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Spatiotemporal synchronization in lattices of locally coupled chaotic oscillators
Mathematics and Computers in Simulation 58 (2002) 477–492
Spatiotemporal synchronization in lattices of locally coupled chaotic oscillators V.N. Belykh a,∗ , I.V. Belykh b , K.V. Nelvidin a a
Mathematics Department, Volga State Academy, 5 Nesterov Street, Nizhny Novgorod 603600, Russia b Department of Differential Equations, Institute of Applied Mathematics and Cybernetics, 10 Ul’yanov Street, Nizhny Novgorod 603005, Russia
Abstract The paper combines theoretical analyses with computer simulation studies of spatiotemporal synchronization regimes arising in a two-dimensional (2D) lattice of diffusively coupled identical oscillators with complicated individual dynamics. The existence of linear invariant manifolds, defining different modes of spatiotemporal synchronization, is examined. The set of possible modes of cluster synchronization is stated. The appearance and order of stabilization of the cluster synchronization regimes with increasing coupling between the oscillators are revealed for 2D lattices of coupled Lur’e systems and of coupled Rössler oscillators. © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Cluster; Synchronization; Manifolds; Chaos
1. Introduction In the last decade, there has been significant interest in the behavior of coupled chaotic systems. Numerous studies of the spatiotemporal dynamics of ensembles of coupled oscillators have found different types of coherent structures, patterns and synchronization phenomena. The simplest mode of spatiotemporal behavior that can arise in a discrete medium of coupled identical oscillators with chaotic behavior is full synchronization [1–14]. Here, all oscillators of the lattice acquire identical chaotic behavior even though their initial conditions may be different. Cluster synchronization (or clustering) is observed when the oscillators of the ensemble completely synchronize in groups but there is no synchronization among the groups [15–25]. Clustering of chaotic oscillators is considered to be particularly significant in the biological sciences where one often encounters coupled cells or functional units that each displays a complicated nonlinear behavior [15–18]. The phenomenon of chaos synchronization has many different potential applications. In engineering, for example, it is being considered as a tool for transmitting information by means of chaotic signals ∗
Corresponding author. E-mail address: [email protected] (V.N. Belykh). 0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 8 5 - 8
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[9,10]. The general idea for such a transmission is the following. An information carrying signal is injected into the transmitter system that shows chaotic behavior. It generates an output signal of chaotic nature which is transmitted. The receiver is driven by this signal and performs the inverse operation of the transmitter, it retrieves the information signal. Thus, the transmitter mixes in some way the information with chaos so that it is not possible, or very difficult, to extract the information from the transmitted signal. This provides secure communication. Another useful property of the transmitted chaotic signal is its broadband nature, as in the case of spread spectrum transmission. It might be difficult to imagine how the receiver system can extract from a chaotic signal the hidden information. Often, the receiver system is identical to the transmitter system and its behavior is chaotic. To retrieve the information, it is necessary to synchronize the two chaotic systems [9]. We have recently described a family of embedded linear invariant manifolds of a 1D array of diffusively coupled identical oscillators, whether chaotic or periodic [23–25]. Depending on an essential manner on the number of oscillators contained in the array and on boundary conditions, these invariant manifolds define the strict set of possible modes of cluster synchronization. The purpose of the present paper is to extend our results on the existence and stability of different cluster synchronization regimes for a chain of coupled identical systems to the case of 2D lattices of coupled oscillators with nontrivial individual dynamics. We study a 2D lattice of diffusively coupled identical oscillators that is described by the following dynamical system X˙ i,j = F (Xi,j ) + ε1 P (Xi+1,j − 2Xi,j + Xi−1,j ) + ε2 P (Xi,j +1 − 2Xi,j + Xi,j −1 ), i = 1, N1 ,
j = 1, N2
(1.1)
with zero-flux boundary conditions Xi,0 ≡ Xi,1 , X0,j ≡ X1,j , Xi,N2 +1 ≡ Xi,N2 , and XN1 +1,j ≡ XN1 ,j . In the system (1.1), Xi,j is the m-dimensional vector of the (i, j )th oscillator variables. F (Xi,j ): R m → R m is a vector function, m the dimension of the individual oscillator, B = N1 × N2 × m the dimension of the whole lattice system, ε1 and ε2 are coupling parameters determining the coupling strength between the oscillators in two directions of the lattice, and P is the m × m matrix whose elements determine by which variables the oscillators are coupled. For simplicity, we shall consider 1 0 0 P = 0 0 0, 0 0 0 i.e. the case of a simple scalar diffusive coupling. N1 and N2 define the size of the lattice. Note that the system (1.1) represents a class of 2D reaction–diffusion cellular neural networks (CNN) having no inputs (i.e. autonomous) that may be considered as a universal active substrate or medium for modeling and generating many pattern formation and synchronization phenomena from numerous disciplines including biology, chemistry, ecology, engineering, physics, etc. [26,27]. The coupling in the system (1.1) approximates a spatial Laplacian operator. Although the system (1.1) is an excellent approximation to the nonlinear partial differential equations describing reaction–diffusion system if the number of cells is sufficiently large, it can exhibit new phenomena (e.g. propagation failure) that can not be obtained from their limiting partial differential equations. This demonstrates that the autonomous CNN is in some sense more general that its associated nonlinear partial differential equations [27].
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The spatiotemporal synchronization of diffusively coupled oscillators in the lattice system (1.1) is intimately related to invariant synchronization manifolds of the system (1.1). We will discover the existence of various embedded linear manifolds related to cluster synchronization in in-phase and antiphase modes, determine their dimension and discuss their order of appearance with increasing coupling. This study is connected, for example, with one of the most important problems in the biological sciences of how groups of cells can interact with one another to produce different form of coordinated function on a higher organization level. Lattices of locally coupled biological cells have recently received a great deal of attention. In particular, a lattice of locally coupled chaotic -cells producing insulin was considered and different types of in-phase and antiphase synchronization presenting different levels of correlation between the cells were found numerically [28]. Simulations of a lattice of coupled nephrons in the kidney were performed. The main purpose of this lattice model was to look for possible phase transition in temporal and spatial structures as the coupling between the nephrons is changed. In particular, phase-locking in separate groups of nephrons for a week coupling and a “chessboard effect” (locking with a -phase lag between neighboring cycles) for high values of the coupling were discovered [29]. 2. Existence of cluster synchronization manifolds Let us introduce an N1 × N2 matrix A = [ai,j ], where the symbol ai,j denotes the oscillator standing in the ith line and j th column of the lattice (1.1). We shall exploit symbol equalities such as ai1 ,j1 = ai2 ,j2 implying that two oscillators are synchronized (Xi1 ,j1 = Xi2 ,j2 ). We introduce the notation M0 (N ) for the N1 × N2 × m-dimensional phase space of the system (1.1) with the coordinate matrix Xi,j , and M(d) ≡ M(n1 , n2 )(d = n1 × n2 ) for the d × m-dimensional Euclidean space with the coordinate matrix (U1 , U2 , . . . , Ud ) ∈ R d×m , where Ul = Xil ,jl , l = 1, 2, . . . , d. Definition. If there exists a map T : M(d) → M0 (N) such that the system (1.1) under the action of T attains a form of d × m compatible equations, then the system (1.1) has an invariant synchronization manifold M(d) corresponding to d clusters of synchronized oscillators. This definition implies that the d base elements of the matrix A disagree with each other and all the other r = N1 × N2 − d elements coincide with these d base elements. At the same time, r × m equations in (1.1) become linearly dependent of the d × m base ones. The numbers q 1 , q2 , . . . , qd of oscillators in each cluster are defined by the map T and satisfy the natural condition dp=1 qp = N1 × N2 . We present the cluster synchronization in the lattice by a distribution of base cluster elements along the phase space M0 (N), where each cluster element is encountered qp times and situated at sites determined by the map T . Hereafter, we will use the following notation for the distribution: ˜ ˜ 1 , n2 ) = [{ri1 , . . . , ril , . . .}; {cj1 , . . . , cjl , . . .}], M(d) ≡ M(n
N1
N2
where each symbol ril , l = 1, n1 ; cjl , l = 1, n2 stands at the place defined by the map T , i.e. the place of each symbol ril , cjl (l = 1, n1 ; l = 1, n2 ) in the subsets defines the place of each oscillator in the ith row and in the j th column. Oscillators involved in one cluster along one direction of the lattice are denoted by a symbol with the same index. As an example, we introduce two well-known clusters.
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1. For d = 1, the lattice (1.1) has the full synchronization manifold M(1) = {Xi,j ≡ U1 }. Its distribution ˜ reads M(1) = [{r1 , r1 , . . . , r1 }; {c1 , c1 , . . . , c1 }]. Dynamics in the hyperplane M(1) is defined by the single oscillator and, therefore, we have a single homogeneous cluster exhibiting full synchronization in the lattice when the spatial sites are all doing the same thing at the same moment, even though their dynamics are chaotic. 2. For d = N1 × N2 , we have the invariant manifold M0 (N), being the phase space of the system ˜ (1.1), with the following distribution of cluster elements along the lattice M(N) = [{r1 , r2 , . . . , rN1 }; {c1 , c2 , . . . , cN2 }]. Obviously, the lattice (1.1) has d = N1 ×N2 clusters defined by N1 ×N2 independent oscillators. The coupling in the system (1.1) is free of crossing terms and it appears that the hierarchy and embeddings of cluster synchronization manifolds, discovered for the 1D array of diffusively coupled dynamical systems [23–25], play the crucial role in the study of 2D lattices of coupled oscillators. The 2D lattice A may be considered as a 1D array of rows R i and columns C j in each of two lattice directions, respectively, where the elements (R 1 , R 2 , . . . , R N1 ) = A introduce a set of 1D chains. Applying directly the results from [23–25] for the set of 1D chains, we obtain the following assertion. Statement 1. 1. The system (1.1) has a symmetrical 1D array invariant manifold Msr (n1 , N2 ) with n1 = int((N1 +1)/2) defined by the map Tsr : {XN1 −i+1,j = Xi,j = Ui,j , i = 1, 2, . . . , int((N1 + 1)/2), j = 1, 2, . . . , N2 }. The distribution of the manifold Msr (n1 , N2 ) has the form M˜ sr (n1 , N2 ) = [{r1 , . . . , rn1 −1 , rn , rn1 −1 , . . . , r1 }; {c1 , c2 , . . . , cN2 }] for odd number of oscillators N1 = 2n1 − 1 composing the lattice in the ith direction, and the form M˜ sr (n1 , N2 ) = [{r1 , . . . , rn1 , rn1 , . . . , r1 }; {c1 , c2 , . . . , cN2 }] for even N1 = 2n1 . 2. The system (1.1) has an asymmetrical invariant manifold Mar (n1 , N2 ), where N1 = p1 n1 (p1 and n1 p are arbitrary integers), defined by Ta : {XNi+2n1 l,j = X−i+1+2n1 l,j = Xi,j = Ui,j , i = 1, 2, . . . , n1 , l = 1, 2, . . . , p1 , j = 1, 2, . . . , N2 }. The distribution of the manifold Mar (n1 , N2 ) has the form M˜ ar (n1 , N2 ) = [{r1 , . . . , rn1 , rn1 , . . . , r1 , r1 , . . . , rn1 , . . . }; {c1 , c2 , . . . , cN2 }]. 3. The system (1.1) has similar invariant manifolds in the j th lattice direction: the symmetrical manifold Msc (N1 , n2 ), where n2 = int((N2 + 1)/2), and the asymmetrical synchronization manifold Mac (N1 , n2 ), where N2 = p2 n2 . The separate oscillators of the 1D array, involved in the cluster synchronization regime, are introduced here by the 1D rows and columns of the lattice (1.1). That is, for example, in the case of odd N1 = 2n1 −1, the dynamics in the manifold Msr (n1 , 1) defines a cluster synchronization regime under which the rows of completely synchronized oscillators are synchronized in pairs relatively to the middle row R n1 of the lattice. × Statement 2. The system (1.1) has a family of intersection synchronization manifolds Ms,a (n1 , n2 ) = r c Ms,a (n1 , N2 ) Ms,a (N1 , n2 ), being the intersection of any row and any column synchronization mani× (n1 , n2 ) is defined by the system of folds existing in the case of the 1D chain. Thus, the manifold M r c × r ˜ ˜ maps (T , T ) and has the spatial distribution M (n1 , n2 ) = M M˜ c . The dimension of the manifold × is dimM = n1 n2 m and Ui,j , with i = 1, n1 , j = 1, n2 , are coordinates in M × (n1 , n2 ).
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Example 1. Let us consider a 2D lattice containing N1 = 3 and N2 = 5 oscillators in the two lattice directions and let us use natural numbers as the symbols ai,j standing in the matrix A such that the oscillators denoted by the same digit have identical dynamics. We present the distribution of main cluster synchronization manifolds which exist for these N1 and N2 . The distribution of the synchronization manifold Msr (1, 5) and the cluster synchronization manifold Msr (2, 5) have the following forms: 1 3 5 7 9 1 2 3 4 5 1 2 3 4 5 and 2 4 6 8 10 , 1 3 5 7 9 1 2 3 4 5 respectively. The manifold Msr (1, 5) corresponds to full synchronization of the rows of the lattice and desynchronization of the columns, and the manifold Msr (2, 5) illustrates the two-cluster synchronization of the rows and a desynchronization pattern between the columns. The distributions of the intersection synchronization manifolds Ms× (1, 3) and Ms× (2, 3), also existing in this case, attain the forms 1 3 5 3 1 1 2 3 2 1 1 2 3 2 1 and 2 4 6 4 2 , 1 3 5 3 1 1 2 3 2 1 respectively. The manifold Ms× (1, 3) defines full synchronization of the rows and cluster one of the columns, where 1 × 3 = 3 is the number of clusters, and the manifold Ms× (3, 2) presents cluster synchronization both in the rows and columns and the number of clusters is 2 × 3 = 6. The existence of the intersection synchronization manifolds M × (n1 , n2 ) implies that the cluster manifolds of the 2D lattice (1.1) have the structure of a topological product of cluster synchronization manifolds existing in the two lattice directions. For different composed numbers N1 and N2 , the manifolds for R i and C j may be rather complicated, hence the complexity of the 2D lattice clusters increases as a product. Example 2. Let N1 = 3 and N2 = 6. The distributions of the main intersection invariant manifolds Ms× (2, 3) and Ma× (2, 2) have the following forms: 1 2 2 1 1 2 1 2 3 3 2 1 4 5 6 6 5 4 and 3 4 4 3 3 4 , 1 2 2 1 1 2 1 2 3 3 2 1 respectively. Let us now consider the system (1.1) under the additional conditions on the function F (Xi,j ) to be odd: F (−Xi,j ) = −F (Xi,j ). Here, the system (1.1) is centrally symmetric with respect to the zero point Xi,j = 0 that is an equilibrium point of (1.1). In this case, the 1D array of diffusively coupled systems (1.1) for ε1 = 0 (or ε2 = 0) has transversal invariant synchronization manifolds, defining the existence of antiphase synchronization. Such antiphase synchronization is observed, for example, in a system of two coupled oscillators where all corresponding variables of the two individual oscillators are equal with opposite sign. For ε1 = 0 (ε2 = 0), we have the straightforward transfer of the appropriate statements to the existence of transversal row (column) synchronization manifolds of the 2D lattice.
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Statement 3. Under the condition F (−Xi,j ) = −F (Xi,j ) 1. the system (1.1) has a transversal row manifold Mtr (n1 , N2 ) with n1 = int((N1 + 1)/2) defined by the map Tter : {−XN1 −i+1,j = Xi,j = Ui,j , i = 1, 2, . . . , n1 , j = 1, 2, . . . , N2 } for even N1 = 2n1 and by the map Ttor : {−XN1 −i+1,j = Xi,j = Ui,j , i = 1, 2, . . . , n1 , Xn1 +1,j ≡ 0, j = 1, 2, . . . , N2 } for odd N1 = 2n1 + 1. 2. The system (1.1) has the similar transversal column manifold Mtc (N1 , n2 ) with n2 = int((N2 + 1)/2). Example 3. Let us use positive numbers for in-phase synchronization and negative ones for antiphase cluster as the symbols ai,j in the matrix A, “0” denotes Xi,j = 0. Let N1 = 3, N2 = 4 and N1 = 3, N2 = 5. The transversal manifolds Mtc (3, 2) and Mtc (3, 3), existing in accordance with Statement 3, have the following distribution clusters along the lattice 1 4 0 −4 −1 1 4 −4 −1 2 5 −5 −2 and 2 5 0 −5 −2 , 3 6 0 −6 −3 3 6 −6 −3 respectively. Let us consider now a particular case of the system (1.1) with N1 = N2 = N and ε1 = ε2 = ε. Thus, we consider the square lattice (1.1) with equal coupling strength in two lattice directions. Under these conditions, the following hold. Statement 4. 1. The system (1.1) is invariant under permutation of discrete variables (i, j, N − i + 1, N − j + 1) and, therefore, it has additional invariant manifolds M + (d) and M − (d), where d = N(N + 1)/2 is the number of clusters. The manifold M + (d) is defined by the map T1 : {Xj,i = Xi,j = Ui,j , i = 1, N, N ≥ i ≥ i}, and M − (d) is defined by the map T2 : {XN−j +1,N−i+1 = Xi,j = Ui,j , i = 1, N, 1 ≤ j ≤ N − i + 1}. 2. There exists an intersection manifold Ms∗ (d) with number of clusters (n + 1)(n + 2) , for N = 2n + 1, 2 d = n(n + 1) , for N = 2n, 2 being a particular case of the intersection manifold Ms× (n1 , n2 ). The dynamics in the cluster manifolds M + (d) and M − (d) defines a simple symmetry of the synchronized oscillators with respect to the principal and secondary diagonal − of the lattice, respectively. ± + Obviously, the intersection symmetrical manifold M (ds ) = M (d) M (d), where (n + 1)2 , for odd N = 2n + 1, ds = n(n + 1), for even N = 2n, is also invariant. The invariant manifold Ms∗ (d) = M ± (ds ) Ms× (n1 , n2 ) defines simultaneously the symmetries of synchronized oscillators with respect to the two diagonals and to the middles of the rows and columns of the lattice.
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Example 4. Let the square lattice (1.1) be composed from N = 5. For this case, we present the distributions of the invariant manifolds M + (15), M ± (9), and Ms∗ (6)
1
2 3 4 5
2
3
4
6
7
8
7
10
11
8
11
13
9
12
14
5
9 12 , 14 15
1
2 3 4 5
2
3
4
6
7
8
7
9
7
8
7
6
4
3
2
5
4 3 2 1
and
1
2 3 2 1
2
3
2
4
5
4
5
6
5
4
5
4
2
3
2
1
2 3 , 2 1
respectively. Let us study now the embeddings of the invariant synchronization manifolds and thus to define possible routes of transition from completely unsynchronized pattern to a single homogeneous cluster, defining full synchronization, with increasing coupling between oscillators. First, we consider the N × N lattice (1.1) with ε1 = ε2 = ε and with the prime number N = 2n + 1. Below, we list linear in-phase synchronization manifolds existing in this case. 1. M + (d), M − (d), and M ± (ds ) defining the symmetries of synchronized oscillators of the square lattice with respect to the principal and secondary diagonals of the lattice. 2. M ∗ ≡ Ms∗ ((n + 1)(n + 2)/2). 3. M cl/des ≡ Msr (n + 1, N) with the distribution M˜ sr =[{r1 , . . . , rn+1 , . . . , r1 }; {c1 , . . . , cN }] defining the (n + 1)-cluster synchronization of the oscillators in the vertical columns of the lattice and a desynchronization of the oscillators in the horizontal rows. 4. M syn/des ≡ M r (1, N), its distribution reads [{r1 , r1 , r1 , . . . }; {c1 , . . . , cN }]. 5. M cl/cl ≡ Ms× (n + 1, n + 1) with the distribution [{r1 , . . . , rn+1 , . . . , r1 }; {c1 , . . . , cn+1 , . . . , c1 }] defining two (n + 1)-cluster synchronization regimes of the horizontal and vertical lines of the lattice, respectively. 6. M syn/cl ≡ Msr (1, n + 1) with the distribution [{r1 , . . . , r1 , . . . , r1 }; {c1 , . . . , cn+1 , . . . , c1 }] corresponding to full synchronization of the horizontal rows and the cluster synchronization of the vertical columns. 7. M(1) defining synchronization of all oscillators of the lattice. These manifolds are embedded as follows:
(2.1)
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Remark. Obviously, the horizontal and vertical lines of the 2D lattice may be inverted and the same clusters and the enclosure may be written with respect to the vertical columns. Let the N × N lattice (1.1) be consisted from the composed N = pn1 , where p and n1 are arbitrary integers greater than 1. By reference to Statement 1, we note that the system (1.1) has the additional asymmetrical invariant manifolds Mar (n1 , N ) and Mac (N, n1 ). Therefore, the collection of possible modes of cluster synchronization in the 2D lattice is even richer and additional intersection invariant manifolds may be obtained as the topological product of the cluster patterns, and the enclosure (2.1) is extendable. These enclosures may determine the order of appearance of cluster synchronization regimes with changing coupling between the individual oscillators of the lattice (1.1). Thus, after having considered the existence of embedded linear invariant manifolds, the main problem is to study their stability and the order of stabilization for concrete coupled chaotic oscillators. That is the topic of next sections.
3. Lattice of coupled Lur’e systems As the first example of 2D lattices of coupled concrete systems, we consider the system (1.1) with the Lur’e system (the classical automatic control system) as the individual oscillator of the lattice. Thus, we consider the 2D lattice (1.1) of x-coupled Lur’e systems, where the individual dynamics of each oscillator of the lattice is described by the following dynamical system: x˙ = y − αz − αP (x), y˙ = a11 y + a12 z + b1 P (x), (3.1) z˙ = a21 y + a22 z + b2 P (x), where α = 10.5, b1 = −110.17, b2 = 10.5, a11 = −3.8, a12 = −35.25, a21 = 0.36, a22 = 2.78 are fixed parameters and the nonlinear smooth function P (x) = 5(x 3 − 9x/4)/100. This form of the function P (x) and the corresponding parameters values are chosen to provide the nontrivial dynamics of the individual oscillator of the lattice. We study numerically the spatiotemporal dynamics of the lattice and the order of appearance of possible cluster synchronization modes when changing the coupling parameter ε from zero. 3.1. 6 × 6 Lattice With increasing coupling, the cluster synchronization M cl/des (3, 6) first becomes stable (ε = 0.47). All the next manifolds from the enclosure (2.1) remain unstable, and the corresponding cluster synchronization regime is observed in the lattice (see Fig. 1a). The oscillators synchronize in pairs around the middle of the vertical columns and desynchronize in the horizontal rows of the lattice. For ε = 0.68, the invariant manifolds M + (21) and M − (21) become stable and define cluster synchronizations with respect to the principal and secondary diagonal, respectively. Realization of a particular of these two cluster regimes depends on the initial conditions. For initial conditions taken from the basin of attraction of M − (21), the corresponding spatiotemporal regime arises (see Fig. 1b) and remains stable in some region of coupling parameter.
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Fig. 1. Lattice of coupled Lur’e systems (6 × 6). Snapshots of the established cluster synchronization regimes. Different shades of gray represent different ranges of amplitudes of xi,j (t): (a) ε = 0.47; and (b) ε = 0.68.
The individual Lur’e system has odd symmetry. This means that the lattice system (1.1) with a Lur’e system as the individual oscillator is invariant under the involution (xi,j , yi,j , zi,j ) → (−xi,j , −yi,j , −zi,j ). This fact allows the lattice system (1.1) to have antiphase cluster synchronized oscillations, defined, according to Statement 3, by the transversal synchronization manifolds Mt× (n1 , n2 ). For ε = 0.89, the transversal cluster manifold Mtr (1, 3) is indeed stable and defines a spatiotemporal regime under which the oscillators of the lattice are completely in-phase synchronized in the vertical columns and the columns are antiphase synchronized in pairs. That is, the first vertical column is antiphase synchronized with the last one, and so forth (see Fig. 2). In Fig. 2, different shades of gray from black to white represent different ranges of amplitudes xi,j (t) such that, for example, the first and the sixth antiphase synchronized columns (xi,1 = −xi,6 , yi,1 = −yi,6 , i = 1, 6) are depicted by white and black, respectively.
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Fig. 2. (a) Snapshot of the established antiphase cluster synchronization (ε = 0.89). The vertical columns of synchronized oscillators are antiphase synchronized in pairs. The bar displays the range of changing of xi,j -amplitudes from black to white. (b) Antiphase synchronization between the (1, 1) and (6, 6) oscillators.
With further increase in coupling ε = 1.2, the cluster in-phase synchronization manifold M syn/des ≡ M r (1, 6) becomes stable and defines full synchronization of the rows of the lattice and desynchronization of the columns (see Fig. 3a). Fig. 3b shows a multiperiodic attractor defining the temporal behavior of the (3, 3) oscillator in the regime of the six-cluster synchronization. With final increase of the coupling, the manifold M(1) becomes stable and synchronization of all oscillators of the lattice takes place. 3.2. 7 × 7 lattice In this subsection, we present the results of our computer simulations of the spatiotemporal behavior for the square lattice of coupled Lur’e systems with N = 7 to show that the realization of cluster synchronization modes in this case differs from that of the previous case N = 6.
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Fig. 3. (a) Snapshot of the established cluster synchronization (ε = 1.2). The vertical columns of synchronized oscillators are desynchronized. (b) Attractor defining the dynamics of the (3, 3) oscillator in the regime of the six-cluster synchronization.
For ε = 0.83, the cluster synchronization manifold M ± (16) becomes stable and defines the cluster synchronization of the oscillators with respect to the principal and secondary diagonals of the square lattice (see Fig. 4a). With further increase in coupling (ε = 0.96), this cluster pattern decays and the invariant manifold M des/syn ≡ M c (7, 1) becomes stable (see Fig. 4b) and defines synchronization of the columns of the lattice and desynchronization of the rows. Under a further increase of the coupling (ε = 1.01), this spatiotemporal synchronization regime gradually develops into the cluster defined by the next manifold M cl/syn ≡ M c (4, 1) from the enclosure (2.1). Here, the oscillators are completely synchronized in the rows and the rows are synchronized in pairs around the middle row of the lattice (see Fig. 4b). Finally, for a sufficiently large ε, full synchronization occurs in the system.
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Fig. 4. Lattice of coupled Lur’e systems (7 × 7). Snapshots of the established cluster synchronization. (a) Oscillators synchronize with respect to the principal and secondary diagonals of the lattice (ε = 0.83). (b) Oscillators synchronize in rows (ε = 0.96). (c) The rows are synchronized in pairs around the forth unsynchronized row (ε = 1.01).
4. Lattice of coupled Rössler oscillators Let us consider, as the second example of 2D lattices of coupled dynamical systems with individual chaotic behavior, the 2D lattice (1.1) of x-coupled Rössler oscillators with zero-flux boundary conditions. The individual dynamics of each oscillator of the lattice is described by the dynamical system
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x˙ = −(y + z) y˙ = x + ay z˙ = b + (x − c)z.
489
(4.1)
The parameters are a = 0.2, b = 0.2 and c = 5.7. We study numericaly the synchronization pattern formation in the 7 × 7 lattice (1.1) with the individual systems (4.1). Fig. 5 presents snapshots of the established cluster chaos synchronization regimes existing in the coupled system for different coupling strength and their order of stabilization. With increasing coupling from 0, the full synchronization manifold M(1) first becomes locally stable and attracts the trajectories from its small neighborhood (global stability is impossible due to pecularities of the individual Rössler system and the use of x-coupling [23]), and a spatially homogeneous cluster of chaotic oscillators
Fig. 5. Lattice of coupled Rössler systems (7×7). Snapshots of the established cluster synchronization: (a) ε = 0.25; (b) ε = 0.4; and (c) ε = 0.55.
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is observed in the system. Thus, the cluster synchronization appearance, that may be defined by the enclosure (2.1) for a prime number N, has the reverse order. In fact, with further increased coupling, the full synchronization manifold M(1) loses its local stability but the next embedded manifold of a greater dimension from the enclosure (2.1), the manifold M cl/syn (4, 1), remains stable (ε = 0.25), and the lattice exhibits full synchronization of the vertical columns and the four-cluster synchronization of the horizontal rows (Fig. 5a). With increasing coupling, the manifold M cl/syn in turn loses its stability and there exists a stable cluster regime (ε = 0.4) under which the oscillators remain synchronized in the horizontal rows of the lattice but now there is no cluster synchronization between the rows (Fig. 5b). Realization of this cluster regime is defined by the stability of the invariant manifold M des/syn (7, 1) and the simultaneous instability of all other invariant manifolds of
Fig. 6. (a) Snapshot of the established 10-cluster synchronization (ε = 0.58). (b) Chaotic attractor defining the temporal behavior of the (2, 2) oscillator in the regime of cluster synchronization.
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less dimensions embedded into it, namely, the invariant manifolds M(1) and M cl/syn (4, 1). For ε = 0.55, the manifold M des/syn (7, 1) loses its stability and a cluster synchronization mode, defined by the next manifold M des/cl (7, 4) of a greater dimension from the enclosure (2.1), arises (Fig. 5c). For ε = 0.58, the invariant manifold M ∗ (10) becomes stable. Fig. 6a presents the established cluster synchronization regime defined by the manifold M ∗ (10). This spatiotemporal pattern, with chaotic time dependent amplitudes of the individual oscillators, defines 10 clusters and synchronization in nine groups of elements. Fig. 6b shows a chaotic attractor defining the temporal behavior of the (2, 2) oscillator in the regime of cluster synchronization. Finally, with gradual increase of coupling, this cluster synchronization regime becomes unstable and develops into a completely unsynchronized pattern defining 7×7 independent clusters. Thus, the sequence of bifurcations corresponding to the increase of the dimension of cluster synchronization (the number of independent clusters) appeares to be broken at this place. 5. Conclusions We have described the family of embedded cluster synchronization manifolds for a 2D lattice of diffusively coupled dynamical systems and discussed the question of stability of the manifolds. These embedded manifolds come from the symmetries of coupling, boundary conditions and, in particular, from the square lattice symmetry, independent of the dynamics of the individual element, and they serve as a frame for the possible dynamical behaviors of the coupled systems in phase space, defining in particular the strict set of cluster synchronized modes that can occur in the lattice. We note that all invariant synchronization manifolds, described in the paper, are actually hyperplanes. But we preserved the term “manifold” in order to emphasize the property of persistence of the stable manifolds under small perturbations such that the perturbed synchronization hyperplanes become nonlinear manifolds. These perturbations arise in the case where a small parameter mismatch between the oscillators exists [3,11,23]. Acknowledgements The authors would like to thank Erik Mosekilde and Martin Hasler for many helpful discussions and useful comments. This work was supported in part by RFFI (Grants Nos. 99-01-01126 and 99-01-00231) and by the Grant “Universities of Russia”. References [1] [2] [3] [4] [5] [6] [7] [8]
H. Fujisaka, T. Yamada, Prog. Theor. Phys. 69 (1983) 32. H. Fujisaka, T. Yamada, Prog. Theor. Phys. 72 (1984) 885. V.S. Afraimovich, N.N. Verichev, M.I. Rabinovich, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 29 (1986) 795. L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. V.N. Belykh, N.N. Verichev, L.J. Kocarev, L.O. Chua, J. Circuits Syst. Comput. 3 (1993) 579. V.S. Anishenko, T.E. Vadivasova, D.E. Postnov, M.A. Safonova, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2 (1992) 633. J.F. Heagy, L.M. Pecora, T.L Carroll, Phys. Rev. Lett. 74 (1994) 4185. J.F. Heagy, T.L. Carroll, L.M. Pecora, Phys. Rev. E 52 (1995) 1253.
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V.N. Belykh et al. / Mathematics and Computers in Simulation 58 (2002) 477–492 M. Hasler, Phil. Trans. R. Soc., London 353 (1995) 115. N.F. Rulkov, Chaos 6 (1996) 262. G.A. Johnson, D.J. Mar, T.L. Carroll, L.M. Pecora, Phys. Rev. Lett. 80 (1998) 3956. A.L. Fradkov, A.Y. Markov, IEEE Trans. Circuits Syst. I. Fundam. Theory Appl. 44 (1997) 905. A.S. Dmitriev, M. Shirokov, S.O. Starkov, IEEE Trans. Circuits Syst. I. Fundam. Theory Appl. 44 (1997) 918. Yu.L. Maistrenko, V.L. Maistrenko, A. Popovich, E. Mosekilde, Phys. Rev. Lett. 80 (1998) 1638. K. Kaneko, Phys. D 41 (1990) 137. K. Kaneko, Phys. D 54 (1991) 5. K. Kaneko, Phys. D 55 (1992) 368. K. Kaneko, Phys. D 75 (1994) 55. V.N. Belykh, E. Mosekilde, Phys. Rev. E 54 (1996) 3196. D.H. Zanette, A.S. Mikhailov, Phys. Rev. E 57 (1998) 276. M. Hasler, Yu. Maistrenko, O. Popovich, Phys. Rev. E 58 (1998) 6843. Z. Liu, S. Chen, B. Hu, Phys. Rev. E 59 (1999) 2817. V.N. Belykh, I.V. Belykh, M. Hasler, Phys. Rev. E 62 (5) (2000) 6332. V.N. Belykh, I.V. Belykh, N.L. Komrakov, E. Mosekilde, Discrete Dynamics in Nature and Society, Vol. 4, Gordon and Breach, London, 2000, p. 245. I.V. Belykh, V.N. Belykh, in: Proceedings of the International Conference COC-2000, Vol. 2, St. Peterburg, 2000, p. 346. L.O. Chua, L. Yang, IEEE Trans. Circuits Syst. I 35 (1988) 1257. L.O. Chua, M. Hasler, G.S. Moschytz, J. Neirynek, IEEE Trans. Circuits Syst. I 42 (1995) 559. A. Sherman, Bull. Math. Biol. 56 (5) (1994) 811. H. Bohr, K.S. Jensen, T. Petersen, B. Rathjen, E. Mosekilde, N.-H. Holstein-Rathlou, Parallel Comput. 12 (1989) 113.
Spatiotemporal synchronization in lattices of locally coupled chaotic oscillators V.N. Belykh a,∗ , I.V. Belykh b , K.V. Nelvidin a a
Mathematics Department, Volga State Academy, 5 Nesterov Street, Nizhny Novgorod 603600, Russia b Department of Differential Equations, Institute of Applied Mathematics and Cybernetics, 10 Ul’yanov Street, Nizhny Novgorod 603005, Russia
Abstract The paper combines theoretical analyses with computer simulation studies of spatiotemporal synchronization regimes arising in a two-dimensional (2D) lattice of diffusively coupled identical oscillators with complicated individual dynamics. The existence of linear invariant manifolds, defining different modes of spatiotemporal synchronization, is examined. The set of possible modes of cluster synchronization is stated. The appearance and order of stabilization of the cluster synchronization regimes with increasing coupling between the oscillators are revealed for 2D lattices of coupled Lur’e systems and of coupled Rössler oscillators. © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Cluster; Synchronization; Manifolds; Chaos
1. Introduction In the last decade, there has been significant interest in the behavior of coupled chaotic systems. Numerous studies of the spatiotemporal dynamics of ensembles of coupled oscillators have found different types of coherent structures, patterns and synchronization phenomena. The simplest mode of spatiotemporal behavior that can arise in a discrete medium of coupled identical oscillators with chaotic behavior is full synchronization [1–14]. Here, all oscillators of the lattice acquire identical chaotic behavior even though their initial conditions may be different. Cluster synchronization (or clustering) is observed when the oscillators of the ensemble completely synchronize in groups but there is no synchronization among the groups [15–25]. Clustering of chaotic oscillators is considered to be particularly significant in the biological sciences where one often encounters coupled cells or functional units that each displays a complicated nonlinear behavior [15–18]. The phenomenon of chaos synchronization has many different potential applications. In engineering, for example, it is being considered as a tool for transmitting information by means of chaotic signals ∗
Corresponding author. E-mail address: [email protected] (V.N. Belykh). 0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 8 5 - 8
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[9,10]. The general idea for such a transmission is the following. An information carrying signal is injected into the transmitter system that shows chaotic behavior. It generates an output signal of chaotic nature which is transmitted. The receiver is driven by this signal and performs the inverse operation of the transmitter, it retrieves the information signal. Thus, the transmitter mixes in some way the information with chaos so that it is not possible, or very difficult, to extract the information from the transmitted signal. This provides secure communication. Another useful property of the transmitted chaotic signal is its broadband nature, as in the case of spread spectrum transmission. It might be difficult to imagine how the receiver system can extract from a chaotic signal the hidden information. Often, the receiver system is identical to the transmitter system and its behavior is chaotic. To retrieve the information, it is necessary to synchronize the two chaotic systems [9]. We have recently described a family of embedded linear invariant manifolds of a 1D array of diffusively coupled identical oscillators, whether chaotic or periodic [23–25]. Depending on an essential manner on the number of oscillators contained in the array and on boundary conditions, these invariant manifolds define the strict set of possible modes of cluster synchronization. The purpose of the present paper is to extend our results on the existence and stability of different cluster synchronization regimes for a chain of coupled identical systems to the case of 2D lattices of coupled oscillators with nontrivial individual dynamics. We study a 2D lattice of diffusively coupled identical oscillators that is described by the following dynamical system X˙ i,j = F (Xi,j ) + ε1 P (Xi+1,j − 2Xi,j + Xi−1,j ) + ε2 P (Xi,j +1 − 2Xi,j + Xi,j −1 ), i = 1, N1 ,
j = 1, N2
(1.1)
with zero-flux boundary conditions Xi,0 ≡ Xi,1 , X0,j ≡ X1,j , Xi,N2 +1 ≡ Xi,N2 , and XN1 +1,j ≡ XN1 ,j . In the system (1.1), Xi,j is the m-dimensional vector of the (i, j )th oscillator variables. F (Xi,j ): R m → R m is a vector function, m the dimension of the individual oscillator, B = N1 × N2 × m the dimension of the whole lattice system, ε1 and ε2 are coupling parameters determining the coupling strength between the oscillators in two directions of the lattice, and P is the m × m matrix whose elements determine by which variables the oscillators are coupled. For simplicity, we shall consider 1 0 0 P = 0 0 0, 0 0 0 i.e. the case of a simple scalar diffusive coupling. N1 and N2 define the size of the lattice. Note that the system (1.1) represents a class of 2D reaction–diffusion cellular neural networks (CNN) having no inputs (i.e. autonomous) that may be considered as a universal active substrate or medium for modeling and generating many pattern formation and synchronization phenomena from numerous disciplines including biology, chemistry, ecology, engineering, physics, etc. [26,27]. The coupling in the system (1.1) approximates a spatial Laplacian operator. Although the system (1.1) is an excellent approximation to the nonlinear partial differential equations describing reaction–diffusion system if the number of cells is sufficiently large, it can exhibit new phenomena (e.g. propagation failure) that can not be obtained from their limiting partial differential equations. This demonstrates that the autonomous CNN is in some sense more general that its associated nonlinear partial differential equations [27].
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The spatiotemporal synchronization of diffusively coupled oscillators in the lattice system (1.1) is intimately related to invariant synchronization manifolds of the system (1.1). We will discover the existence of various embedded linear manifolds related to cluster synchronization in in-phase and antiphase modes, determine their dimension and discuss their order of appearance with increasing coupling. This study is connected, for example, with one of the most important problems in the biological sciences of how groups of cells can interact with one another to produce different form of coordinated function on a higher organization level. Lattices of locally coupled biological cells have recently received a great deal of attention. In particular, a lattice of locally coupled chaotic -cells producing insulin was considered and different types of in-phase and antiphase synchronization presenting different levels of correlation between the cells were found numerically [28]. Simulations of a lattice of coupled nephrons in the kidney were performed. The main purpose of this lattice model was to look for possible phase transition in temporal and spatial structures as the coupling between the nephrons is changed. In particular, phase-locking in separate groups of nephrons for a week coupling and a “chessboard effect” (locking with a -phase lag between neighboring cycles) for high values of the coupling were discovered [29]. 2. Existence of cluster synchronization manifolds Let us introduce an N1 × N2 matrix A = [ai,j ], where the symbol ai,j denotes the oscillator standing in the ith line and j th column of the lattice (1.1). We shall exploit symbol equalities such as ai1 ,j1 = ai2 ,j2 implying that two oscillators are synchronized (Xi1 ,j1 = Xi2 ,j2 ). We introduce the notation M0 (N ) for the N1 × N2 × m-dimensional phase space of the system (1.1) with the coordinate matrix Xi,j , and M(d) ≡ M(n1 , n2 )(d = n1 × n2 ) for the d × m-dimensional Euclidean space with the coordinate matrix (U1 , U2 , . . . , Ud ) ∈ R d×m , where Ul = Xil ,jl , l = 1, 2, . . . , d. Definition. If there exists a map T : M(d) → M0 (N) such that the system (1.1) under the action of T attains a form of d × m compatible equations, then the system (1.1) has an invariant synchronization manifold M(d) corresponding to d clusters of synchronized oscillators. This definition implies that the d base elements of the matrix A disagree with each other and all the other r = N1 × N2 − d elements coincide with these d base elements. At the same time, r × m equations in (1.1) become linearly dependent of the d × m base ones. The numbers q 1 , q2 , . . . , qd of oscillators in each cluster are defined by the map T and satisfy the natural condition dp=1 qp = N1 × N2 . We present the cluster synchronization in the lattice by a distribution of base cluster elements along the phase space M0 (N), where each cluster element is encountered qp times and situated at sites determined by the map T . Hereafter, we will use the following notation for the distribution: ˜ ˜ 1 , n2 ) = [{ri1 , . . . , ril , . . .}; {cj1 , . . . , cjl , . . .}], M(d) ≡ M(n
N1
N2
where each symbol ril , l = 1, n1 ; cjl , l = 1, n2 stands at the place defined by the map T , i.e. the place of each symbol ril , cjl (l = 1, n1 ; l = 1, n2 ) in the subsets defines the place of each oscillator in the ith row and in the j th column. Oscillators involved in one cluster along one direction of the lattice are denoted by a symbol with the same index. As an example, we introduce two well-known clusters.
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1. For d = 1, the lattice (1.1) has the full synchronization manifold M(1) = {Xi,j ≡ U1 }. Its distribution ˜ reads M(1) = [{r1 , r1 , . . . , r1 }; {c1 , c1 , . . . , c1 }]. Dynamics in the hyperplane M(1) is defined by the single oscillator and, therefore, we have a single homogeneous cluster exhibiting full synchronization in the lattice when the spatial sites are all doing the same thing at the same moment, even though their dynamics are chaotic. 2. For d = N1 × N2 , we have the invariant manifold M0 (N), being the phase space of the system ˜ (1.1), with the following distribution of cluster elements along the lattice M(N) = [{r1 , r2 , . . . , rN1 }; {c1 , c2 , . . . , cN2 }]. Obviously, the lattice (1.1) has d = N1 ×N2 clusters defined by N1 ×N2 independent oscillators. The coupling in the system (1.1) is free of crossing terms and it appears that the hierarchy and embeddings of cluster synchronization manifolds, discovered for the 1D array of diffusively coupled dynamical systems [23–25], play the crucial role in the study of 2D lattices of coupled oscillators. The 2D lattice A may be considered as a 1D array of rows R i and columns C j in each of two lattice directions, respectively, where the elements (R 1 , R 2 , . . . , R N1 ) = A introduce a set of 1D chains. Applying directly the results from [23–25] for the set of 1D chains, we obtain the following assertion. Statement 1. 1. The system (1.1) has a symmetrical 1D array invariant manifold Msr (n1 , N2 ) with n1 = int((N1 +1)/2) defined by the map Tsr : {XN1 −i+1,j = Xi,j = Ui,j , i = 1, 2, . . . , int((N1 + 1)/2), j = 1, 2, . . . , N2 }. The distribution of the manifold Msr (n1 , N2 ) has the form M˜ sr (n1 , N2 ) = [{r1 , . . . , rn1 −1 , rn , rn1 −1 , . . . , r1 }; {c1 , c2 , . . . , cN2 }] for odd number of oscillators N1 = 2n1 − 1 composing the lattice in the ith direction, and the form M˜ sr (n1 , N2 ) = [{r1 , . . . , rn1 , rn1 , . . . , r1 }; {c1 , c2 , . . . , cN2 }] for even N1 = 2n1 . 2. The system (1.1) has an asymmetrical invariant manifold Mar (n1 , N2 ), where N1 = p1 n1 (p1 and n1 p are arbitrary integers), defined by Ta : {XNi+2n1 l,j = X−i+1+2n1 l,j = Xi,j = Ui,j , i = 1, 2, . . . , n1 , l = 1, 2, . . . , p1 , j = 1, 2, . . . , N2 }. The distribution of the manifold Mar (n1 , N2 ) has the form M˜ ar (n1 , N2 ) = [{r1 , . . . , rn1 , rn1 , . . . , r1 , r1 , . . . , rn1 , . . . }; {c1 , c2 , . . . , cN2 }]. 3. The system (1.1) has similar invariant manifolds in the j th lattice direction: the symmetrical manifold Msc (N1 , n2 ), where n2 = int((N2 + 1)/2), and the asymmetrical synchronization manifold Mac (N1 , n2 ), where N2 = p2 n2 . The separate oscillators of the 1D array, involved in the cluster synchronization regime, are introduced here by the 1D rows and columns of the lattice (1.1). That is, for example, in the case of odd N1 = 2n1 −1, the dynamics in the manifold Msr (n1 , 1) defines a cluster synchronization regime under which the rows of completely synchronized oscillators are synchronized in pairs relatively to the middle row R n1 of the lattice. × Statement 2. The system (1.1) has a family of intersection synchronization manifolds Ms,a (n1 , n2 ) = r c Ms,a (n1 , N2 ) Ms,a (N1 , n2 ), being the intersection of any row and any column synchronization mani× (n1 , n2 ) is defined by the system of folds existing in the case of the 1D chain. Thus, the manifold M r c × r ˜ ˜ maps (T , T ) and has the spatial distribution M (n1 , n2 ) = M M˜ c . The dimension of the manifold × is dimM = n1 n2 m and Ui,j , with i = 1, n1 , j = 1, n2 , are coordinates in M × (n1 , n2 ).
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Example 1. Let us consider a 2D lattice containing N1 = 3 and N2 = 5 oscillators in the two lattice directions and let us use natural numbers as the symbols ai,j standing in the matrix A such that the oscillators denoted by the same digit have identical dynamics. We present the distribution of main cluster synchronization manifolds which exist for these N1 and N2 . The distribution of the synchronization manifold Msr (1, 5) and the cluster synchronization manifold Msr (2, 5) have the following forms: 1 3 5 7 9 1 2 3 4 5 1 2 3 4 5 and 2 4 6 8 10 , 1 3 5 7 9 1 2 3 4 5 respectively. The manifold Msr (1, 5) corresponds to full synchronization of the rows of the lattice and desynchronization of the columns, and the manifold Msr (2, 5) illustrates the two-cluster synchronization of the rows and a desynchronization pattern between the columns. The distributions of the intersection synchronization manifolds Ms× (1, 3) and Ms× (2, 3), also existing in this case, attain the forms 1 3 5 3 1 1 2 3 2 1 1 2 3 2 1 and 2 4 6 4 2 , 1 3 5 3 1 1 2 3 2 1 respectively. The manifold Ms× (1, 3) defines full synchronization of the rows and cluster one of the columns, where 1 × 3 = 3 is the number of clusters, and the manifold Ms× (3, 2) presents cluster synchronization both in the rows and columns and the number of clusters is 2 × 3 = 6. The existence of the intersection synchronization manifolds M × (n1 , n2 ) implies that the cluster manifolds of the 2D lattice (1.1) have the structure of a topological product of cluster synchronization manifolds existing in the two lattice directions. For different composed numbers N1 and N2 , the manifolds for R i and C j may be rather complicated, hence the complexity of the 2D lattice clusters increases as a product. Example 2. Let N1 = 3 and N2 = 6. The distributions of the main intersection invariant manifolds Ms× (2, 3) and Ma× (2, 2) have the following forms: 1 2 2 1 1 2 1 2 3 3 2 1 4 5 6 6 5 4 and 3 4 4 3 3 4 , 1 2 2 1 1 2 1 2 3 3 2 1 respectively. Let us now consider the system (1.1) under the additional conditions on the function F (Xi,j ) to be odd: F (−Xi,j ) = −F (Xi,j ). Here, the system (1.1) is centrally symmetric with respect to the zero point Xi,j = 0 that is an equilibrium point of (1.1). In this case, the 1D array of diffusively coupled systems (1.1) for ε1 = 0 (or ε2 = 0) has transversal invariant synchronization manifolds, defining the existence of antiphase synchronization. Such antiphase synchronization is observed, for example, in a system of two coupled oscillators where all corresponding variables of the two individual oscillators are equal with opposite sign. For ε1 = 0 (ε2 = 0), we have the straightforward transfer of the appropriate statements to the existence of transversal row (column) synchronization manifolds of the 2D lattice.
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Statement 3. Under the condition F (−Xi,j ) = −F (Xi,j ) 1. the system (1.1) has a transversal row manifold Mtr (n1 , N2 ) with n1 = int((N1 + 1)/2) defined by the map Tter : {−XN1 −i+1,j = Xi,j = Ui,j , i = 1, 2, . . . , n1 , j = 1, 2, . . . , N2 } for even N1 = 2n1 and by the map Ttor : {−XN1 −i+1,j = Xi,j = Ui,j , i = 1, 2, . . . , n1 , Xn1 +1,j ≡ 0, j = 1, 2, . . . , N2 } for odd N1 = 2n1 + 1. 2. The system (1.1) has the similar transversal column manifold Mtc (N1 , n2 ) with n2 = int((N2 + 1)/2). Example 3. Let us use positive numbers for in-phase synchronization and negative ones for antiphase cluster as the symbols ai,j in the matrix A, “0” denotes Xi,j = 0. Let N1 = 3, N2 = 4 and N1 = 3, N2 = 5. The transversal manifolds Mtc (3, 2) and Mtc (3, 3), existing in accordance with Statement 3, have the following distribution clusters along the lattice 1 4 0 −4 −1 1 4 −4 −1 2 5 −5 −2 and 2 5 0 −5 −2 , 3 6 0 −6 −3 3 6 −6 −3 respectively. Let us consider now a particular case of the system (1.1) with N1 = N2 = N and ε1 = ε2 = ε. Thus, we consider the square lattice (1.1) with equal coupling strength in two lattice directions. Under these conditions, the following hold. Statement 4. 1. The system (1.1) is invariant under permutation of discrete variables (i, j, N − i + 1, N − j + 1) and, therefore, it has additional invariant manifolds M + (d) and M − (d), where d = N(N + 1)/2 is the number of clusters. The manifold M + (d) is defined by the map T1 : {Xj,i = Xi,j = Ui,j , i = 1, N, N ≥ i ≥ i}, and M − (d) is defined by the map T2 : {XN−j +1,N−i+1 = Xi,j = Ui,j , i = 1, N, 1 ≤ j ≤ N − i + 1}. 2. There exists an intersection manifold Ms∗ (d) with number of clusters (n + 1)(n + 2) , for N = 2n + 1, 2 d = n(n + 1) , for N = 2n, 2 being a particular case of the intersection manifold Ms× (n1 , n2 ). The dynamics in the cluster manifolds M + (d) and M − (d) defines a simple symmetry of the synchronized oscillators with respect to the principal and secondary diagonal − of the lattice, respectively. ± + Obviously, the intersection symmetrical manifold M (ds ) = M (d) M (d), where (n + 1)2 , for odd N = 2n + 1, ds = n(n + 1), for even N = 2n, is also invariant. The invariant manifold Ms∗ (d) = M ± (ds ) Ms× (n1 , n2 ) defines simultaneously the symmetries of synchronized oscillators with respect to the two diagonals and to the middles of the rows and columns of the lattice.
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Example 4. Let the square lattice (1.1) be composed from N = 5. For this case, we present the distributions of the invariant manifolds M + (15), M ± (9), and Ms∗ (6)
1
2 3 4 5
2
3
4
6
7
8
7
10
11
8
11
13
9
12
14
5
9 12 , 14 15
1
2 3 4 5
2
3
4
6
7
8
7
9
7
8
7
6
4
3
2
5
4 3 2 1
and
1
2 3 2 1
2
3
2
4
5
4
5
6
5
4
5
4
2
3
2
1
2 3 , 2 1
respectively. Let us study now the embeddings of the invariant synchronization manifolds and thus to define possible routes of transition from completely unsynchronized pattern to a single homogeneous cluster, defining full synchronization, with increasing coupling between oscillators. First, we consider the N × N lattice (1.1) with ε1 = ε2 = ε and with the prime number N = 2n + 1. Below, we list linear in-phase synchronization manifolds existing in this case. 1. M + (d), M − (d), and M ± (ds ) defining the symmetries of synchronized oscillators of the square lattice with respect to the principal and secondary diagonals of the lattice. 2. M ∗ ≡ Ms∗ ((n + 1)(n + 2)/2). 3. M cl/des ≡ Msr (n + 1, N) with the distribution M˜ sr =[{r1 , . . . , rn+1 , . . . , r1 }; {c1 , . . . , cN }] defining the (n + 1)-cluster synchronization of the oscillators in the vertical columns of the lattice and a desynchronization of the oscillators in the horizontal rows. 4. M syn/des ≡ M r (1, N), its distribution reads [{r1 , r1 , r1 , . . . }; {c1 , . . . , cN }]. 5. M cl/cl ≡ Ms× (n + 1, n + 1) with the distribution [{r1 , . . . , rn+1 , . . . , r1 }; {c1 , . . . , cn+1 , . . . , c1 }] defining two (n + 1)-cluster synchronization regimes of the horizontal and vertical lines of the lattice, respectively. 6. M syn/cl ≡ Msr (1, n + 1) with the distribution [{r1 , . . . , r1 , . . . , r1 }; {c1 , . . . , cn+1 , . . . , c1 }] corresponding to full synchronization of the horizontal rows and the cluster synchronization of the vertical columns. 7. M(1) defining synchronization of all oscillators of the lattice. These manifolds are embedded as follows:
(2.1)
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Remark. Obviously, the horizontal and vertical lines of the 2D lattice may be inverted and the same clusters and the enclosure may be written with respect to the vertical columns. Let the N × N lattice (1.1) be consisted from the composed N = pn1 , where p and n1 are arbitrary integers greater than 1. By reference to Statement 1, we note that the system (1.1) has the additional asymmetrical invariant manifolds Mar (n1 , N ) and Mac (N, n1 ). Therefore, the collection of possible modes of cluster synchronization in the 2D lattice is even richer and additional intersection invariant manifolds may be obtained as the topological product of the cluster patterns, and the enclosure (2.1) is extendable. These enclosures may determine the order of appearance of cluster synchronization regimes with changing coupling between the individual oscillators of the lattice (1.1). Thus, after having considered the existence of embedded linear invariant manifolds, the main problem is to study their stability and the order of stabilization for concrete coupled chaotic oscillators. That is the topic of next sections.
3. Lattice of coupled Lur’e systems As the first example of 2D lattices of coupled concrete systems, we consider the system (1.1) with the Lur’e system (the classical automatic control system) as the individual oscillator of the lattice. Thus, we consider the 2D lattice (1.1) of x-coupled Lur’e systems, where the individual dynamics of each oscillator of the lattice is described by the following dynamical system: x˙ = y − αz − αP (x), y˙ = a11 y + a12 z + b1 P (x), (3.1) z˙ = a21 y + a22 z + b2 P (x), where α = 10.5, b1 = −110.17, b2 = 10.5, a11 = −3.8, a12 = −35.25, a21 = 0.36, a22 = 2.78 are fixed parameters and the nonlinear smooth function P (x) = 5(x 3 − 9x/4)/100. This form of the function P (x) and the corresponding parameters values are chosen to provide the nontrivial dynamics of the individual oscillator of the lattice. We study numerically the spatiotemporal dynamics of the lattice and the order of appearance of possible cluster synchronization modes when changing the coupling parameter ε from zero. 3.1. 6 × 6 Lattice With increasing coupling, the cluster synchronization M cl/des (3, 6) first becomes stable (ε = 0.47). All the next manifolds from the enclosure (2.1) remain unstable, and the corresponding cluster synchronization regime is observed in the lattice (see Fig. 1a). The oscillators synchronize in pairs around the middle of the vertical columns and desynchronize in the horizontal rows of the lattice. For ε = 0.68, the invariant manifolds M + (21) and M − (21) become stable and define cluster synchronizations with respect to the principal and secondary diagonal, respectively. Realization of a particular of these two cluster regimes depends on the initial conditions. For initial conditions taken from the basin of attraction of M − (21), the corresponding spatiotemporal regime arises (see Fig. 1b) and remains stable in some region of coupling parameter.
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Fig. 1. Lattice of coupled Lur’e systems (6 × 6). Snapshots of the established cluster synchronization regimes. Different shades of gray represent different ranges of amplitudes of xi,j (t): (a) ε = 0.47; and (b) ε = 0.68.
The individual Lur’e system has odd symmetry. This means that the lattice system (1.1) with a Lur’e system as the individual oscillator is invariant under the involution (xi,j , yi,j , zi,j ) → (−xi,j , −yi,j , −zi,j ). This fact allows the lattice system (1.1) to have antiphase cluster synchronized oscillations, defined, according to Statement 3, by the transversal synchronization manifolds Mt× (n1 , n2 ). For ε = 0.89, the transversal cluster manifold Mtr (1, 3) is indeed stable and defines a spatiotemporal regime under which the oscillators of the lattice are completely in-phase synchronized in the vertical columns and the columns are antiphase synchronized in pairs. That is, the first vertical column is antiphase synchronized with the last one, and so forth (see Fig. 2). In Fig. 2, different shades of gray from black to white represent different ranges of amplitudes xi,j (t) such that, for example, the first and the sixth antiphase synchronized columns (xi,1 = −xi,6 , yi,1 = −yi,6 , i = 1, 6) are depicted by white and black, respectively.
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Fig. 2. (a) Snapshot of the established antiphase cluster synchronization (ε = 0.89). The vertical columns of synchronized oscillators are antiphase synchronized in pairs. The bar displays the range of changing of xi,j -amplitudes from black to white. (b) Antiphase synchronization between the (1, 1) and (6, 6) oscillators.
With further increase in coupling ε = 1.2, the cluster in-phase synchronization manifold M syn/des ≡ M r (1, 6) becomes stable and defines full synchronization of the rows of the lattice and desynchronization of the columns (see Fig. 3a). Fig. 3b shows a multiperiodic attractor defining the temporal behavior of the (3, 3) oscillator in the regime of the six-cluster synchronization. With final increase of the coupling, the manifold M(1) becomes stable and synchronization of all oscillators of the lattice takes place. 3.2. 7 × 7 lattice In this subsection, we present the results of our computer simulations of the spatiotemporal behavior for the square lattice of coupled Lur’e systems with N = 7 to show that the realization of cluster synchronization modes in this case differs from that of the previous case N = 6.
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Fig. 3. (a) Snapshot of the established cluster synchronization (ε = 1.2). The vertical columns of synchronized oscillators are desynchronized. (b) Attractor defining the dynamics of the (3, 3) oscillator in the regime of the six-cluster synchronization.
For ε = 0.83, the cluster synchronization manifold M ± (16) becomes stable and defines the cluster synchronization of the oscillators with respect to the principal and secondary diagonals of the square lattice (see Fig. 4a). With further increase in coupling (ε = 0.96), this cluster pattern decays and the invariant manifold M des/syn ≡ M c (7, 1) becomes stable (see Fig. 4b) and defines synchronization of the columns of the lattice and desynchronization of the rows. Under a further increase of the coupling (ε = 1.01), this spatiotemporal synchronization regime gradually develops into the cluster defined by the next manifold M cl/syn ≡ M c (4, 1) from the enclosure (2.1). Here, the oscillators are completely synchronized in the rows and the rows are synchronized in pairs around the middle row of the lattice (see Fig. 4b). Finally, for a sufficiently large ε, full synchronization occurs in the system.
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Fig. 4. Lattice of coupled Lur’e systems (7 × 7). Snapshots of the established cluster synchronization. (a) Oscillators synchronize with respect to the principal and secondary diagonals of the lattice (ε = 0.83). (b) Oscillators synchronize in rows (ε = 0.96). (c) The rows are synchronized in pairs around the forth unsynchronized row (ε = 1.01).
4. Lattice of coupled Rössler oscillators Let us consider, as the second example of 2D lattices of coupled dynamical systems with individual chaotic behavior, the 2D lattice (1.1) of x-coupled Rössler oscillators with zero-flux boundary conditions. The individual dynamics of each oscillator of the lattice is described by the dynamical system
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x˙ = −(y + z) y˙ = x + ay z˙ = b + (x − c)z.
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(4.1)
The parameters are a = 0.2, b = 0.2 and c = 5.7. We study numericaly the synchronization pattern formation in the 7 × 7 lattice (1.1) with the individual systems (4.1). Fig. 5 presents snapshots of the established cluster chaos synchronization regimes existing in the coupled system for different coupling strength and their order of stabilization. With increasing coupling from 0, the full synchronization manifold M(1) first becomes locally stable and attracts the trajectories from its small neighborhood (global stability is impossible due to pecularities of the individual Rössler system and the use of x-coupling [23]), and a spatially homogeneous cluster of chaotic oscillators
Fig. 5. Lattice of coupled Rössler systems (7×7). Snapshots of the established cluster synchronization: (a) ε = 0.25; (b) ε = 0.4; and (c) ε = 0.55.
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is observed in the system. Thus, the cluster synchronization appearance, that may be defined by the enclosure (2.1) for a prime number N, has the reverse order. In fact, with further increased coupling, the full synchronization manifold M(1) loses its local stability but the next embedded manifold of a greater dimension from the enclosure (2.1), the manifold M cl/syn (4, 1), remains stable (ε = 0.25), and the lattice exhibits full synchronization of the vertical columns and the four-cluster synchronization of the horizontal rows (Fig. 5a). With increasing coupling, the manifold M cl/syn in turn loses its stability and there exists a stable cluster regime (ε = 0.4) under which the oscillators remain synchronized in the horizontal rows of the lattice but now there is no cluster synchronization between the rows (Fig. 5b). Realization of this cluster regime is defined by the stability of the invariant manifold M des/syn (7, 1) and the simultaneous instability of all other invariant manifolds of
Fig. 6. (a) Snapshot of the established 10-cluster synchronization (ε = 0.58). (b) Chaotic attractor defining the temporal behavior of the (2, 2) oscillator in the regime of cluster synchronization.
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less dimensions embedded into it, namely, the invariant manifolds M(1) and M cl/syn (4, 1). For ε = 0.55, the manifold M des/syn (7, 1) loses its stability and a cluster synchronization mode, defined by the next manifold M des/cl (7, 4) of a greater dimension from the enclosure (2.1), arises (Fig. 5c). For ε = 0.58, the invariant manifold M ∗ (10) becomes stable. Fig. 6a presents the established cluster synchronization regime defined by the manifold M ∗ (10). This spatiotemporal pattern, with chaotic time dependent amplitudes of the individual oscillators, defines 10 clusters and synchronization in nine groups of elements. Fig. 6b shows a chaotic attractor defining the temporal behavior of the (2, 2) oscillator in the regime of cluster synchronization. Finally, with gradual increase of coupling, this cluster synchronization regime becomes unstable and develops into a completely unsynchronized pattern defining 7×7 independent clusters. Thus, the sequence of bifurcations corresponding to the increase of the dimension of cluster synchronization (the number of independent clusters) appeares to be broken at this place. 5. Conclusions We have described the family of embedded cluster synchronization manifolds for a 2D lattice of diffusively coupled dynamical systems and discussed the question of stability of the manifolds. These embedded manifolds come from the symmetries of coupling, boundary conditions and, in particular, from the square lattice symmetry, independent of the dynamics of the individual element, and they serve as a frame for the possible dynamical behaviors of the coupled systems in phase space, defining in particular the strict set of cluster synchronized modes that can occur in the lattice. We note that all invariant synchronization manifolds, described in the paper, are actually hyperplanes. But we preserved the term “manifold” in order to emphasize the property of persistence of the stable manifolds under small perturbations such that the perturbed synchronization hyperplanes become nonlinear manifolds. These perturbations arise in the case where a small parameter mismatch between the oscillators exists [3,11,23]. Acknowledgements The authors would like to thank Erik Mosekilde and Martin Hasler for many helpful discussions and useful comments. This work was supported in part by RFFI (Grants Nos. 99-01-01126 and 99-01-00231) and by the Grant “Universities of Russia”. References [1] [2] [3] [4] [5] [6] [7] [8]
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