Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing and Pattern Formation <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> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing and Pattern Formation <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> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Phase Balancing and Pattern Formation <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> BIB003
As compared with phase synchronization, only few results are known about phase balancing. This asymmetry may be caused by the fact that phase synchrony is studied in more applications than phase balancing. Moreover, the phase-synchronized set Arc n (0) admits a very simple geometric characterization, whereas the phasebalanced set Bal n has a complicated structure consisting of multiple disjoint subsets. The number of these subsets grows combinatorially with the number of nodes n. Consider the coupled oscillator model (29). By inverting the direction of time, we obtaiṅ In what follows, we say that an undirected graph G(V, E, A) is circulant if the adjacency matrix A = A T is a circulant matrix. Circulant graphs are highly symmetric graphs; examples include complete graphs, complete bipartite graphs, and ring graphs. 8 For circulant and uniformly weighted graphs, the coupled oscillator model (34) achieves phase balancing. The following theorem summarizes several results originally presented in , Theorem 1) and (Sepulchre et al., 2008, Theorem 2) . is complete, then the region of attraction of the stable phase-balanced set Bal n is almost all of T n . The proof of Theorem 5.3 is similar to that of Theorem 5.1: convergence is established by a potential function argument and the local (in)stability of equilibria is established by a Jacobian argument. An illustrative simulation is shown in Fig. 12(b) . The analysis leading to BIB003 , Theorem 2) suggests also almost global stability of the phase-balanced set for arbitrary circulant graphs, but a complete proof is not available. For non-circulant graphs, the conclusions of Theorem 5.3 are not true. As a remedy to achieve almost globally stable phase balancing, higher order models with dynamic coupling can be considered, see BIB001 BIB003 for further details. Alternatively, phase balancing can also be achieved by coupling functions with higher-order harmonics. For example, a modification of model (34) iṡ where K ∈ R are appropriate gains, and m ∈ N divides n. The dynamics (35) are again a gradient system whose critical points include symmetric balanced (m, n)-patterns; recall Fig. 9 for a schematic illustration. The following result is given in , Theorem 7) and BIB003 , Theorem 7). Theorem 5.4 (Pattern formation) Consider the coupled oscillator model (35) with a connected, undirected, uniformly weighted, and circulant graph G(V, E, A). Let m ∈ N be a divisor of n, let K > 0 for ∈ {1, . . . , m − 1}, and let K m < 0 be sufficiently small. Then each symmetric balanced (m, n)-pattern is a locally exponentially stable equilibrium manifold. Two representative simulations of the pattern-forming model (35) are shown in Fig. 12(c) and Fig. 12(d) . Theorem 5.4 can also be extended to non-circulant (and directed) graphs through dynamic coupling, see (Sepulchre et al., 2008, Theorem 8) . Notice that Theorem 5.4 establishes only the local stability of (m, n) patterns; the dynamics (35) may feature also other stable equilibria. For the complete graph, BIB002 conjecture almost global convergence to the set of symmetric balanced (m, n)-patterns.