Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <s> Systems that are competitive and possess symmetric interactions admit a global Lyapunov function. However, a global Lyapunov function whose equilibrium set can be effectively analyzed has not yet been discovered. It remains an open question whether the Lyapunov function approach, which requires a study of equilibrium points, or an alternative global approach, such as the Lyapunov functional approach, which sidesteps a direct study of equilibrium points will ultimately handle all of the physically important cases. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <s> 1 Introduction.- 2 Bifurcations in Neuron Dynamics.- 3 Neural Networks.- 4 Introduction to Canonical Models.- 5 Local Analysis of WCNNs.- 6 Local Analysis of Singularly Perturbed WCNNs.- 7 Local Analysis of Weakly Connected Maps.- 8 Saddle-Node on a Limit Cycle.- 9 Weakly Connected Oscillators.- 10 Multiple Andronov-Hopf Bifurcation.- 11 Multiple Cusp Bifurcation.- 12 Quasi-Static Bifurcations.- 13 Synaptic Organizations of the Brain.- References. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <s> We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <s> In this article we study global stability properties of the Kuramoto model of sinusoidally coupled oscillators. We base our analysis on previous results by the control community that analyze local properties of the consensus point of different kinds of Kuramoto models. We prove that for the complete symmetric case, the consensus point is almost globally stable, that is, the set of trajectories that do not converge to it has zero measure. We present a counter-example of that when the completeness hypothesis is removed. We also show that the general non-symmetric case is more complex and we analyze the particular case of oscillators coupled in a ring structure, where we can establish some global stability properties. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <s> In this paper, we study the behavior of a network of N agents, each evolving on the circle. We propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the (possibly time-varying and unidirectional) communication graphs. The global convergence analysis on the N-torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the Euclidean space. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <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> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <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> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <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> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <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> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Synchronization <s> Kuramoto oscillator networks represent a standard model of synchronization analysis. Extensive studies have shown that static interactions can only bring about a partial synchronization (phase locking). This paper investigates the complete synchronization, where the oscillator outputs follow exactly a common trajectory. It is shown that oscillators with individual natural frequencies can be synchronized by dynamic couplings with integrator dynamics. A necessary and sufficient synchronization condition is derived, which shows that oscillators with positive parameters are completely synchronized for arbitrary coupling strength. <s> BIB010
It can be easily verified that for non-zero and dissimilar natural frequencies ω ∈ 1 ⊥ n , the coupled oscillator model (1) does not admit a phase-synchronized solution of the form θ i (t) = θ j (t) for all i, j ∈ {1, . . . , n}. On the contrary, if all natural frequencies are identical, ω i ≡ ω 0 for all i ∈ {1, . . . , n}, then a transformation of the oscillator network (1) to a rotating frame with frequency ω 0 yieldṡ An elegant analysis of the coupled oscillator model (29) follows the insights developed in Subsections 4.1 and 4.2. System (29) is a negative gradient floẇ θ = −∂U (θ)/∂θ defined by the smooth function U (θ) with compact sublevel sets. Hence, the LaSalle Invariance Principle (Khalil, 2002, Theorem 4.4) asserts that every solution converges to set of critical points {θ ∈ T n | ∂U (θ)/∂θ = 0 n }. This basic convergence result using potential functions and the LaSalle Invariance Principle has long been known in the neuroscience community, see, for example, BIB001 BIB002 . Recent research efforts focus predominantly on establishing almost global synchronization. By Lemma 4.1, the phasesynchronized equilibrium manifold [θ] ∈ ∆ G (0) is locally exponentially stable, and for a S 1 -synchronizing graph, all other equilibria are unstable. We collect these observations in the following result presented, among others, in BIB003 BIB004 BIB005 BIB006 . Theorem 5.1 (Phase synchronization) Consider the coupled oscillator model (1) with a connected graph G(V, E, A) and with natural frequencies ω ∈ R n . The following statements are equivalent: (i) Homogeneity: there exists a constant ω 0 ∈ R such that ω i = ω 0 for all i ∈ {1, . . . , n}; and (ii) Local phase sync: there exists a locally exponentially stable phase synchronization manifold ∆ G (0). If the two equivalent cases (i) and (ii) are true, the following statements hold: 1) Global convergence: For all initial angles θ(0) ∈ T n , the frequenciesθ(t) converge to ω 0 1 n and the phases θ(t) converge to {θ ∈ T n | ∂U (θ)/∂θ = 0 n }; and 2) Almost global stability: If G(V, E, A) is S 1 -synchronizing, the region of attraction of the phase synchronization manifold ∆ G (0) is almost all of T n . A representative simulation is shown in Fig. 12(a) below. The corresponding discrete-time analog to Theorem 5.1 can be found in BIB007 BIB008 BIB005 . If higher order models with dynamic coupling are considered, then almost globally stable phase synchronization can be achieved for arbitrary connected (and also directed) graphs; see BIB005 BIB009 BIB010 for details.