Synchronization in complex networks of phase oscillators: A survey <s> A Simple yet Illustrative Example <s> A complex high-dimensional chaotic behavior, phase chaos, is found in the finite-dimensional Kuramoto model of coupled phase oscillators. This type of chaos is characterized by half of the spectrum of Lyapunov exponents being positive and the Lyapunov dimension equaling almost the total system dimension. Intriguingly, the strongest phase chaos occurs for intermediate-size ensembles. Phase chaos is a common property of networks of oscillators of very different natures, such as phase oscillators, limit-cycle oscillators, and chaotic oscillators, e.g., Rössler systems. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> A Simple yet Illustrative Example <s> We derive a master stability function (MSF) for synchronization in networks of coupled dynamical systems with small but arbitrary parametric variations. Analogous to the MSF for identical systems, our generalized MSF simultaneously solves the linear-stability problem for near-synchronous states (NSS) for all possible connectivity structures. We also derive a general sufficient condition for stable near-synchronization and show that the synchronization error scales linearly with the magnitude of parameter variations. Our analysis underlines the significant role played by the Laplacian eigenvectors in the study of network synchronization of near-identical systems. <s> BIB002
The following example illustrates different notions of synchronization and points out various important geometric subtleties occurring on the compact state space T 2 . Consider n = 2 oscillators with ω 2 ≥ 0 ≥ ω 1 = −ω 2 . We restrict our attention to angles contained in Arc n (π): for angles θ 1 , θ 2 with |θ 2 −θ 1 | < π, the angular difference θ 2 − θ 1 is the number in ]−π, π[ with magnitude equal to the geodesic distance |θ 2 − θ 1 | and with positive sign if and only if the counter-clockwise path length from θ 1 to θ 2 is smaller than the clockwise path length. With this definition the two-dimensional oscillator dynamics (θ 1 ,θ 2 ) can be reduced to the scalar difference dynamicṡ θ 2 −θ 1 . After scaling time as t → t(ω 2 − ω 1 ) and intro- The one-parameter family of dynamical systems (15) can be analyzed graphically by plotting the scalar vector field Fig. 7(a) . The vector field features a saddle-node bifurcation at κ = 1. For κ < 1 no equilibria exist. For κ > 1 we have an asymptotically stable equilibrium θ stable = arcsin(κ −1 ) ∈ ]0, π/2[ and an unstable equilibrium θ unstable = arcsin(κ trajectories converge to θ stable , that is, the oscillators synchronize and remain phase cohesive (or arc invariant). For (θ 2 (0) − θ 1 (0)) ∈ [0, θ unstable [ the difference θ 2 (t) − θ 1 (t) increases beyond π, and θ 2 (t) − θ 1 (t) converges asymptotically to the equilibrium θ stable in the set where θ 2 − θ 1 < 0. Equivalently, in the configuration space S 1 the oscillators revolve once around the circle before converging to [θ stable ]. Since sin(θ stable ) = sin(θ unstable ) = κ −1 , in the limit κ → ∞ the oscillators achieve phase synchronization from every initial condition in an open semi-circle Arc 2 (π). In the critical case, κ = 1, the saddle equilibrium manifold at [θ saddle ] is globally attractive but not stable, see Fig. 7 (b). Fig. 7 . Plot of the vector field (15) for θ2 − θ1 > 0 and various values of κ and a trajectory θ(t) ∈ T 2 for the critical case κ = 1, where the dashed line is the saddle equilibrium manifold and and • depict θ(0) and limt→∞ θ(t). In conclusion, the simple but already rich 2-dimensional case shows that two oscillators are phase cohesive and synchronize if and only if κ > 1, that is, if and only if the coupling dominates the heterogeneity as 2a 12 > ω 2 −ω 1 . The ratio 1/κ determines the asymptotic phase cohesiveness as well as the set of admissible initial conditions. More general oscillator networks display the same phenomenology, but the threshold from incoherence to synchrony is generally unknown. Finally, we remark that for oscillator networks of dimension n ≥ 3, this loss of synchrony via a saddle-node bifurcation is only the starting point of a series of bifurcations occurring if the coupling is further decreased, see  BIB001 BIB002 .