Synchronization in complex networks of phase oscillators: A survey <s> 7 <s> Deals with several notions of incremental stability. In other words, the focus is on stability of trajectories with respect to one another, rather than with respect to some attractor. The aim is to present a framework for understanding such questions fully compatible with the well-known input-to-state stability approach. Applications of the newly introduced stability notions are also discussed. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> 7 <s> We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> 7 <s> Two related problems are treated in continuous time. First, the state agreement problem is studied for coupled nonlinear differential equations. The vector fields can switch within a finite family. Associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems. Generalizing the work of Moreau, under the assumption that the vector fields satisfy a certain subtangentiality condition, it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time. The proof uses nonsmooth analysis. Second, the rendezvous problem for kinematic point-mass mobile robots is studied when the robots’ fields of view have a fixed radius. The circumcenter control law of Ando e [IEEE Trans. Robotics Automation, 15 (1999), pp. 818-828] is shown to solve the problem. The rendezvous problem is a kind of state agreement problem, but the interaction structure is state dependent. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> 7 <s> An important task for multi-agent systems (MAS) is to reach a consensus, e.g. to align their velocity vectors. Recent results propose appropriate consensus protocols to achieve such tasks, but most of them do not consider the effect of communication constraints such as the presence of time-delays in the exchange of information between the agents. In this article, we provide conditions for a non-linear, locally passive MAS of arbitrary size to reach a consensus, when the agents communicate over a packet-switched network that is characterised by a given topology. Both the cases of constant and switching topologies are considered. The nature of the communication channel imposes constraints that are modelled using stochastic delays of arbitrary distribution. We first embed this model in another, distributed but deterministic delay model and provide conditions for the error introduced by this simplification. In our main result, we provide conditions for the locally passive MAS with distributed delays to reach a consensus. In the case of a fixed topology, the underlying directed graph has to contain a spanning tree. In the case of a switching topology, only the union graph of all graphs that persist over time is required to contain a spanning tree. These conditions are independent of the distribution and the size of the packet delays. To show attractivity of the consensus set, we use an invariance principle for systems described by functional differential equations based on an appropriate Lyapunov–Razumikhin function. This methodological approach is the main contribution of this work and can also be applied to other consensus problems with delays. We illustrate our results by numerical simulations showing synchronisation of non-linear Kuramoto oscillators over a digital network. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> 7 <s> Abstract We discuss the asymptotic complete phase–frequency synchronization for the Kuramoto phase model with a finite size N . We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N , hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> 7 <s> We investigate frequency synchronization and phase agreement in networks of non-identical Kuramoto oscillators with delayed coupling. We provide delay-dependent lower bounds for the coupling gain so that a network with all-to-all coupling and constant heterogeneous delays achieves synchronization. The given bounds are determined explicitly by the system parameters. Furthermore, we show that frequency synchronization and phase agreement are possible in general directed networks of non-identical oscillators which contain a spanning tree if the delays satisfy a self-consistency condition. Networks with delays differ at this point notably from networks without delays, because phase agreement is not possible in undelayed non-identical Kuramoto oscillator networks. In addition to the sufficient conditions for synchronization, we investigate the connection between phase agreement and the existence of solutions of a necessary self-consistency condition. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> 7 <s> Abstract In this Letter, Kuramoto model in a high-dimensional linear space is investigated. Some results on the equilibria and synchronization of the classical Kuramoto model are generalized to the high-dimensional Kuramoto model. It is proved that, if the interconnection graph is connected and all the initial states lie in a half part of the state space, the synchronization can be achieved. Finally, numerical simulations are given to validate the obtained theoretical results. <s> BIB007
Theorem 5.2 also holds for directed and time-variant graphs, it applies to more general interaction functions, and it can be extended to time-delayed systems. Applications to oscillator networks and extensions can be found in BIB003 BIB002 BIB004 BIB005  BIB005 Bullo, 2011, 2012b; BIB006 . We will revisit this literature in Section 6. An elegant generalization of the above analysis to oscillator networks on nspheres (rather than on S 1 ) can be found in BIB007 , and diffusively-coupled Lienard-type oscillators can be analyzed using similar ideas in the phase plane (Tuna, 2012). Remark 2 (A control-theoretical perspective on synchronization) As established in Theorems 5.1 and 5.2, the phase-synchronized set ∆ G (0) = Arc n (0) is locally exponentially stable provided that all natural frequencies are identical. While phase synchronization is not possible for dissimilar natural frequencies, a certain degree of phase cohesiveness can still be achieved. Indeed, the coupled oscillator model (1) can be regarded as an exponentially stable system subject to the disturbance ω ∈ 1 ⊥ n and synchronization can be studied using classic control-theoretical concepts such as input-to-state stability, practical stability, ultimate boundedness  or their incremental versions BIB001 . In control-theoretical terminology, phase cohesiveness can then be described as "practical phase synchronization." Compared to prototypical nonlinear control examples, various additional challenges arise in the analysis of the coupled oscillator model (1) due to the bounded and nonmonotone sinusoidal coupling, the compact state space, and the coexistence of multiple equilibria.