On the Global Synchronization of Pulse-coupled Oscillators Interacting on Chain and Directed Tree Graphs
OOn the Global Synchronization of Pulse-coupled OscillatorsInteracting on Chain and Directed Tree Graphs (cid:63)
Huan Gao, Yongqiang Wang Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634, United States
Abstract
Driven by increased applications in biological networks and wireless sensor networks, synchronization of pulse-coupled oscillators (PCOs)has gained increased popularity. However, most existing results address the local synchronization of PCOs with initial phases constrainedin a half cycle, and results on global synchronization from any initial condition are very sparse. In this paper, we address global
PCO synchronization from an arbitrary phase distribution under chain or directed tree graphs. Our results differ from existing globalsynchronization studies on decentralized PCO networks in two key aspects: first, our work allows heterogeneous coupling functions, andwe analyze the behavior of oscillators with perturbations on their natural frequencies; secondly, rather than requiring a large enoughcoupling strength, our results hold under any coupling strength between zero and one, which is crucial because a large coupling strengthhas been shown to be detrimental to the robustness of PCO synchronization to disturbances.
Key words:
Global synchronization; pulse-coupled oscillators; hybrid systems.
Pulse-coupled oscillators (PCOs) are limit cycle oscillatorscoupled through exchanging pulses at discrete time instants.They were originally proposed to model the synchronizationphenomena in biological systems, such as contracting car-diac cells, flashing fireflies, and firing neurons [1–3]. Dueto their amazing scalability, simplicity, and robustness, re-cently they have found applications in wireless sensor net-works [4–7], image processing [8], and motion coordination[9].Early results on PCO synchronization were motivated bybiological applications, and normally assume a fixed inter-action or coupling mechanism [1, 2]. In engineering appli-cations, such restrictions do not exist any more. In fact, theinteraction mechanism becomes a design variable that pro-vides opportunities to achieve desired performance. For ex- (cid:63)
The work was supported in part by the National Science Foun-dation under Grant 1738902. (cid:63)(cid:63)
This paper has been accepted to Automatica as a full pa-per. Please cite this article as: H. Gao and Y. Wang, On theglobal synchronization of pulse-coupled oscillators interact-ing on chain and directed tree graphs. Automatica (2019),https://doi.org/10.1016/j.automatica.2019.02.059.
Email addresses: [email protected] (Huan Gao), [email protected] (Yongqiang Wang). Corresponding author. ample, [10] and [11] designed the interaction to improve therobustness to communication delays. Our prior work [12] op-timized the interaction, i.e., phase response function (PRF),to improve the speed of synchronization. However, most ofthese results are for local synchronization assuming that theinitial phases are restricted within a half cycle [3, 11–30].Assuming restricted initial phase distribution severely hin-ders the application of PCO based synchronization, since indistributed systems it is hard to control the initial phase dis-tribution. Recently, efforts have emerged to address globalPCO synchronization from an arbitrary initial phase distri-bution. However, these results focus on special graphs, suchas all-to-all graph [31,32,41,42], cycle graph [33], strongly-rooted graph [32], or master/slave graph [34]. Moreover,they rely on sufficiently large coupling strengths, which maynot be desirable as large coupling strengths are detrimentalto robustness to disturbances [4].In this paper, we address the global synchronization of PCOsunder arbitrary initial conditions and heterogeneous cou-pling functions (PRFs). Our main focus is on the globalsynchronization of PCOs under undirected chain graphs,but the results are easily extendable to PCO synchroniza-tion under directed chain/tree graphs. Note that the chainor directed tree graphs are basic elements for constructingmore complicated graphs and are desirable in engineeringapplications where reducing the number of connections isimportant to save energy consumption and cost in deploy-
Article published in Automatica 8 March 2019 a r X i v : . [ m a t h . D S ] M a r able 1. Comparison our results with other results.Homogeneous coupling Heterogeneous couplingPCO network having(at least) a global node DecentralizedPCO networks PCO network having(at least) a global node DecentralizedPCO networksNon-globalsynchronization Localsynchronization [3, 13–19, 28, 31, 32] [11, 12, 20–28, 32, 33] [34] [29, 30]Almost globalsynchronizationor synchronizationwith probability one [2, 35, 36] [37–39] (cid:31) (cid:31)
Globalsynchronization Discrete statesynchronization [40] [39] (cid:31) (cid:31) (Continuous) phasesynchronization [31, 32, 41, 42] [32, 33, 43 ] [34] This paper A node is called as a global node if it is directly connected to all the other nodes. Note that when the maximum degree of an undirected tree graph is not over , [43] obtained global synchronization results for theconventional phase-only PCO model, though results were also obtained under general undirected tree graphs for a more complicatedPCO model with multiple additional state variables. ment/maintenance. Furthermore, the chain graph has beenregarded as the worst-case scenario for synchronization dueto its minimum number of connections [44]. We also con-sider oscillators with perturbations on their natural frequen-cies. Compared with existing results including our priorwork (cf. Table 1), this paper has the following contribu-tions: 1) Different from most existing results which focuson local PCO synchronization and assume that the initialphases of oscillators are restricted within a half cycle, ourwork addresses global synchronization from an arbitrary ini-tial phase distribution; 2) Different from existing global syn-chronization studies on decentralized PCO networks, ourwork allows heterogeneous phase response functions, andwe analyze the behavior of oscillators with perturbations ontheir natural frequencies. These scenarios, to our knowledge,have not been considered in any existing global synchro-nization results on decentralized PCO networks; 3) In con-trast to existing global PCO synchronization results requir-ing a strong enough coupling strength, our results guaranteeglobal synchronization under any coupling strength betweenzero and one, which is more desirable since a very strongcoupling strength, although can bring fast convergence, hasbeen shown to be detrimental to the robustness of synchro-nization to disturbances [4].It is worth noting that even in the theoretical derivation pointof view, this paper also differs significantly from our priorwork [32–34]: 1) Different from our prior work [32–34]whose proofs are essentially based on local synchronizationanalysis, this work presents a direct global analyzing ap-proach. More specifically, to obtain global synchronizationresults, our prior work [32–34] used strong enough cou-pling strengths to reduce the network to a state where allphases are contained in a half cycle, and then achieved globalsynchronization based on local synchronization analysis. Incomparison, this work studies the systematic evolution ofphases even when they are not restricted in a half cycle, andhence can allow the coupling strength to be any value be-tween zero and one; 2) Although the Lyapunov candidate function seems similar to the one used in our prior work[33], the analysis here is much more complicated due to theconsidered more complicated scenarios (arbitrary couplingstrength between zero and one and heterogeneous PRFs). Infact, to address synchronization under such scenarios, wehad to introduce Invariance Principle, which is not neededin our prior results [32–34] due to their simple dynamicsbrought by strong and homogeneous coupling.The outline of this paper is as follows. Section 2 introducespreliminary concepts. A hybrid model for PCO networksand its dynamical properties are presented in Section 3. InSection 4, we analyze global synchronization on both chainand directed tree graphs and provide robustness analysisunder frequency perturbations. Numerical experiments aregiven in Section 5. Finally, we conclude the paper in Section6. R , R ≥ , and Z ≥ denote real numbers, nonnegative realnumbers, and nonnegative integers, respectively. R n denotesthe Euclidean space of dimension n , and R n × n denotes theset of n × n square matrices with real coefficients. B denotesthe closed unit ball in the Euclidean norm. A set-valuedmap M : A ⇒ B associates an element α ∈ A with a set M ( α ) ⊆ B ; the graph of M is defined as graph( M ) := { ( α, β ) ∈ A × B : β ∈ M ( α ) } . M is outer-semicontinuousif and only if its graph is closed [45]. The range of a function f : R n → R m is denoted as rge f . The closure of set A is denoted as A . The distance of a vector x ∈ R n to aclosed set A ⊂ R n is denoted as | x | A = inf y ∈A | x − y | .The µ -level set of function V : dom V → R is denoted as V − ( µ ) = { x ∈ dom V : V ( x ) = µ } [46].2 .2 Hybrid Systems We use hybrid systems framework with state x ∈ R n [46] H : (cid:26) ˙ x = f ( x ) , x ∈ C x + ∈ G ( x ) , x ∈ D (1)where f , C , G , and D are the flow map , flow set , jumpmap , and jump set , respectively. The hybrid system can berepresented by H = ( C , f, D , G ) . In hybrid system, a hybridtime point ( t, j ) ∈ E is parameterized by both t , the amountof time passed since initiation, and j , the number of jumpsthat have occurred. A subset E ⊂ R ≥ × Z ≥ is a hybridtime domain if it is the union of a finite or infinite sequenceof interval [ t k , t k +1 ] × { k } . A solution to H is a function φ : E → R n where φ satisfies the dynamics of H , E is a hybridtime domain, and for each j ∈ N , the function t (cid:55)→ φ ( t, j ) is locally absolutely continuous on I j = { t : ( t, j ) ∈ E } . φ ( t, j ) is called a hybrid arc . A hybrid arc φ is nontrivial if its domain contains at least two points, is maximal if it isnot the truncation of another solution, and is complete if itsdomain is unbounded. Moreover, a hybrid arc φ is Zeno ifit is complete and sup t dom φ < ∞ , is continuous if it isnontrivial and dom φ ⊂ R ≥ ×{ } , is eventually continuous if J = sup j dom φ < ∞ and dom φ ∩ ( R ≥ × { J } ) containsat least two points, is discrete if it is nontrivial and dom φ ⊂{ } × Z ≥ , and is eventually discrete if T = sup t dom φ < ∞ and dom φ ∩ ( { T } × Z ≥ ) contains at least two points.Given a set M , we denote S H ( M ) the set of all maximalsolutions φ to H with φ (0 , ∈ M .Some notions and results for the hybrid system H from [46]which will be used in this paper are given as follows. Definition 1 H = ( C , f, D , G ) satisfies the hybrid basicconditions if: 1) C and D are closed in R n ; 2) f : R n → R n is continuous and locally bounded on C ⊂ dom f ; and 3) G : R n ⇒ R n is outer-semicontinuous and locally boundedon D ⊂ dom G . Definition 2
A set S ⊂ R n is said to be strongly forwardinvariant if for every φ ∈ S H ( S ) , rge φ ⊂ S . Definition 3
Given a set S ⊂ R n , a hybrid system H on R n is pre-forward complete from S if every φ ∈ S H ( S ) iseither bounded or complete. Definition 4
A compact set
A ⊂ R n is said to be uniformlyattractive from a set S ⊂ R n if every φ ∈ S H ( S ) is boundedand for every ε > there exists τ > such that | φ ( t, j ) | A ≤ ε for every φ ∈ S H ( S ) and ( t, j ) ∈ dom φ with t + j ≥ τ . Definition 5
A compact set
A ⊂ R n is said to be • stable for H if for every ε > there exists δ > suchthat every solution φ to H with | φ (0 , | A ≤ δ satisfies | φ ( t, j ) | A ≤ ε for all ( t, j ) ∈ dom φ ; • locally attractive for H if every maximal solution to H isbounded and complete, and there exists µ > such thatevery solution φ to H with | φ (0 , | A ≤ µ converges to A , i.e., lim t + j →∞ | φ ( t, j ) | A = 0 holds; • locally asymptotically stable for H if it is both stable andlocally attractive for H . Definition 6
Let
A ⊂ R n be locally asymptotically stablefor H . Then the basin of attraction of A , denoted by B A , isthe set of points such that every φ ∈ S H ( B A ) is bounded,complete, and lim t + j →∞ | φ ( t, j ) | A = 0 . Definition 7
Given τ, ε > , two hybrid arcs φ and φ are ( τ, ε ) -close if • ∀ ( t, j ) ∈ dom φ with t + j ≤ τ there exists s such that ( s, j ) ∈ dom φ , | t − s | < ε and | φ ( t, j ) − φ ( s, j ) | < ε ; • ∀ ( t, j ) ∈ dom φ with t + j ≤ τ there exists s such that ( s, j ) ∈ dom φ , | t − s | < ε and | φ ( t, j ) − φ ( s, j ) | < ε . Lemma 1 (Theorem 8.2 in [46]) Consider a continuousfunction V : R n → R , any functions u C , u D : R n → [ −∞ , ∞ ] , and a set U ⊂ R n such that u C ( z ) ≤ , u D ( z ) ≤ for every z ∈ U and such that the growth of V alongsolutions to H is bounded by u C , u D on U . Let a precom-pact solution φ ∗ ∈ S H be such that rge φ ∗ ⊂ U . Then, forsome r ∈ V ( U ) , φ ∗ approaches the nonempty set that is thelargest weakly invariant subset of V − ( r ) ∩ U ∩ (cid:2) u − C (0) ∪ (cid:0) u − D (0) ∩ G ( u − D (0)) (cid:1)(cid:3) . Lemma 2 (Proposition 7.5 in [46]) Let H be nominallywell-posed. Suppose that a compact set A ⊂ R n has thefollowing properties: 1) it is strongly forward invariant, and2) it is uniformly attractive from a neighborhood of itself, i.e.,there exists µ > such that A is uniformly attractive from A + µ B . Then the compact set A is locally asymptoticallystable. Lemma 3 (Proposition 6.34 in [46]) Let H be well-posed.Suppose that H is pre-forward complete from a compact set K ⊂ R n and ρ : R n → R ≥ . Then for every ε > and τ ≥ , there exists δ > with the following property: forevery solution φ δ to H δρ with φ δ (0 , ∈ K + δ B , thereexists a solution φ to H with φ (0 , ∈ K such that φ δ and φ are ( τ, ε ) -close.2.3 Communication Graph We use a graph G = ( V , E , W ) to represent the interactionpattern of PCOs, where the node set V = { , , . . . , N } denotes all oscillators. E ⊆ V × V is the edge set, whoseelements are such that ( i, j ) ∈ E holds if and only if node j can receive messages from node i . We assume that noself edge exists, i.e., ( i, i ) / ∈ E . W = [ w ij ] ∈ R N × N isthe weighted adjacency matrix of G with w ij ≥ , where w ij > if and only if ( i, j ) ∈ E holds. The out-neighborset of node i , which represents the set of nodes that can3 Fig. 1. Illustration of graphs: (a) undirected chain graph with sixnodes; (b) directed chain graph with six nodes; (c) directed treegraph with ten nodes. receive messages from node i , is denoted as N outi := { j ∈V : ( i, j ) ∈ E} .We focus on chain graphs (both undirected and directed)and directed tree graphs which are defined as follows: Definition 8
An undirected chain graph G is a graph whosenodes can be indexed such that there exist two edges ( i, i +1) and ( i +1 , i ) between nodes i and i +1 for i = 1 , , . . . , N − . Definition 9
A directed chain graph G is a graph whosenodes can be indexed such that there is only one edge be-tween nodes i and i +1 for i = 1 , , . . . , N − and all edgesare directed in the same direction. Without loss of general-ity, we suppose that the edge between nodes i and i + 1 is ( i, i + 1) . Definition 10
A directed tree graph G is a cycle-free graphwith a designated node as a root such that the root hasexactly one directed chain to every other node. Examples of undirected chain graph, directed chain graph,and directed tree graph are given in Fig. 1.
We consider N PCOs interacting on a graph G =( V , E , W ) . Each oscillator is characterized by a phase vari-able x i ∈ [0 , π ] for each i ∈ V . Each phase variable x i evolves from to π according to integrate-and-fire dynam-ics, i.e., ˙ x i = ω , where ω ∈ R > is the natural frequency ofthe oscillators. When x i reaches π , oscillator i fires (emitsa pulse) and resets x i to , after which the cycle repeats.When a neighboring oscillator j receives the pulse from os-cillator i , it shifts its phase according to its coupling strength l j ∈ (0 , (a scalar value) and its phase response function(PRF) F j [3, 15, 18, 19, 30, 47], i.e., x + j = x j + l j F j ( x j ) ,where x + j denotes the phase right after phase shift. Due to the hybrid behavior of PCOs similar to [32, 33,48], we model them as a hybrid system H with state x =[ x , . . . , x N ] T . To this end, we define the flow set C and theflow map f ( x ) as follows C = [0 , π ] N , f ( x ) = ω N ∀ x ∈ C (2)According to [32, 33], the jump set D and the jump map G ( x ) can be defined as the union of the individual jump sets D i and individual jump maps G i ( x ) , respectively D := (cid:91) i ∈V D i , G ( x ) := (cid:91) i ∈V : x ∈D i G i ( x ) (3)where D i is defined as D i = { x ∈ C : x i = 2 π } and ∀ x ∈ D i , G i ( x ) is given by G i ( x ) = { x + : x + i = 0 , x + j ∈ x j + w ij F j ( x j ) ∀ j (cid:54) = i } (4)Note w ij = l j ∈ (0 , if j ∈ N outi ; otherwise, w ij = 0 .To make H an accurate description of PCOs, we make thefollowing assumptions on the PRF F j . Assumption 1
The graph of F j for j ∈ V is such that graph( F j ) ⊆ { ( x j , y j ) : x j ∈ [0 , π ] , − x j ≤ y j ≤ π − x j } . This assumption ensures that G ( D ) ⊂ C ∪ D = C since l j ∈ (0 , holds, which avoids the existence of solutionsending in finite time due to jumping outside C . Assumption 2
The PRF F j for j ∈ V is an outer-semicontinuous set-valued map with F j (0) = F j (2 π ) = 0 . The constraint F j (0) = F j (2 π ) = 0 rules out discrete andeventually discrete solutions, meaning that PCOs will notfire continuously without rest [32, 34]. In fact, there are atmost N consecutive jumps with no flow in between becausean incoming pulse cannot trigger an oscillator who just firedto fire again under the constraint F j (0) = F j (2 π ) = 0 .The dynamical properties of H are characterized as follows. Proposition 1
Under Assumptions 1 and 2, we have1) H satisfies the hybrid basic conditions in Definition 1;2) For every initial condition ξ ∈ C ∪ D = C , there existsat least one nontrivial solution to H . In particular, everysolution φ ∈ S H ( C ) is maximal, complete, and non-Zeno;3) For every solution φ ∈ S H ( C ) , sup j dom φ = ∞ holds,which rules out the existence of continuous and eventu-ally continuous solutions. roof : First we prove statement 1). According to the hybridmodel in (2)–(4), C and D are closed, and f is continuousand locally bounded on C . Also G is locally bounded sincethe PRF F j satisfies Assumption 1. To prove G is outer-semicontinuous on D , it suffices to show that graph( G ) = (cid:83) i ∈V { ( x, x + ) : x ∈ D i , x + ∈ G i ( x ) } is closed. Accordingto [32–34], the outer-semicontinuity of F j in Assumption 2ensures that { ( x, x + ) : x ∈ D i , x + ∈ G i ( x ) } is closed for i ∈ V , and hence G is outer-semicontinuous on D . There-fore, H satisfies the hybrid basic conditions in Definition 1.Next we prove statement 2). Since H satisfies the hybridbasic conditions, according to Proposition . in [46], thereexists at least one nontrivial solution to H for every initialcondition ξ ∈ C ∪ D = C , and every solution φ ∈ S H ( C ) iscomplete due to the facts that G ( D ) ⊂ C ∪ D = C holds and C is compact, which also implies that φ is maximal. Since G ( D ) ⊂ C holds, we have rge φ ⊂ C for every φ ∈ S H ( C ) .So, according to Definition 2, C is strongly forward invariant.Since the constraint F j (0) = F j (2 π ) = 0 in Assumption 2rules out complete discrete solutions, from Proposition . in [46] we have that S H ( C ) is uniformly non-Zeno, whichmeans that every φ ∈ S H ( C ) is non-Zeno.Finally we prove statement 3). Since every φ ∈ S H ( C ) iscomplete and the length of each flow interval is at most πω ,we have sup j dom φ = ∞ . So the existence of continuousand eventually continuous solutions is ruled out. (cid:4) Remark 1
As indicated in [33], such hybrid model H isable to handle multiple simultaneous pulses, i.e., if an os-cillator receives multiple pulses simultaneously, it will re-spond to these pulses sequentially (in whatever order), butthe oscillation behavior is the same as if the components of x jumped simultaneously.3.3 General Delay-Advance PRF In this paper, we consider general delay-advance PRFs.
Assumption 3
A delay-advance PRF F j is such that F j ( x j ) = F (1) j ( x j ) , if x j ∈ [0 , π ) (cid:8) F (1) j ( π ) , F (2) j ( π ) (cid:9) , if x j = πF (2) j ( x j ) , if x j ∈ ( π, π ] (5) where F (1) j ( x j ) and F (2) j ( x j ) are continuous functions on [0 , π ] and [ π, π ] , respectively, and satisfy (cid:40) F (1) j (0) = 0 , F (1) j ( x j ) ∈ [ − x j , if x j ∈ (0 , π ] F (2) j (2 π ) = 0 , F (2) j ( x j ) ∈ (0 , π − x j ] if x j ∈ [ π, π ) (6)Similar to [32–34], F j is an outer-semicontinuous set-valuedmap. Note that oscillators with phases in (0 , π ) will be de-layed after receiving a pulse, meaning that their phases will Phase x j π π π π P R F F j ( x j ) - π -0.5 π ππ Phase x j π π π π P R F F j ( x j ) - π -0.5 π ππ Phase x j π π π π P R F F j ( x j ) - π -0.5 π ππ Phase x j π π π π P R F F j ( x j ) - π -0.5 π ππ (a) (b)(c) (d) Fig. 2. Examples of the general delay-advance PRF F j ( x j ) . be pushed closer to zero by each pulse received, whereas os-cillators with phases in ( π, π ) will be advanced, meaningthat their phases will be pushed toward π by each pulse. Ifan oscillator has phase (or π ) upon receiving a pulse, itsphase is unchanged by the pulse.Since Assumption 3 implies Assumptions 1 and 2, the prop-erties of H in Proposition 1 still hold. Several examples ofdelay-advance PRF are illustrated in Fig. 2. Remark 2
It is worth noting that our PRF can be heteroge-neous and is also very general. In fact, it includes the PRFsused in [12, 24–26, 32, 33, 49] as special cases. Therefore,our work has broad potential applications in engineered sys-tems [50] as well as biological systems [47].
In this section, we analyze global PCO synchronization onboth chain and directed tree graphs, and provide robustnessanalysis in the presence of frequency perturbations.To this end, we first define the synchronization set A : A = { x ∈ C : | x i − x j | = 0 or | x i − x j | = 2 π, ∀ i, j ∈ V} (7)The PCO network synchronizes if the state x converges tothe synchronization set A . Note that A is compact since itis closed and bounded (included in C that is bounded).In the following, we refer to an arc as a connected subset of [0 , π ] where and π are associated with each other. Sophase difference ∆ i that measures the length of the shorterarc between x i and x i +1 on the unit cycle is given by ∆ i = min {| x i − x i +1 | , π − | x i − x i +1 |} (8)where x N +1 is mapped to x in ∆ N . It is straightforwardto show that ∆ i satisfies ≤ ∆ i ≤ π .5o measure the degree of synchronization, we define L as L = N (cid:88) i =1 ∆ i (9)Since ≤ ∆ i ≤ π holds, we have ≤ L ≤ N π . Notethat both ∆ i for i ∈ V and L are dependent on x , and L ispositive definite with respect to A on C ∪ D = C because L = 0 holds if and only if ∆ = ∆ = · · · = ∆ N = 0 holds. Therefore, in order to prove synchronization, we onlyneed to show that L will converge to . It is worth notingthat L is continuous in x ∈ C but not differentiable withrespect to it. Lemma 4
For N PCOs interacting on an undirected chain,if the PRF F j ( x j ) satisfies Assumption 3 and l j ∈ (0 , holds for all j ∈ V , then L in (9) is nonincreasing alongany solution φ ∈ S H ( C ) .Proof : Since there is no interaction among oscillators duringflows and all oscillators have the same natural frequency,we have that L is constant during flows and its dynamicsonly depends on jumps. Without loss of generality, we as-sume that at time ( t ∗ i , k ∗ i ) , we have x ( t ∗ i , k ∗ i ) ∈ D i , i.e., x i ( t ∗ i , k ∗ i ) = 2 π . (In the following, we omit time index ( t ∗ i , k ∗ i ) to simplify the notation.) When oscillator i firesand resets its phase to x + i = 0 , an oscillator j ∈ N outi has x + j ∈ x j + l j F j ( x j ) but an oscillator j / ∈ N outi still has x + j = x j .For the undirected chain graph, we call oscillator i − asthe left-neighbor of oscillator i for i = 2 , , . . . , N , andcall oscillator i + 1 as the right-neighbor of oscillator i for i = 1 , , . . . , N − . Upon the firing of oscillator i , if theleft-neighbor oscillator i − exists, it will update its phaseand affect ∆ i − and ∆ i − . Note that for i = 2 , ∆ i − ismapped to ∆ N . Similarly, if the right-neighbor oscillator i + 1 exists, ∆ i and ∆ i +1 will be affected. No other ∆ k swill be affected by this pulse, i.e., ∆ + k = ∆ k holds for k / ∈ { i − , i − , i, i + 1 } where ∆ + k denotes the phasedifference between oscillators k and k + 1 after the jump.Therefore, we only need to consider two situations whenoscillator i fires, i.e., how ∆ i − and ∆ i − change if theleft-neighbor oscillator i − exists and how ∆ i and ∆ i +1 change if the right-neighbor oscillator i + 1 exists. Situation I:
If the left-neighbor oscillator i − exists, from(4) and (5) we have x + i − = (cid:40) x i − + l i − F (1) i − ( x i − ) , if x i − ∈ [0 , π ] x i − + l i − F (2) i − ( x i − ) , if x i − ∈ [ π, π ] (10)To facilitate the proof, we use an nonnegative variable δ i − to denote the jump magnitude of oscillator i − . According (e) (f) (g) (h) (a) (b) (c) (d) i d - i d - i x - p p i d - p p p p p i d - i x - i x - i x i x - i x - i x i x - i x - i x i x - p i x p p i d - p p i d - p p i d - p p i d - i x i x - i x - i x i x - i x - i x i x - i x - i x i x - i x - Fig. 3. Illustration of Situation I. to (6) and l i − ∈ (0 , , δ i − is determined by δ i − = (cid:40) − l i − F (1) i − ( x i − ) , if x i − ∈ [0 , π ] l i − F (2) i − ( x i − ) , if x i − ∈ [ π, π ] (11)Since x i = 2 π and x + i = 0 hold, from (10) and (11) weknow that oscillator i − jumps δ i − towards oscillator i ,as illustrated in Fig. 3. So we have ∆ + i − = ∆ i − − δ i − .Now we analyze how ∆ i − changes upon oscillator i ’s fir-ing. Note that x + i − = x i − holds as i − / ∈ N outi . Accord-ing to the direction of oscillator i − ’s jump and the re-lationship between δ i − and ∆ i − , we have four followingcases: Case 1:
If oscillator i − jumps δ i − towards oscillator i − and δ i − ≤ ∆ i − holds (cf. Fig. 3 (a) and (e)), we have ∆ + i − = ∆ i − − δ i − , which leads to ∆ + i − +∆ + i − = ∆ i − +∆ i − − δ i − ≤ ∆ i − +∆ i − (12)Note that the equality holds if and only if δ i − = 0 exists,i.e., ∆ + i − + ∆ + i − = ∆ i − + ∆ i − holds if and only if ∆ + i − = ∆ i − − δ i − = ∆ i − + δ i − holds. Case 2:
If oscillator i − jumps δ i − towards oscillator i − and δ i − > ∆ i − holds (cf. Fig. 3 (b) and (f)), we have ∆ + i − = δ i − − ∆ i − . So it follows ∆ + i − + ∆ + i − = ∆ i − − ∆ i − ≤ ∆ i − + ∆ i − (13)where the equality occurs when ∆ i − = 0 , i.e., ∆ + i − +∆ + i − = ∆ i − + ∆ i − holds if and only if ∆ + i − = δ i − − ∆ i − = ∆ i − + δ i − holds. Case 3:
If oscillator i − jumps δ i − away from oscillator i − and ∆ i − + δ i − ≤ π holds (cf. Fig. 3 (c) and (g)),we have ∆ + i − = ∆ i − + δ i − , which leads to ∆ + i − + ∆ + i − = ∆ i − + ∆ i − (14)6 ase 4: If oscillator i − jumps δ i − away from oscillator i − and ∆ i − + δ i − > π holds (cf. Fig. 3 (d) and (h)), wehave ∆ + i − = 2 π − ∆ i − − δ i − < π < ∆ i − + δ i − and ∆ + i − + ∆ + i − < (∆ i − − δ i − ) + (∆ i − + δ i − )= ∆ i − + ∆ i − (15)Summarizing the above four cases, we have ∆ + i − + ∆ + i − ≤ ∆ i − + ∆ i − (16)where the equality occurs when ∆ + i − = ∆ i − + δ i − . Situation II:
If the right-neighbor oscillator i + 1 exists, itwill update its phase according to (4) and (5) as follows x + i +1 = (cid:40) x i +1 + l i +1 F (1) i +1 ( x i +1 ) , if x i +1 ∈ [0 , π ] x i +1 + l i +1 F (2) i +1 ( x i +1 ) , if x i +1 ∈ [ π, π ] (17)Also the nonnegative magnitude of oscillator i + 1 ’s phasejump (denoted by δ i +1 ) is given as δ i +1 = (cid:40) − l i +1 F (1) i +1 ( x i +1 ) , if x i +1 ∈ [0 , π ] l i +1 F (2) i +1 ( x i +1 ) , if x i +1 ∈ [ π, π ] (18)Since x i = 2 π and x + i = 0 hold, and oscillator i + 1 jumps δ i +1 towards oscillator i , we have ∆ + i = ∆ i − δ i +1 .According to the relationship between δ i +1 and ∆ i +1 , thereare also four cases on the change of ∆ i +1 . Similar to Situ-ation I, we can obtain the following result ∆ + i + ∆ + i +1 ≤ ∆ i + ∆ i +1 (19)where the equality occurs when ∆ + i +1 = ∆ i +1 + δ i +1 .Summarizing Situation I and Situation II, we can see that L will not increase during jumps. Therefore, L is nonincreas-ing along any solution φ ∈ S H ( C ) . (cid:4) Now we are in position to introduce our results for globalsynchronization on undirected chain graphs.
Theorem 1
For N PCOs interacting on an undirectedchain, if the PRF F j ( x j ) satisfies Assumption 3 and l j ∈ (0 , holds for all j ∈ V , then the synchronizationset A in (7) is globally asymptotically stable, i.e., globalsynchronization can be achieved from an arbitrary initialcondition.Proof : According to the derivation in Lemma 4, the contin-uous function L in (9) is constant during flows and will notincrease during jumps, which implies that L ( g ) − L ( x ) ≤ holds for all x ∈ D and g ∈ G ( x ) . Defining u C ( x ) = 0 for each x ∈ C and u C ( x ) = −∞ otherwise; u D ( x ) =max g ∈ G ( x ) { L ( g ) − L ( x ) } ≤ for each x ∈ D and u D ( x ) = −∞ otherwise, we can bound the growth of L along so-lutions by u C and u D on C [46]. According to Proposi-tion 1, every solution φ ∈ S H ( C ) is precompact, i.e., com-plete and bounded, and satisfies rge φ ⊂ C ∪ D = C . FromLemma 1, for some r ∈ L ( C ) = [0 , N π ] , φ approachesthe nonempty set that is the largest weakly invariant subsetof L − ( r ) ∩ C ∩ (cid:2) u − C (0) ∪ (cid:0) u − D (0) ∩ G ( u − D (0)) (cid:1)(cid:3) where L − ( r ) denotes the r -level set of L defined in Subsection2.1 (note that Lemma 1 does not need L to be continu-ously differentiable in x ∈ C [46]). Since u − C (0) = C and u − D (0) ∩ G ( u − D (0)) ⊂ D hold, we have L − ( r ) ∩ C ∩ (cid:2) u − C (0) ∪ (cid:0) u − D (0) ∩ G ( u − D (0)) (cid:1)(cid:3) = L − ( r ) ∩ C .According to Lemma 5 in Appendix A, L cannot be retainedat any nonzero value along a complete solution φ . So thelargest weakly invariant subset of L − ( r ) ∩ C is empty forevery r ∈ (0 , N π ] , which implies that every solution φ ∈S H ( C ) approaches L − (0) ∩ C = A .Next we show that A is locally asymptotically stable. Sinceevery solution φ ∈ S H ( C ) approaches A , from Definition4, A is uniformly attractive from C . As Assumption 2 guar-antees that rge φ ⊂ A for every φ ∈ S H ( A ) , A is stronglyforward invariant according to Definition 2. Therefore, fromLemma 2, A is locally asymptotically stable.To show A is globally asymptotically stable, it suffices toshow that A ’s basin of attraction B A contains C ∪ D = C .Since we have shown that the largest weakly invariant subsetof L − ( r ) ∩ C is empty for every r ∈ (0 , N π ] and everysolution φ ∈ S H ( C ) approaches A , according to Definition6, A ’s basin of attraction B A contains C . Therefore, A isglobally asymptotically stable.In summary, A is globally asymptotically stable, meaningthat global synchronization can be achieved from an arbi-trary initial condition. (cid:4) Remark 3
Because using four phase differences ( ∆ i − , ∆ i − , ∆ i , and ∆ i +1 , which requires N ≥ ) is essential todescribe and characterize the dynamics of a general numberof N oscillators in a uniform manner, we assumed N ≥ in the proof. However, the results are also applicable to N = 2 and N = 3 . In fact, following the analysis in Lemma4, we can obtain that L is non-increasing when N = 2 or . Then using the Invariance Principle based derivation inTheorem 1 gives the convergence of L to 0 and thus theachievement of global synchronization for N = 2 and . Remark 4
Compared with existing results in [28] whichshow that local synchronization on chain graphs can beobtained as long as the coupling is not too strong, our resultscan guarantee global synchronization under any couplingstrength between zero and one. emark 5 It is worth noting that different from local PCOsynchronization analysis [4, 28] and global PCO synchro-nization analysis under all-to-all topology [32,42] where thefiring order is time-invariant, the coupling strength l ∈ (0 , cannot guarantee invariant firing order in our consideredscenarios, as confirmed by numerical simulations in Fig. 5.4.2 Global Synchronization on Directed Chain and TreeGraphs In this subsection, we extend the global synchronization re-sults to directed chain and tree graphs.
Corollary 1
For N PCOs interacting on a directed chain,if the PRF F j ( x j ) satisfies Assumption 3 and l j ∈ (0 , holds for all j ∈ V , then the synchronization set A in (7) isglobally asymptotically stable, i.e., global synchronizationcan be achieved from an arbitrary initial condition.Proof : The proof is similar to Theorem 1 and omitted. (cid:4) Remark 6
Different from the cycle graph in [33] where astrong enough coupling strength is required, global synchro-nization can be achieved here under any coupling strengthbetween zero and one. This is because in the chain case, theabsence of interaction between oscillators and N allows ∆ N to increase freely until it triggers L to decrease; in otherwords, the absence of interaction between oscillators and N breaks the symmetry of the chain graph [51], which is keyto remove undesired equilibria where L keeps unchanged.In comparison, the symmetry of the cycle graph can make L stay at some undesired equilibria under a weak couplingstrength. So a strong enough coupling strength is requiredin the cycle graph case to achieve global synchronization. Corollary 2
For N PCOs interacting on a directed tree, ifthe PRF F j ( x j ) satisfies Assumption 3 and l j ∈ (0 , holdsfor all j ∈ V , then global synchronization can be achievedfrom an arbitrary initial condition.Proof : Suppose in a directed tree graph there are m nodes without any out-neighbors which are represented as v , v , . . . , v m . Take the graph in Fig. 1 (c) as an example,nodes , , , and do not have any out-neighbors. Ac-cording to Definition 10, for every node v i ( i = 1 , , . . . , m )there is a unique directed chain from the root v r to node v i .So the directed tree graph is composed of m directed chains.Note that for every directed chain from the root v r to node v i , it is not affected by oscillators outside the chain. So the m directed chains are decoupled from each other. Accord-ing to Corollary 1, global synchronization can be achievedon the directed chain from an arbitrary initial conditionif F j ( x j ) satisfies Assumption 3 and if l j ∈ (0 , holds.Adding the fact that the root oscillator v r belongs to all m directed chains implies synchronization of all PCOs. (cid:4) Remark 7
Different from the arguments in the proofs ofCorollaries 1 and 2, an alternative approach to proving global synchronization on direct chain (and tree) graphsis using inductive reasoning based on the following twofacts: first, a parent node can affect its child node but achild node never affects its parent node; secondly, under thegiven piecewise continuous delay-advance PRF (with valuesbeing nonzero in (0 , π ) ), the phases of all oscillators on adirected chain will be reduced to within a half cycle, whichalways leads to synchronization (cf. Theorem 2 in [29]). Remark 8
Different from the “probability-one synchroniza-tion” in [37–39] where oscillators synchronize with proba-bility one under a stochastic phase-responding mechanismand the “almost global synchronization” in [2,35,36] wheresynchronization is guaranteed for all initial conditions ex-cept a set of Lebesgue-measure zero, our studied global syn-chronization is achieved in a deterministic manner from anyinitial condition, which is not only important theoreticallybut also mandatory in many safety-critical applications. Atypical application justifying the necessity of deterministicglobal synchronization is synchronization based motion co-ordination of AUV (autonomous underwater vehicles) [52]and UAV (unmanned aerial vehicles) [53]. In such an ap-plication, even one single failure in synchronization mightbe too costly in money, time, energy, or even lives (cf. themulti-UAV based target engagement problem in [54]).4.3 Robustness Analysis for Frequency Perturbations
In this subsection, we analyze the robustness property ofPCOs under small frequency perturbations on the naturalfrequency ω . It is worth noting that robustness is importantsince frequency perturbations are unavoidable and under aninappropriate synchronization mechanism, even a small dif-ference in natural frequency may accumulate and lead tolarge phase differences. The hybrid systems model with fre-quency perturbations is given as follows: H p : (cid:26) ˙ x = ω N + p, x ∈ C x + ∈ G ( x ) , x ∈ D (20)where p = [ p , . . . , p N ] T represents the frequency pertur-bations. Using the notion of ( τ, ε ) -closeness given in Defi-nition 7 in Subsection 2.2, we have the following result: Theorem 2
Consider N PCOs with frequency perturba-tions as described by H p in (20). For every ε > , τ ≥ ,and ρ : R N → R ≥ , there exists a scalar σ > such thatunder any p ∈ σρ ( x ) B every solution φ p to H p from C is ( τ, ε ) -close to a solution φ to the perturbation-free dynam-ics H .Proof : According to Proposition 1 in Subsection 3.2, H satis-fies the hybrid basic conditions, and is pre-forward completefrom the compact set C since every φ ∈ S H ( C ) is complete(see Definition 3). So from Lemma 3, for every ε > , τ ≥ ,and ρ : R N → R ≥ , there exists a scalar σ > with the fol-lowing property: for every solution φ σ to H σρ from C , there8xists a solution φ to H from C such that φ σ and φ are ( τ, ε ) -close, where H σρ = ( C , f σρ , D , G ) is the σρ -perturbationof H and f σρ ( x ) = f ( x ) + σρ ( x ) B = ω N + σρ ( x ) B forevery x ∈ C . Note that if p ∈ σρ ( x ) B , every solution φ p to H p from C is in fact the solution to H σρ , which implies that φ p and φ are ( τ, ε ) -close. (cid:4) According to Theorem 2, the behavior of perturbed PCOs isclose to the perturbation-free case, i.e., the solutions to theperturbed PCOs converge to the neighborhood of the syn-chronization set A . Therefore, the phases of oscillators willremain close to each other under small frequency perturba-tions. We first considered the unperturbed case, i.e., all oscillatorshad an identical frequency ω = 2 π .First we considered N = 6 PCOs on an undirected chaingraph. Oscillators , . . . , adopted the PRFs (a), (b), (c), (d),(a), and (b) in Fig. 2, respectively. The respective analyticalexpressions of these PRFs are given below. ( a ) : F j ( x j ) = − . x j , if x j ∈ [0 , π ) (cid:8) − . π, . π (cid:9) , if x j = π . π − x j ) , if x j ∈ ( π, π ] (21) ( b ) : F j ( x j ) = − . x j , if x j ∈ [0 , π − . π, if x j ∈ [ π , π ) (cid:8) − . π, . π (cid:9) , if x j = π . π, if x j ∈ ( π, π . π − x j ) , if x j ∈ ( 3 π , π ] (22) ( c ) : F j ( x j ) = − . . x j ) , if x j ∈ [0 , π ) (cid:8) − . , . (cid:9) , if x j = π . . x j ) , if x j ∈ ( π, π ] (23) ( d ) : F j ( x j ) = − x j /π + x j /π − . x j , if x j ∈ [0 , π ) (cid:8) − . π, . π (cid:9) , if x j = π − x j /π + 5 x j /π − . x j + 5 . π, if x j ∈ ( π, π ] (24) The coupling strength l , . . . , l were set to . , . , . , . , . , and . , respectively. The initial phase x (0 , wasrandomly chosen from C ∪ D . Fig. 4 shows the evolutionsof phases and L . It can be seen that L converged to , whichconfirmed Theorem 1. Time [s] x i [ r a d ] Time [s] L a nd V c [ r a d ] L Contianing arc V c Fig. 4. Evolutions of phases and L for PCOs on an undirectedchain graph. Firing event index
PCO 5PCO 1PCO 3PCO 6PCO 2PCO 4
Fig. 5. Firing order of PCOs on the undirected chain graph.
Time [s] x i [ r a d ] Time [s] L , L , L , a nd L [ r a d ] L L L L Fig. 6. Evolutions of phases, L , L , L , and L for PCOs on adirected tree graph. PCOs synchronized as L , L , L , and L converged to . From the lower plot of Fig. 4, we can also see that the lengthof the shortest containing arc V c , which is widely used as aLyapunov function in local synchronization analysis [24,29,32, 34], is not appropriate for global PCO synchronizationas it may not decrease monotonically. Along the same line,the firing order which is invariant in [4, 28, 42], and [32], isnot constant in the considered dynamics as exemplified in9 ime [s] x i [ r a d ] Time [s] L [ r a d ] L Fig. 7. Evolutions of phases and L for PCOs on an undirectedchain graph under frequency perturbations. Fig. 5. These unique properties of chain and directed treePCOs corroborate the novelty and importance of our results.Then we considered N = 10 PCOs on a directed tree graph,as illustrated in Fig. 1 (c). There are directed chains inthis graph, namely, oscillators → → , oscillators → → → , oscillators → → → , and oscil-lators → → → → . The same as (9), L , L , L , and L were defined to measure the degree of synchro-nization corresponding to the directed chains, respectively.Oscillators , . . . , adopted the PRFs (a), (b), (c), (d), (a),(b), (c), (d), (a), and (b) in Fig. 2, respectively. The couplingstrength l , . . . , l were set to . , . , . , . , . , . , . , . , . , and . , respectively. The initial phase x (0 , was randomly chosen from C ∪ D . The convergence of L i ( i = 1 , . . . , ) to zero in Fig. 6 implies the synchroniza-tion of the i th directed chain, which confirmed Corollary 1.The simultaneous synchronization of all four directed chainsalso means synchronization of the entire directed tree graph,which confirmed Corollary 2. We considered N = 6 PCOs on an undirected chain graphwith frequency perturbations on oscillator k set to p k =0 . πt + 2 πk/N ) . The other settings were the sameas the undirected chain case. The evolutions of phases and L were shown in Fig. 7. It can be seen that the perturbedbehaviors did not differ too much from the unperturbed casein Fig. 4, and the solution converged to a neighborhood ofthe synchronization set A as L approached a ball containingzero, which confirmed Theorem 2. The global synchronization of PCOs interacting on chainand directed tree graphs was addressed. It was proven thatPCOs can be synchronized from an arbitrary initial phasedistribution under heterogeneous phase response functionsand coupling strengths. The results are also applicable when oscillators are heterogeneous and subject to time-varyingperturbations on their natural frequencies. Note that differ-ent from existing global synchronization results, the cou-pling strengths in our results can be freely chosen betweenzero and one, which is desirable since a very strong cou-pling strength, although can bring fast convergence, has beenshown to be detrimental to the robustness of synchroniza-tion to disturbances. Given that a very weak coupling maynot be desirable either due to low convergence speed whichmay allow disturbances to accumulate, the results give flex-ibility in meeting versatile requirements in practical PCOapplications.
Acknowledgements
The authors would like to thank Francesco Ferrante for dis-cussions and feedback which greatly strengthened the paper.
Appendix A: Lemma 5Lemma 5
For N PCOs interacting on an undirected chain,if the PRF F j ( x j ) satisfies Assumption 3 and l j ∈ (0 , holds for all j ∈ V , then L in (9) cannot be retained at anynonzero value along a complete solution φ .Proof : We use proof of contradiction. Since L ∈ [0 , N π ] holds, we suppose that for some r ∈ (0 , N π ] , L is retainedat r along a complete solution φ . From Lemma 4, to keep L at r , we must have ∆ + i − = ∆ i − − δ i − , ∆ + i − = ∆ i − + δ i − (25)or ∆ + i = ∆ i − δ i +1 , ∆ + i +1 = ∆ i +1 + δ i +1 (26)if the left-neighbor oscillator i − or right-neighbor oscillator i + 1 exists when oscillator i fires, respectively. Next weshow that ∆ N will exceed π , which contradicts the constraint ≤ ∆ i ≤ π for i ∈ V .Given / ∈ N outi and N / ∈ N outi for i = 3 , , . . . , N − , both x +1 = x and x + N = x N hold when oscillators , , . . . , N − fire, which leads to ∆ + N = ∆ N . Similarly, N / ∈ N out (resp. / ∈ N outN ) implies x + N = x N (resp. x +1 = x ) whenoscillator (resp. N ) fires, which leads to ∆ + N = ∆ N whenoscillator or N fires.So we focus on the evolution of ∆ N when oscillators and N − fire. According to Lemma 6 in Appendix B, neitheroscillator nor oscillator N − will stop firing. Withoutloss of generality, we assume that oscillator fires at time ( t ∗ , k ∗ ) . From (25) we have ∆ + N = ∆ N + δ . Similarly,from (26) we have ∆ + N = ∆ N + δ N when oscillator N − fires. Since δ and δ N are nonnegative, we have ∆ + N ≥ ∆ N .To prove that ∆ N will surpass π , we need to show that atleast one of the following statements is true:10 .... ... p x x q x q x + Fig. 8. Illustration of a set of q ≥ neighboring oscillators beingsynchronized. δ = 0 cannot always hold when oscillator fires;2) δ N = 0 cannot always hold when oscillator N − fires.Proof of statement 1): Given l ∈ (0 , , according to (6) and(11), δ = 0 holds if and only if x = 0 or x = 2 π holds,which means that oscillators and are synchronized whenoscillator fires. So we need to show that oscillators and cannot always be synchronized when oscillator fires. Moregenerally, we assume that there is a set of q ≥ oscillators , , . . . , q being synchronized and having phases differentfrom oscillator q + 1 . According to Lemma 6 in AppendixB, oscillator q + 1 will not stop firing in this situation. Weassume that oscillator q + 1 fires at time ( t ∗ q +1 , k ∗ q +1 ) , and x = . . . = x q ∈ [ π, π ) holds when oscillator q + 1 fires,as illustrated in Fig. 8. Note that the case of x = . . . = x q ∈ (0 , π ] can be proved by following the same line of reasoning.Given < l q < , from (6) and (11) we have < δ q < π − x q . Since oscillator q is the left-neighbor of oscillator q + 1 , according to (25), when oscillator q + 1 fires we have ∆ + q = ∆ q − δ q = 2 π − x q − δ q > and ∆ + q − = ∆ q − + δ q = 0 + δ q > . So oscillator q escapes from the set ofsynchronized oscillators due to ∆ + q − > and will fire next.Similarly, when oscillator q fires, the left-neighbor oscillator q − will escape from the set of synchronized oscillatorsand fire next. Iterating this argument, when oscillator fires,the left-neighbor oscillator will escape from the set ofsynchronized oscillators and fire next. So we have x (cid:54) = x ,i.e., oscillators and are not synchronized when oscillator fires. Therefore, δ = 0 cannot always hold when oscillator fires.Similarly, we can prove statement 2), i.e., δ N cannot alwaysbe when oscillator N − fires, and thus ∆ N will keepincreasing. Since δ and δ N will not converge to unlesssynchronization is achieved, ∆ N will surpass π , which con-tradicts the constraint ≤ ∆ i ≤ π for i ∈ V . Therefore, L cannot be retained at any nonzero value along a completesolution φ . (cid:4) Appendix B: Lemma 6Lemma 6
For N PCOs interacting on an undirected chain,if the PRF F j ( x j ) satisfies Assumption 3 and l j ∈ (0 , holds for all j ∈ V , we have the following results:1) Neither oscillator nor oscillator N − will stop firing; 2) Oscillator q + 1 will not stop firing if oscillators , . . . , q ( ≤ q ≤ N − ) have been synchronized and oscillator q + 1 is not synchronized with these q oscillators. Simi-larly, oscillator N − q will not stop firing if oscillators N − q + 1 , . . . , N have been synchronized and oscillator N − q is not synchronized with these q oscillators.Proof : We first use proof of contradiction to prove statement1). Suppose that oscillator stops firing after time instant ( t (cid:48) , k (cid:48) ) , then x will stay in [0 , π ] . This is because if x ∈ ( π, π ) holds, it will evolve continuously to π and fire, andreceiving pulses from other oscillators can only expedite thisprocess under the PRFs in Assumption 3. Since oscillator only receives pulses from oscillators and , without loss ofgenerality, we suppose at time ( t (cid:48) , k (cid:48) ) that oscillator firesand resets its phase to . Note that oscillator will fire ata period of T = 2 π/ω since its only neighbor oscillator stops firing. After receiving the pulse, oscillator updates itsphase to x +2 = x + l F (1)2 ( x ) ∈ [0 , π ) . If oscillator doesnot receive any other pulse before its phase surpasses π , itwill fire, which contradicts the assumption. So we supposethat oscillator fires at time ( t (cid:48) , k (cid:48) ) before x surpasses π ,which implies t (cid:48) − t (cid:48) ≤ π/ω . Since the time it takes forphase evolving from to π is at least π/ω and after reaching π oscillator will not fire immediately even if it receives apulse under given PRFs and coupling strengths, the length ofoscillator ’s firing period T satisfies T > π/ω . There aretwo cases in this situation, t (cid:48) = t (cid:48) and t (cid:48) < t (cid:48) , respectively:Case 1: If t (cid:48) = t (cid:48) holds, then the length of time intervalfor oscillator receiving the next pulse after ( t (cid:48) , k (cid:48) + 1) isgreater than π/ω . Since x ( t (cid:48) , k (cid:48) +1) ≥ holds, x will begreater than π when receiving the next pulse. So oscillator will fire again, which contradicts the assumption.Case 2: If t (cid:48) < t (cid:48) holds, then we have x ( t (cid:48) , k (cid:48) + 1) > due to x ( t (cid:48) , k (cid:48) ) = x ( t (cid:48) , k (cid:48) + 1) + ω ( t (cid:48) − t (cid:48) ) > undergiven PRFs and coupling strengths. Since t (cid:48) − t (cid:48) ≤ π/ω holds, after time interval [ π − x ( t (cid:48) , k (cid:48) + 1)] /ω which isless than π/ω , we have x < π , x < π , and x = π .So x will be greater than π when receiving the next pulse,and thus oscillator will fire again, which contradicts theassumption.Therefore, oscillator will not stop firing. Similarly, we canprove that oscillator N − will not stop firing either.Next we prove statement 2). Suppose that oscillator q + 1 stops firing after time ( t (cid:48) q +1 , k (cid:48) q +1 ) . Since oscillators , . . . , q will not receive any pulses from other oscillators,they will remain synchronized and oscillator q will fire witha period of T q = 2 π/ω . The same as statement 1), the lengthof oscillator q + 2 ’s firing period T q +2 satisfies T q +2 > π/ω and oscillator q + 1 will not stop firing if oscillator q + 1 has a phase different from synchronized oscillators , . . . , q .Similarly, we can prove that oscillator N − q will not stopfiring either if oscillator N − q has a phase different fromsynchronized oscillators N − q + 1 , . . . , N . (cid:4) eferences [1] C. S. Peskin. Mathematical aspects of heart physiology . CourantInstitute of Mathematical Science, New York University, 1975.[2] R. Mirollo and S. Strogatz. Synchronization of pulse-coupledbiological oscillators.
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