51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA Kick synchronization versus diffusive synchronization Alexandre Mauroy, Pierre Sacré, and Rodolphe Sepulchre Abstract— The paper provides an introductory discussion analysis [3], dissipativity analysis [4], [29], [65], [70], [71], about two fundamental models of oscillator synchronization: and—to a growing extent—contraction analysis [59], [60], the (continuous-time) diffusive model, that dominates the math- [64], [68], [69], [80], provide natural system theoretic tools ematical literature on synchronization, and the (hybrid) kick model, that accounts for most popular examples of synchroniza- to study synchronization. The literature of synchronization tion, but for which only few theoretical results exist. The paper (closely related to consensus theory and coordination theory) stresses fundamental differences between the two models, such is growing and the topic has attracted many systems and as the different contraction measures underlying the analysis, control researchers in the recent years. as well as important analogies that can be drawn in the limit The kick model is largely dominant in natural manifes- of weak coupling. tations of synchronization. Popular examples include syn- I. INTRODUCTION chronization of metronomes [84], clocks [5], [33], heart beats [61], flashing fireflies [11], neurons [23], earth- Synchronization is a pervasive concept in science and quakes [56], and in fact most if not all spiking oscillators. engineering. Currently, it is perhaps the most widely studied In addition, kick synchronization is a source of inspira- dynamical concept across systems biology [24], [26], [83], tion for engineering applications (e.g. synchronization in neuroscience [32], [35], chemistry [39], physics [34], [62], wireless sensor networks [31], unsupervised classification astronomy [6], and engineering [53], [73]. Because synchro- problems [63]). Despite the widespread occurrence of the nization involves interconnection at its core, the relevance of phenomenon, the mathematical literature on kick synchro- systems theory to model, understand, and control synchro- nization is rather sparse compared to the literature on diffu- nization is obvious and was recognized early, e.g. [52]. sive synchronization. Two fundamental mathematical models of synchronization Primarily motivated by the recent thesis [44], the present have emerged across the literature: the diffusive model paper aims at comparing and contrasting the diffusive model and the kick model (a nickname throughout the paper for and the kick model for the synchronization of periodic pulse-coupled synchronization model). The diffusive model oscillators. We stress both the differences and the analogies analyzes synchronization as the result of diffusive coupling: between the two models, with a particular emphasis on the interconnection has the input–output interpretation of their global stability properties. The discussion is tutorial in a static diffusive passive map. Owing to the fundamental nature and focuses on simple examples, such as the coupling homogenization nature of diffusion, diffusive interconnec- of van der Pol oscillators, which provides an insightful tions tend to reduce differences between the time-course illustration of diffusive synchronization in the weakly non- of interconnected variables, thereby favoring synchronized linear oscillation regime and of kick synchronization in the behavior if they are strong enough. In contrast, the kick relaxation oscillation regime. The diffusive model is studied model analyzes synchronization as the result of mutual in continuous-time models while the kick model, hybrid rhythmic locking by short and weak pulses, akin to the in nature, is typically studied in discrete time. Ultimately, physical phenomenon of resonance. The impulsive nature of synchronization is always proven by showing that a certain the coupling combined with the continuous-time flow of the distance between trajectories contracts over time, but the model between the pulses results in a hybrid model, see [54] contraction measure is distinctively different in diffusive and for a rigorous description of the kick model as a hybrid kick models. model. While the diffusive and kick models of oscillator syn- The diffusive model is largely dominant in the mathemat- chronization are fundamentally different, they also exhibit ical literature of synchronization. Synchronization between a remarkable analogy in the limit of weak coupling. This trajectories of state-space models is analyzed as an incre- is because arbitrary oscillator models all reduce to one- mental stability property [43] (i.e. the trajectories converge dimensional phase models when the interconnection is suffi- to one another rather than being attracted toward some ciently weak to maintain system trajectories in the neigh- equilibrium position). The leading concepts of Lyapunov borhood of the limit cycle oscillations of the uncoupled oscillators. Rooted in the seminal contributions of Win- A. Mauroy is with the Department of Mechanical Engineering, Uni- versity of California Santa Barbara, Santa Barbara, CA 93106, USA. free [82] and Kuramoto [38], phase models of interconnected [email protected] oscillators have a universal structure entirely characterized P. Sacré and R. Sepulchre are with the Department of Electrical by their coupling function, which is strongly related to Engineering and Computer Science (Montefiore Institute, B28), Univer- sity of Liège, 4000 Liège, Belgium. [email protected], the phase response curve of the oscillators (i.e. a function [email protected] which corresponds to the phase sensitivity of the uncoupled 978-1-4673-2064-1/12/$31.00978-1-4673-2066-5/12/$31.00 ©2012 IEEE 7171 strong coupling section III section IV Fig. 2. The asymptotic phase map Θ: B(γ) → S1 assigns to each point q in the basin B(γ) a single scalar phase θ on the unit circle S1, such that γ limt→+∞ kΦ(t, q, 0) − Φ(t, p, 0)k2 = 0 where p = x (θ/ω). The set of all points q characterized by the same phase θ is the isochron Iθ. weak coupling section V A. State-space models We consider open dynamical systems described by non- linear time-invariant state-space models diffusive coupling kick coupling x˙ = F(x) + G(x)u, x Rn, u R, (1a) ∈ ∈ R Fig. 1. The paper is organized according to the coupling models. y = H(x), y , (1b) Sections III and IV focus on (possibly strong) diffusive and kick synchro- ∈ nization, respectively. Section V deals with the phase models encountered where the vector fields F and G, and the measurement in the limit of weak coupling. map H support all usual smoothness conditions that are necessary for existence and uniqueness of solutions. We write Φ(t, x0, u) for the solution of the initial value problem (1a) oscillators to an external perturbation). As a result, the with x(0) = x0. fundamental difference between diffusive synchronization An oscillator is an open dynamical system whose zero- and kick synchronization is entirely coded in the shape of input steady-state behavior is periodic rather than constant. the coupling function, a phase map defined on the nonlinear Formally, we assume that the zero-input system x˙ = F(x) unit circle. It is typically harmonic in the weak coupling admits a (locally hyperbolic) stable periodic orbit γ with pe- limit of diffusive synchronization and typically monotone riod T (and angular frequency ω = 2π/T ). Picking an initial γ (and hence discontinuous) in the weak coupling limit of kick condition x0 on the periodic orbit γ, this latter is described γ γ synchronization. Again, the synchronization mechanisms and by the (nonconstant) periodic trajectory Φ(t, x0 , 0) = x (t), the contraction measure are distinctively different even in the such that xγ (t) = xγ (t + T ). The basin of attraction of γ weak coupling limit, despite the shared model structure. is the maximal open set (γ) from which the periodic orbit B The paper structure is illustrated in Figure 1. Section II attracts. reviews state-space models of oscillators and their phase re- Since the periodic orbit γ is homeomorphic to the unit duction. The next sections present global stability results for circle S1, it is naturally parametrized by a single scalar phase. the different synchronization models. Section III focuses on Any point p γ is associated with a phase θ S1, such ∈ ∈ (possibly strong) diffusive synchronization while Section IV that focuses (possibly strong) kick synchronization. Section V p = xγ (θ/ω) deals with the phase models encountered in the limit of weak γ coupling. Section VI provides concluding remarks. (where x0 is by convention associated with the phase θ = 0). For hyperbolic periodic orbits, the notion of phase is extended to any point q in the basin of attraction (γ) through the concept of asymptotic phase. The asymptoticB II. OPEN OSCILLATOR MODELS phase map Θ: (γ) S1 assigns to each point q in the basin (γ) its asymptoticB → phase θ S1, such that This section provides a short introduction to oscillators B ∈ viewed as open dynamical systems, that is, as dynamical lim Φ(t, q, 0) Φ(t, xγ (θ/ω), 0) = 0. (2) t→+∞ k − k2 systems that interact with their environment [67]. We first γ recall basic definitions about stable periodic orbits in n- This mapping is constructed such that the image of x0 is dimensional state-space models (see [21], [28] for details). equal to 0 and such that the progression along any orbit We then introduce (finite and infinitesimal) phase response in (γ) (in absence of perturbation) produces a constant B d curves as fundamental mathematical information required for increase in θ, that is, dt Θ(Φ(t, x0, 0)) = ω. the reduction. We finally show how to reduce n-dimensional An isochron is a level set of the asymptotic phase map Θ, state-space models into one-dimensional phase models de- that is, the set of all points in the basin of attraction of γ pending on the nature of the input. characterized by a same asymptotic phase. 7172 7173 7174 7175 7176 A −1 oscillator 1 oscillator 2 x−1.5 −2 0 5 10 t 15 20 25 B 0 5 10 t 15 20 25 Fig.