchronization error of the coupled-based internal clock synchronization algorithm has been carried out, showing how convergence time and the synchronization error depends on the coupling factor and the local view size. Moreover, the variation of the synchronization error with respect to churn and the impact of a sudden variation of the number of nodes have been analyzed to show the stability of the algorithm. In all these contexts, the algorithm shows nice performance and very good self-organizing properties. Finally, we showed how the assumption on the existence of a uniform peer-sampling service is instrumental for the good behavior of the algorithm and how, in system models where network delays are unbounded, a mean-based convergence function reaches a lower synchronization error than median-based convergence functions exploiting the number of averaged clock values. <s> BIB027 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> This paper presents a novel synchronizing method for a human-robot ensemble using coupled oscillators. We define an ensemble as a synchronized performance produced through interactions between independent players. To attain better synchronized performance, the robot should predict the human's behavior to reduce the difference between the human's and robot's onset timings. Existing studies in such synchronization only adapts to onset intervals, thus, need a considerable time to synchronize. We use a coupled oscillator model to predict the human's behavior. Experimental results show that our method reduces the average of onset time errors; when we use a metronome, a tempo-varying metronome or a human drummer, errors are reduced by 38%, 10% or 14% on the average, respectively. These results mean that the prediction of human's behaviors is effective for the synchronized performance. <s> BIB028 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> The Kuramoto model captures various synchronization phenomena in biological and man-made systems of coupled oscillators. It is well-known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength for finite and infinite-dimensional, and for first and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. Fourth and finally, we extend our analysis to multi-rate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that the multi-rate Kuramoto model is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies, and we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization. Interestingly, these conditions do not depend on the inertiae which contradicts prior observations on the role of inertiae in synchronization of second-order Kuramoto models. <s> BIB029 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> We consider the problem of desynchronizing a network of synchronized, globally (all-to-all) coupled neurons using an input to a single neuron. This is done by applying the discrete time dynamic programming method to reduced phase models for neural populations. This technique numerically minimizes a certain cost function over the whole state space, and is applied to a Kuramoto model and a reduced phase model for Hodgkin-Huxley neurons with electrotonic coupling. We evaluate the effectiveness of control inputs obtained by averaging over results obtained for different coupling strengths. We also investigate the applicability of this method to Hodgkin-Huxley models driven by multiplicative stimuli. <s> BIB030 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Abstract Motivated by synchronous communications, we propose a consensus based carrier synchronization algorithm involving two transceiving units. Our algorithm achieves frequency lock globally and exponentially. Further it also achieves phase synchronization in the following sense. Asymptotically it induces the two transmitters to be either in phase, or out of phase by 180 degrees. We provide a simple method for the transmitters to know if they are out phase by 180 degrees. This constitutes a significant advance over existing carrier synchronization technology which is largely based on Phase Locked Loops that only achieve lock locally. <s> BIB031 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> The idea that synchronized oscillations are important in cognitive tasks is receiving significant attention. In this view, single neurons are no longer elementary computational units. Rather, coherent oscillating groups of neurons are seen as nodes of networks performing cognitive tasks. From this assumption, we develop a model of stimulus-pattern learning and recognition. The three most salient features of our model are: 1) a new definition of synchronization; 2) demonstrated robustness in the presence of noise; and 3) pattern learning. <s> BIB032 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Motivated by neuroscience applications, and in particular by the deep brain stimulation treatment for Parkinson’s disease, we have recently derived a simplified model of an interconnected neuronal population under the effect of its mean-field proportional feedback. In this paper, we rely on that model to propose conditions under which proportional mean-field feedback achieves either oscillation inhibition or desynchronization. More precisely, we show that for small natural frequencies, this scalar control signal induces an inhibition of the collective oscillation. For the closed-loop system, this situation corresponds to a fixed point which is shown to be almost globally asymptotically stable in the fictitious case of zero natural frequencies and all-to-all coupling and feedback. In the case of an odd number of oscillators, this property is shown to be robust to small natural frequencies and heterogencities in both the coupling and feedback topology. On the contrary, for large natural frequencies, we show that scalar proportional mean-field feedback is able to induce desynchronization. After having recalled a formal definition for desynchronization, we show how it can be induced in a network of originally synchronized oscillators. <s> BIB033 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> Motivated by the recent and growing interest in smart grid technology, we study the operation of DC/AC inverters in an inductive microgrid. We show that a network of loads and DC/AC inverters equipped with power-frequency droop controllers can be cast as a Kuramoto model of phase-coupled oscillators. This novel description, together with results from the theory of coupled oscillators, allows us to characterize the behavior of the network of inverters and loads. Specifically, we provide a necessary and sufficient condition for the existence of a synchronized solution that is unique and locally exponentially stable. We present a selection of controller gains leading to a desirable sharing of power among the inverters, and specify the set of loads which can be serviced without violating given actuation constraints. Moreover, we propose a distributed integral controller based on averaging algorithms which dynamically regulates the system frequency in the presence of a time-varying load. Remarkably, this distributed-averaging integral controller has the additional property that it maintains the power sharing properties of the primary droop controller. Our results hold without assumptions on identical line characteristics or voltage magnitudes. <s> BIB034 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> The aim of this paper is to describe a coupled oscillator model for Bayesian inference. The coupled oscillator model comprises of a large number of oscillators with mean-field coupling. The collective dynamics of the oscillators are used to solve an inference problem: the empirical distribution of the population encodes a ‘belief state' (posterior distribution) that is continuously updated based on noisy measurements. In effect, the coupled oscillator model works as a particle filter. The framework is described here with the aid of a model problem involving estimation of a walking gait cycle. For this problem, the coupled oscillator particle filter is developed, and demonstrated on experimental data taken from an Ankle-foot Orthosis (AFO) device. <s> BIB035 </s> Synchronization in complex networks of phase oscillators: A survey <s> Applications in engineering: <s> We explore different approaches for generating music from the flocking dynamics of groups of mobile autonomous agents following a simple decentralized control rule. By developing software that links these dynamics to a set of sound wave generators, we study how each approach reflects sonically the transition to collective order and which produces musically interesting results. We consider three qualitatively different ways of translating flocking dynamics into music: (1) A direct approach that maps agent positions to sounds, (2) a synchronization approach where each agent carries an oscillator that couples to neighboring agents, and (3) a physics-inspired approach that mimics the sound that would result from an effective friction between neighboring agents. We find that all approaches allow the listener to discriminate between different phases in the system, but that the second and third can yield more musically interesting and appealing results. <s> BIB036
Technological applications of the coupled oscillator model (1) and its generalization (4) include deep brain stimulation BIB010 BIB030 BIB033 , locking in solid-state circuit oscillators BIB001 BIB015 , planar vehicle coordination BIB016 BIB019 BIB020 BIB021 , carrier synchronization without phase-locked loops BIB031 , synchronization in semiconductor laser arrays BIB004 , and microwave oscillator arrays BIB003 . Since alternating current (AC) circuits are naturally modeled by equations similar to (1), some electric applications are found in structure-preserving BIB002 BIB005 and networkreduced power system models  BIB026 , and droop-controlled inverters in microgrids BIB034 . Algorithmic applications of the coupled oscillator model (1) include limit-cycle estimation through particle filters BIB035 , clock synchronization in decentralized computing networks BIB022 BIB027 , central pattern generators for robotic locomotion BIB011 BIB012 BIB023 , decentralized maximum likelihood estimation BIB013 , and human-robot interaction BIB028 . Further envisioned applications of oscillator networks obeying equations similar to (1) include generating music BIB036 , signal processing  , pattern recognition BIB032 , and neuro-computing through micromechanical BIB008 or laser BIB006 BIB017 oscillators. Theoretical investigations: Coupled oscillator models of the form (1) are studied from a purely theoretical perspective in the physics, dynamical systems, and control communities. At the heart of the coupled oscillator dynamics is the transition from incoherence to synchrony. In this article we will be particularly interested in the notion of frequency synchronization, that is, in the property of certain solutions to reach equal frequencieṡ θ i (t) among all oscillators. For infinitely many oscillators this notion can be relaxed to a subset of oscillators reaching frequency synchronization. We will also study conditions under which the angles θ i (t) themselves synchronize, or they are tightly clustered (in a single or multiple groups), or they are spread evenly in regular patterns over the circle. We refer to the surveys and tutorials  BIB007 BIB009  BIB014 BIB024 BIB018 BIB029 BIB025  for an incomplete set of recent theoretic research activities. We will review and attribute relevant theoretical results throughout the course of this article.