Nonlinear Synchronization on Connected Undirected Networks
aa r X i v : . [ m a t h . D S ] O c t Nonlinear Synchronization on ConnectedUndirected Networks
S. Orange ∗ and N. Verdi`ere ∗ October 8, 2018
Abstract
This paper gives sufficient conditions for having complete synchroniza-tion of oscillators in connected undirected networks. The considered oscil-lators are not necessarily identical and the synchronization terms can benonlinear. An important problem about oscillators networks is to deter-mine conditions for having complete synchronization that is the stabilityof the synchronous state. The synchronization study requires to take intoaccount the graph topology. In this paper, we extend some results tonon linear cases and we give an existence condition of trajectories. Suffi-cient conditions given in this paper are based on the study of a Lyapunovfunction and the use of a pseudometric which enables us to link networkdynamics and graph theory. Applications of these results are presented.
AMS Subject Classification 2010: 93D20, 93D30, 68R10 . Keywords: Nonlinear systems, Synchronization, Networks, Graph topology, Dy-namical Systems
The study of the dynamics of coupled nonlinear dynamical systems are thesubject of a growing interest in various communities like in theoretical physic,in information technology or in neuronal biology. The literature on this topicshows different kinds of synchronization (see [10]). Classically, two coupledlimit-cycle are said synchronized when their time evolution is periodic with thesame period and perhaps the same phase. From the discover of synchronizationof chaotic systems (see [1, 5, 8]), the word synchronization recovered differentmeanings such as having identical or functional related solutions, eventually witha delay. The definition has also been modulated by considering strong formslike complete, cluster form or weaker forms like phase and lag synchronization(see [11]).An important question about synchronization of a network of oscillators isto determine the stability of the synchronisation state. This question leads toconsider some properties of networks and state vectors of oscillators (see, forexample, [4, 13, 14, 15, 17]). For this purpose, two methods are proposed in ∗ LMAH (Laboratoire de Math´ematiques Appliqu´ees du Havre), Universit´e du Havre, 25rue Philippe Lebon, BP 540, 76058 Le Havre, France.
[email protected],[email protected] master stability function is based on thecomputation of a Lyapunov exponent and the eigenvalues of the connectivitymatrix [9]. However, this method is adapted when the coupling terms are lin-ear and the computation of eigenvalues can become a difficult task. A secondproposed method is the connection graph stability method (see [4]). It links thestudy of a Lyapunov function and the graph topology. This productive methodhas been extended to unbalance and undirected graph (see [2, 3]).The results presented in this paper generalize some results of [4] to the nonlinear synchronization case. For this, we introduce a notion of pseudometric inthe graph. The determination of the sign of the Lyapunov function derivativerequires two steps. The first one is to use assumptions allowing comparisonsbetween oscillators and synchronization terms. The second step consists inusing pseudometrics which enable us to use some graph properties. For thecomplete synchronization, we present two results. The first one gives a conditionon synchronization strength for having a global synchronization of oscillators.The second result is a local versus of the first one, that is when the oscillatorsare closed to the synchronization variety. In these two cases, we give sufficientconditions that insure existence of trajectories.This paper is organized as follows. The problem statements are presented inSection 2. First, we precise the kind of systems and the kind of synchronizationsconsidered. Then, we recall the definition and some properties of pseudomet-rics defined on a graph. In Section 3, after precising the assumptions on thesynchronization term, main results, that is conditions for having complete syn-chronization of the system of oscillators, are presented. These results are appliedin Section 4.
Thereafter, Y T is the transpose of the vector Y = ( Y , . . . , Y m ) ∈ R m . Let G be a connected undirected graph and n its number of vertex. The graph G describes the set of interactions between the oscillators. We denote by E theset of its edges. If G contains an undirected edge from a vertex i to a vertex j ,we denote it by ( i, j ).The considered dynamical systems are defined by the following system ofequations: ˙ X = F ( X , t ) − ǫ X (1 ,j ) ∈E h ( X , X j ) , ...˙ X n = F n ( X n , t ) − ǫ X ( n,j ) ∈E h ( X n , X j ) , (1)where • X i = ( X i , . . . , X di ) T is the vector composed of the d coordinates of the i -th oscillator, • F i = ( F i , . . . , F di ) T is the vectorial function defining one oscillator,2 h = ( h , . . . , h d ) T is the synchronization function which defines the vectorcoupling between oscillators, • the real parameter ǫ corresponds to the synchronization strengthRecall that, for a given initial state of the set of oscillators ( X (0) , X (0) , · · · X n (0)) T , system (1) synchronizes completely if, for all ( i, j ) ∈ 〚 , n 〛 , k X i ( t ) − X j ( t ) k −−−−→ t → + ∞ . This means that the vector ( X , . . . , X n ) approaches the synchronization man-ifold defined by X ( t ) = X ( t ) = · · · = X n ( t ). In particular, this implies thatthe oscillators have the same asymptotic behavior (such as chaotic trajectories,stable and periodic solutions). The complete synchronization of all oscillatorscan occur whatever their initial states are, in this case, the synchronization issaid global; otherwise it is said local.In this paper, we focus naturally on the differences ∆ i,j = X Ti − X Tj andtherefore on the vector∆ = (∆ , , · · · , ∆ ,n , ∆ , , · · · , ∆ ,n , · · · , ∆ n − ,n ) T . Thus, proving the complete synchronization of system (1) is equivalent to provethat k ∆( t ) k −−−−→ t → + ∞ . In the following, we consider pseudometric verifying the ρ -relaxed triangle in-equality for a positive real ρ , that is an application ϕ : D × D → R + , where D is an non empty set, satisfying the following three axioms: • ϕ ( z , z ) = 0; • ϕ ( z , z ) = ϕ ( z , z ) (symmetry property); • ϕ ( z , z ) ≤ ρ ( ϕ ( z , z ) + ϕ ( z , z )) ( ρ -relaxed triangle inequality).Remark that any classical metric is such a pseudometric with ρ = 1.Let ϕ be a pseudometric on a set D . Let’s set, for all m ∈ N ∗ , ρ ( m ) thesmallest real such that ϕ ( z , z m +1 ) ≤ ρ ( m ) [ ϕ ( z , z ) + · · · + ϕ ( z m , z m +1 )] . (2)Note that ρ (1) = 1.In the following examples, expressions of ρ ( m ) appearing in inequalities (2)are direct consequences of the convexity of functions x → ( x ) α and x → x e −| x | . Example 2.1.
1. The application ϕ α : R × R → R + defined by ϕ α (cid:18)(cid:18) x y (cid:19) , (cid:18) x y (cid:19)(cid:19) = (cid:0) ( x − x ) (cid:1) α with α ≥ / is a pseudometric for which ρ ( m ) = m α − . . Let D be the closed ball of center and radius − √ . The application ϕ : D × D → R + defined by ϕ x y z , x y z = ( x − x ) e −| x − x | is a pseudometric for which ρ ( m ) = m . We have the following properties.
Proposition 2.1.
1. The sequence of reals ( ρ ( m )) m ≥ is increasing.2. For all m ∈ N ∗ , we have ρ ( m ) ≤ ρ m − (see [16]).3. Let ϕ and ϕ be two pseudometrics on D and ρ ( m ) and ρ ( m ) bethe smallest respective reals verifying (2). For all α > and β > ,the application α ϕ + β ϕ is a pseudometric on D satisfying ρ ( m ) = M ax { ρ ( m ) , ρ ( m ) } . We now apply pseudometrics to networks of oscillators. Recall that a statevector z i of an oscillator is associated to i -th vertex of G . Let’s consider apseudometric ϕ defined on the set of state vectors of oscillators. This pseu-dometric enables one to define the pseudolength ϕ ( z i , z j ) between vertices i and j and also the pseudolength ϕ ( z i , z i ) + · · · + ϕ ( z i m − , z i m ) of any path P i,j = ( i = i , i , · · · , i m = j ) from vertex i to vertex j .In the following proposition, we bound, up to a multiplicative constant C ( G ),the sum of pseudolengths between any two oscillators by the sum of pseu-dolengths of paths joining any two oscillators. This constant plays an importantrole in Theorems 3.1 and 3.2 since the synchronization strenght ǫ appearing inthese theorems is proportionnal to this constant. Proposition 2.2.
Let G be a connected graph, E be the set of its edges and ϕ bea pseudometric on a set D . For any vertex i , let z i ∈ D be a vector associatedto vertex i . There exists a constant C depending only on G so that we have X i,j ϕ ( z i , z j ) ≤ C X ( i,j ) ∈E ϕ ( z i , z j ) . (3) Moreover, the smallest real C satisfying (3), C ( G ) , is bounded by n ( n − δ ( G ) ρ ( δ ( G )) , (4) where δ ( G ) is the diameter of G .Proof. Let i and j be two vertices of G and let’s denote P i,j = ( i = i , i , · · · , i s +1 = j )a path of G from the vertex i to vertex j (recall that G is connected). Since ϕ is a pseudometric on D , we have ϕ ( z i , z j ) ≤ ρ ( s ) P sℓ =1 ϕ ( z i ℓ , z i ℓ +1 ) . The path P i,j can be chosen so that s ≤ δ ( G ). Suppose that this choice isdone for any vertices i and j ; since the sequence ( ρ ( n )) n ∈ N ∗ is increasing, wehave ρ ( s ) ≤ ρ ( δ ( G )). Consequently, for any vertices i and j , we have ϕ ( z i , z j ) ≤ ρ ( δ ( G )) δ ( G ) M ax ( { ϕ ( z i , z j ) | ( i, j ) ∈ E} ) which implies the result.4n Theorem 3.1, we need to determine the lowest bound C ( G ) of the set ofreals C satisfying inequality (3). The bound (4) of C ( G ) may not lead to a goodestimation of C ( G ) for a particular graph; nevertheless, this bound is valid forany graph with n vertices.In the case of a pseudometric satisfying the classical triangle inequality, i.e.when ρ ( n ) = n for all n ∈ N ∗ , a method taking G as input and returning abound of C ( G ) is proposed in [3]. Its two main steps are:1. for all ( i, j ) with i > j , choose a path P i,j ; this path is usually chosen withminimal length (number of edges in the path);2. for each edge e of the connection graph, determine the sum B ( e ) of thelengths of all chosen paths P i,j containing e . A bound for C ( G ) is then M ax { B ( e ) : e ∈ E} .For each choice of paths, these two steps return a bound for C ( G ). Clearly,the number of possible paths is huge but computations of bounds for C ( G ) arepossible since most of these choices are suboptimal. Up to a slight modificationof the first step, this method can be applied here: its consists in considering,for all path P i,j , the pseudolength ρ ( | P i,j | ) instead of its length | P i,j | . Remark 2.1.
In the case of pseudometrics ϕ satisfying ρ ( m ) = m , explicitbounds of C ( G ) for specific graphs and the method proposed in [4, 3] for com-puting C ( G ) from G can be directly used. This is the case of the second functionin Example 2.1. Afterwards, two cases are considered. The first one is the global completesynchronization for which oscillators X , . . . , X n lies in D = R d . The secondone is the complete synchronization for which oscillators are in a neighborhood D of the variety X = X = · · · = X n .Thereafter, we will suppose the following assumptions on system (1). • For all ( i, j ) ∈ E , there exist some non negative reals a , . . . , a d such that ∀ ( X i , X j ) ∈ D, ϕ ( X i , X j ) = d X k =1 a k ( X ki − X kj ) h k ( X i , X j ) (5)are pseudometrics where h = ( h , . . . , h d ) T is the synchronization func-tion. • For all ( i, j ) ∈ 〚 , n 〛 and, for all t ≥ t where t ∈ R , ∀ ( X i , X j ) ∈ D, d X k =1 a k ( X ki − X kj ) (cid:0) F ki ( X i , t ) − F kj ( X j , t ) (cid:1) ≤ ϕ ( X i , X j ) . (6)5 For all ( i, j ) ∈ 〚 , n 〛 , ∀ ( X i , X j ) ∈ D, ϕ ( X i , X j ) = 0 and/or P dk =1 a k ( X ki − X kj ) (cid:0) F ki ( X i , t ) − F kj ( X j , t ) (cid:1) = 0 ⇒ ( X i = X j ) . (7) Remark 3.1.
1. Notice that hypothesis (5) implies that, ∀ ( i, j ) ∈ E , ∀ ( X i , X j ) ∈ D, h ( X i , X j ) = − h ( X j , X i ) (antisymmetry) . (8)
2. The assumption (7) is necessary for proving the complete synchronisationof system (1) in Theorems 3.1 and 3.2. The condition ϕ ( X i , X j ) = 0 inthis assumption is not always sufficient when it does not imply equalitiesof all the components of oscillators. In this case, the second condition isnecessary for proving the complete synchronization. For practical cases, a first problem is to prove the existence of trajectories ofsystem (1) for a sufficient large t . For this goal, the following proposition enablesus to link existence of trajectories between synchronized and non synchronizedsystems. Proposition 3.1.
For all ( i, j ) ∈ 〚 , n 〛 , suppose that assumptions (5), (6)and (7) are satisfied and that, for all t ≥ t , X Ti F i ( X i , t ) ≤ Ψ( || X i || ) where Ψ satifies the conditions Z + ∞ s = s ds Ψ( t ) = + ∞ and Ψ( s ) > for all s ≥ s ≥ .Then, the Cauchy’s problem defined by system (1) and an initial condition X ( t ) ... X n ( t ) ∈ R nd has a solution on the complete semi-axis [ t ; + ∞ ) .Proof. Let’s set X = X ... X n ∈ R nd and F ( X, t ) = F ( X , t )... F n ( X n , t ) ∈ R nd . Ina first step, we prove that there exists a real β such that the following inequalitybetween the scalar products holds: X T ˙ X ≤ βX T F ( X, t ) . (9)For this, we consider the dn × dn diagonal matrix M = Diag ( a , . . . a d , . . . , a , . . . a d ) . We have: X T M ˙ X = n X i =1 d X k =1 a k X ki F ki ( X i , t ) − ǫ n X i =1 d X k =1 a k X { j | ( i,j ) ∈E} X ki h k ( X i , X j )= X T M F ( X, t ) − ǫ d X k =1 X ( i,j ) ∈E a k X ki h k ( X i , X j )6nd, since to any edge ( i, j ) ∈ E corresponds the edge ( j, i ) ∈ E , we obtain X T M ˙ X = X T M F ( X, t ) − ǫ d X k =1 a k X ( i,j ) ∈E X ki h k ( X i , X j ) + X kj h k ( X j , X i )= X T M F ( X, t ) − ǫ d X k =1 a k X ( i,j ) ∈E ( X ki − X kj ) h k ( X i , X j ) (see equality (8))= X T M F ( X, t ) − ǫ X ( i,j ) ∈E ϕ ( X i , X j ) ≤ X T M F ( X, t ) . (see assumption (5))Inequality (9) is then a direct consequence of the fact that the reals a i are nonnegative.If the conditions of the proposition are verified, inequality (9) shows that wehave, for all t ≥ t , X T ˙ X ≤ e Ψ( || X || )where e Ψ is a application satifying the conditions Z + ∞ s = s ds e Ψ( t ) = + ∞ and e Ψ( s ) > s ≥ s ≥ t ≥ t . Theorem 3.1.
Suppose that the assumptions done in Section 3.1 are satisfiedfor D = ( R d ) . If ǫ > C G n , where C G is the optimal bound such that inequa-lity (3) holds, then system (1) synchronizes completely.Proof. In order to show this result, we will apply the second method of Lya-punov. Let’s consider the Lyapunov candidate function: V = 12 d X k =1 X i ≤ j a k ( X ki − X kj ) . Clearly, this function is non negative if ∆ = −→ −→ V gives:˙ V = d X k =1 a k n X i =1 ∂V∂X ki ˙ X ki = d X k =1 a k n X i =1 ( nX ki − n X j =1 X kj ) ˙ X ki = d X k =1 a k n n X i =1 X ki ˙ X ki − n X j =1 X kj n X i =1 ˙ X ki = d X k =1 a k n n X i =1 X ki F ki ( X i , t ) − ǫ n X i =1 X { j | ( i,j ) ∈E} X ki h k ( X i , X j ) − n X j =1 X kj n X i =1 F ki ( X i , t ) − ǫ n X i =1 X { j | ( i,j ) ∈E} h k ( X i , X j ) = d X k =1 a k n X i =1 nX ki − n X j =1 X kj F ki ( X i , t ) − nǫ X ( i,j ) ∈E X ki h k ( X i , X j ) + ǫ n X j =1 X kj X ( i,j ) ∈E h k ( X i , X j ) = d X k =1 a k n X ( i,j ) ∈ 〚 ,n 〛 (cid:0) X ki − X kj (cid:1) F ki ( X i , t ) − nǫ X ( i,j ) ∈E X ki h k ( X i , X j ) + ǫ n X j =1 X kj X ( i,j ) ∈E h k ( X i , X j ) . Since each edge ( i, j ) ∈ E corresponds to an edge ( j, i ) and using equality (8),we have, for all k ∈ 〚 , n 〛 ,2 X ( i,j ) ∈E h k ( X i , X j ) = X ( i,j ) ∈E h k ( X i , X j ) + X ( i,j ) ∈E h k ( X j , X i )= X ( i,j ) ∈E h k ( X i , X j ) + X ( i,j ) ∈E − h k ( X i , X j )= 0and2 d X k =1 a k X ( i,j ) ∈E X ki h k ( X i , X j ) = d X k =1 a k X ( i,j ) ∈E X ki h k ( X i , X j ) + X ( i,j ) ∈E X kj h k ( X j , X i ) = d X k =1 a k X ( i,j ) ∈E X ki h k ( X i , X j ) + X ( i,j ) ∈E − X kj h k ( X i , X j ) = X ( i,j ) ∈E ϕ ( X i , X j ) (see 5) . X i,j ( X ki − X kj ) F ki ( X i , t ) = X i,j ( X ki − X kj ) F ki ( X i , t ) + X i,j ( X kj − X ki ) F kj ( X j , t )= X i,j ( X ki − X kj )( F ki ( X i , t ) − F kj ( X j , t )) . These three equalities gives˙ V = X i,j d X k =1 a k X ki − X kj ) (cid:0) F ki ( X i , t ) − F kj ( X j , t ) (cid:1) − nǫ X ( i,j ) ∈E ϕ ( X i , X j ) (10)With assumption (6) and inequality (3), we obtain˙ V ≤ X i,j ϕ ( X i , X j ) − nǫ X ( i,j ) ∈E ϕ ( X i , X j ) ≤ (cid:18) C G − nǫ (cid:19) X ( i,j ) ∈E ϕ ( X i , X j )Since ϕ is a pseudometric the right factor of this last expression is non negative.Therefore, if ǫ > C G n then ˙ V ≤
0. To prove that ˙ V is negative definite, itremains to show that if ˙ V = 0 then X = X = · · · = X n . Suppose that ˙ V = 0.Since (cid:0) C G − nǫ (cid:1) <
0, the last inequality implies that we have ϕ ( X i , X j ) = 0 forall ( i, j ) ∈ E . From equality (10), we obtain X i,j d X k =1 a k ( X ki − X kj ) (cid:0) F ki ( X i , t ) − F kj ( X j , t ) (cid:1) = 0 . Consequently, assumption (7) is satisfied and system (1) synchronizes.
Let H be the diagonal matrix Diag ( a , . . . , a d ) and H = H · · · H · · · · · · H the matrix composed with n ( n − matrices H . The application k . k V : R n ( n − d → R + X → q X T H X (11)is a norm since a , . . . , a d are non negative. Let’s set V ( t ) = k ∆( t ) k V = 12 d X k =1 X i 0. This brings to a contradiction.Finally, we have ∀ t ≥ t , ∆( t ) ∈ B and the assumptions of Section 3.1 aresatisfied for any t ≥ t . Now, we can proceed like in the proof of Theorem 3.1to conclude. In this section, we focus on applications of Theorems 3.1 and 3.2 in order tohave a sufficient condition for global synchronization of two systems. The factthat solutions of these two systems are defined on R is a direct consequence ofProposition 3.1. In this section, we apply Theorem 3.1 to a network of neurons satisfying theFitzHugh-Nagumo model (See [6]). Recall that the dynamic of a single neuronis modelised by the equation ˙ X = F ( X ) where • X = (cid:18) xy (cid:19) ; • F ( X ) = (cid:18) − x + x − y + abx − cy − d (cid:19) for some real parameters a , b , c and d .In the following, we suppose that b is positive. Let’s set G the connectedgraph describing the interaction between the oscillators, n its number of verticesand E the set of its edges. For the synchronization terms, we consider thefunction h defined by ∀ ( i, j ) ∈ 〚 , n 〛 , h ( X i , X j ) = (cid:18) α ( x i − x j ) + β p ( x i − x j ) γ ( y i − y j ) (cid:19) α ≥ β ≥ γ ≥ M ax { , − c } . The system of equations for thenetwork of oscillators is then ˙ X = F ( X ) − ǫ X (1 ,j ) ∈E h ( X , X j ) , ...˙ X n = F n ( X n ) − ǫ X ( n,j ) ∈E h ( X n , X j ) . (12)The three hypothesis of Section 3.1 are satisfied with a = 1 and a = 1 /b .Indeed,1. assumption (7) is obvious;2. the fact that the application ϕ corresponding to h , explicitly defined by ϕ ( X i , X j ) = α ( x i − x j ) + β q ( x i − x j ) + γ/b ( y i − y j ) , is a pseudometric satisfying ρ ( m ) = m / is a consequence of Example 2.1and Proposition 2.1. Therefore, assumption (5) is satisfied;3. the following inequalities shows assumption (6), for all ( X i , X j ) ∈ D , P k =1 a k ( X ki − X kj ) (cid:0) F ki ( X i ) − F kj ( X j ) (cid:1) = (cid:18) x i − x j y i − y j (cid:19) . (cid:18) − ( x i − x j ) + ( x i − x j ) − ( y i − y j )( x i − x j ) − c/b ( y i − y j ) (cid:19) = − ( x i − x j )( x i − x j ) + ( x i − x j ) − c/b ( y i − y j ) ≤ ϕ ( X i , X j ) . For any connected graph G with n vertex, inequality (3) is verified for thebound of C ( G ) given by C = n ( n − δ ( G ) ρ ( δ ( G )). Theorem 3.1 shows thenthat, for any connected graph G with n vertex, if ǫ > ( n − δ ( G ) / In this section, we apply Theorem 3.2 to a network of Chua oscillators. Weconsider the simplified version suggested by Chua for these oscillators (see [7]):if we set X = ( x, y, z ) T , the state equation for a single oscillator is given by˙ X = F ( X ) where F ( x, y, z ) = a [ y − x − f ( x )] x − y + z − by − cz ,a > b > c > f is a piece-wise function f ( x ) = dx + 1 / d − e )( | x +1 | − | x − | ) with 2 d < e .Since f is a piece-wise function, a real δ ≥ n f ( x ) − f ( y ) x − y | < | x − y | ≤ o . In the following, we suppose that:11. the set of vertex of G is E = { (1; 2) , (1; 3) , . . . , (1; n ) } . In other words, weconsider a star configuration of oscillators;2. the synchronization function h is given by h (( x i , y i , z i ) , ( x j , y j , z j )) = aδ ( x i − x j ) e −| x i − x j | . The equation for the i -th oscillator of the network is then ˙ x i ˙ y i ˙ z i = a [ y i − x i − f ( x i )] x i − y i + z i − by i − cz i + ǫ X j | ( i,j ) ∈E aδ ( x i − x j ) e −| x i − x j | . Assumptions of Section 3.1 have to be verified in order to apply Theorem 3.2.The first one is obvious. For the second and the third one, let’s set a = 1 /a , a = 1 and a = 1 /b .Let’s consider a closed ball B = n X ∈ R n ( n − d | k X k V ≤ ( √ − √ a o where k . k V is defined by (11) and the norm k . k ˜ V given by k . k ˜ V : R d → R + Y → q Y T HY where H is the diagonal matrix Diag ( a , . . . , a d ). If we have ∆ ∈ B then k ∆ i,j k ˜ V < ( √ − √ a . This implies that | x i − x j | < − √ ϕ corresponding to h satisfies assumption (5).Let’s verify assumption (6). We have P k =1 a k ( X ki − X kj ) (cid:0) F ki ( X i ) − F kj ( X j ) (cid:1) = x i − x j ay i − y j z i − z j b . a [( y i − y j ) − ( x i − x j ) − ( f ( x i ) − f ( x j ))]( x i − x j ) − ( y i − y j ) + ( z i − z j ) − b ( y i − y j ) − c ( z i − z j ) = ( x i − x j )( f ( x i ) − f ( x j )) − ( x i − x j ) − ( y i − y j ) − c/b ( z i − z j ) . By definition of δ , we have ( x i − x j )( f ( x i ) − f ( x j )) ≤ δ ( x i − x j ) e −| x i − x j | . This shows inequality (6).Moreover, if ϕ ( x i , x j ) = 0 and P k =1 a k ( X ki − X kj ) (cid:0) F ki ( X i ) − F kj ( X j ) (cid:1) = 0then we have X i = X j . Consequently, assumption (7) holds.Since the induced pseudometric ϕ satisfies ∀ m ∈ N ∗ , ρ ( m ) = m (see Exam-ple 2.1), the bound C G is given explicitly by 2 n − t ) ∈ ◦ B for an instant t and if ǫ > n − n then system (1) synchronizes.12 Conclusion In this paper, sufficient conditions for proving complete synchronization of os-cillators in a connected undirected network are presented. The contribution ofthis paper lies in the extension of results established in the case of linear syn-chronization to the non linear case. For this, we have introduced pseudometricswhich enable us to link graph topology and minimal synchronization strengthbetween oscillators. Under our assumptions, a criterion proving the existence oftrajectories is given. Two results for proving the complete synchronization arethen proposed: the first one gives a global criterion and the second one dealswith local synchronization, that is when the trajectories lie in a neighborhoodof the synchronization variety. 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