Electromagnetic Modeling of the Propagation Characteristics of Satellite Communications Through Composite Precipitation Layers Science and Technology, 8 (2003) 67-75 © 2003 Sultan Qaboos University m Synchronization and Oscillator Death in Diffusively coupled lattice oscillators Adu A.M. Wasike Department of Mathematics, University of Nairobi, P.O.Box 30187, Nairobi Kenya, Email: aduwasike@yahoo.com. تزامن وموت المتذبذب في المتذبذبات الشبكية المقرنة انتشارياُ واسيكيأدو يهتم هذا البحث بدراسة التزامن وتوقف التذبذب لمتذبذبات مستوية ذات عدد زوجي موجب مقرنة إلى أقرب متذبذبات : خالصة لكل متذبذب حل دوري . شاري خطي ومتماثل ، ثنائية وثالثية من خالل مسار انت في شبكية صحيحة على مستويات آحادية اثبتت الدراسة أن في بعض حاالت التفارق يوجد تزامن متماثل وتزامن غير متماثل يناظر كل حل دوري متماثل وغير . جاذب من غير متماثل يكون عند التفارق الضعيف ، التزامن المتماثل يوجد في حالة التفارق القوى بينهما التزا . متماثل على الترتيب . وهذا يختفي إلى األصل بعد تفارق قوي معين ABSTRACT: We consider the synchronization and cessation of oscillation of a positive even number of planar oscillators that are coupled to their nearest neighbours on one, two, and three dimensional integer lattices via a linear and symmetric diffusion-like path. Each oscillator has a unique periodic solution that is attracting. We show that for certain coupling strength there are both symmetric and antisymmetric synchronization that corresponds to symmetric and antisymmetric non-constant periodic solutions respectively. Symmetric synchronization persists for all coupling strengths while the antisymmetric case exists for only weak coupling strength and disappears to the origin after a certain coupling strength. KEYWORDS: Lattice Differential Equations, Bravais Lattice, Dynchronization and Oscillator Death. 1. Introduction This paper is concerned with the synchronization and mutual extinction of oscillations of a system of ordinary differential equations (ODEs) indexed by points in an -dimensional integer lattice { }, 1, 2,mZ m ∈ 3 . This system is what we shall refer to as lattice differential equations (LDEs).By a lattice oscillator, we mean a system of LDEs with every lattice points having an attracting periodic orbit. They have been great interest in the synchronization of coupled oscillators on a one- dimentional integer lattice. Aside from the mathematical interest in the problem (see for instance Aronson et. al. 1987, and references therein, coupled oscillators occur in other fields. In biology the papers of Ermentrout and Koppel (1990,1992) provide us with examples where synchronization and oscillator death occur. In chemistry Bar-Eli (1985) and Crowley and Epstein (1989) consider the synchronization and oscillator death for some chemical oscillators. LDEs are also found in Cellular neural networks (CNN), (see for example Chua and Roska 1993. Metallurgy is another area where lattice systems can be found (Cahn 1960). (private communication) has considered synchronization of coupled oscillators of subsystems that are described by a second order damped forced Duffing ordinary differential equation. The oscillators are linearly and diffusively coupled both in position and velocity to their nearest neighbours. He found that synchronization that occurred for large values of the coupling 67 mailto:aduwasike@yahoo.com ADU A.M. WASIKE strength for both identical nonidentical oscillators is mainly influenced by the damping in the individual oscillators and dissipation of the coupling component of the system. Much of the work on synchronization and cessation of oscillations is done on a one-dimensional lattice and coupling involves only one variable of each subsystem. This is known in the literature as scalar coupling (Bar-Eli 1985). There is less repertoire on this subject in two and three-dimensional cases and when all variables in the subsystems at each lattice site are involved in the coupling. The problem we consider in this paper is the synchronization and cessation of oscillation of planar oscillators that are linearly and symmetrically coupled via a diffusion-like coupling to their nearest neighbours on one, two and three dimensional lattices. As a special case of the general results, we consider a canonical model where each oscillator is a truncated normal form to cubic order for a general planar oscillator near a Hopf bifurcation point. These types of planar oscillator that were considered by Poincar’e in 1891 (Perko 1998, p. 351) in his works on limit cycles, lend themselves to an easy analytical treatment. The dynamics of each oscillator is governed by the solution of the equation. ( ) 2: , 1, 2, 3,.....,j jz z t R j∈ = ( )2 :j j j j jz A z z z g z= − = , (1.1) where 1 1 , 1 1 A −  =     . denotes the Euclidean norm, and the “dot” denotes differentiation with respect to time t . The solution of Equation (1.1) for any initial condition in {2 \ 0R is a unique limit cycle that attracts the whole of { }2 \ 0 ,R see for instant (Perko 1998, page 356). We couple the differential equations in (1.1) to yield various lattice structures. We shall first consider the dynamics of the system when coupling yields a one-dimensional lattice and then proceed to consider the two and three dimensional lattices cases. 2. One dimensional lattice Now, let us couple the identical subsystems in equation (1.1) to yield n 2≥ ( ) ( ),z B k z f z= + (2.1) where denotes the coordinates of a point on the lattice, ( 1 2, ,....., T nz z z z= ) ( )B k a real symmetric matrix depending on the coupling strength , is given by k 0≥ ( ) 1B k k= ∆ L⊗ with ( ) ( )1 1 1 1 1 0 0 ... 0 0 0 1 2 1 0 ... 0 0 0 1 1 0 1 2 1 ... 0 0 0 , ; 1 1 : : : : ... : : : 0 0 0 0 ... 1 2 1 0 0 0 0 ... 0 1 1 n nR L− × −    −   −    ∆ = − ∈ =         −   −  (2.2) where denotes the Kronecker product, and ⊗ ( ) ( ) ( ) ( )( )1 2, ,..., . T nf z g z g z g z= This type of coupling corresponds to a symmetric nearest neighbour coupling on a linear lattice with Neumann boundary conditions. We observe that if we had taken each subsystem in dR say, then the matrix above could be replaced by a dL d× matrix whose entries are all 1’s. This coupling differs from the one considered in Hale (private communication) where is a d identify matrix. L d× 68 SYNCHRONIZATION AND OSCILLATOR DEATH We say that Equation (2.1) is synchronized or has symmetric solutions if its solution is in the set z { }21 1 2: : ... 0 ,n nz R z z zΘ = ∈ = = = ≠ for all t . We refer to this set as the ‘diagonal ‘ in 0≥ 2nR (Fujisaka and Yamada 1983). Oscillator death occur when there is a mutual cessation of oscillation in all oscillators for certain coupling strengths. That is, when is a constant for all t for some positive values of k The z 0≥ . invariant subspace { }21 : : 0,1n jz R z j n∏ = ∈ ≠ ≤ ≤ −1 , defines antisymmetric solutions to equation (2.1) or antisymmetric synchronization. These two subspace are the most readily observed and are of great interest in practical problems. The analysis of the evolution of the periodic solutions of equation (2.1) with as a parameter is facilitated by analyzing the flow restricted to the invariant manifolds ∏ and k 11 Θ . Proposition 1.1. We let be an even integer. Then there exists a constant for which equation (2.1) defines antisymmetric solutions for all 0 n 0 0k > 0k k< < . Moreover, there is oscillator death for k . 0k> Proof. The proof involves the determinates of the eigenvalues of ( )B k and an appropriate change of coordinates. It is well known (Householder 1964) that the eigenvalues of ∆ are 1 0 0, 2 2 cos ,1 1,s s s n n π λ λ= = − − ≤ ≤ − and they are simple. If is an even number, then clearly n / 2 2nλ = − is an eigenvalue and the corresponding eigenvector is V v where for a given v is a (/ 2 1 2, ,....., ; T n v v= )n 2I1 2 ,I= ± 2 2× identify matrix with v v The eigenvalues of are ( )1 1 , 1, 2,...., j j j j+ = − = 1.−n k L 0 10, 2 .kµ µ= = 2 The set ∏ is a subspace of the generalized eigenspace corresponding to the eigenvalues 1 ,α 0,1.µ α− = We show that for any solution of equation (2.1) with an initial point in , there is oscillator death for all and antisymmetric synchronization for Consider the coordinate transformation given by 1∏ ( )00, .k k∈ 0k k> 1 . 1 1 , 1 1, n j j j j w z z j n y z n+ = = − ≤ ≤ − = j∑ (2.3) If we let w w then w and satisfy the ODEs ( )1 2 1: , ,...., , T nw w −= y ( ) ( ) 2 2 2 1 , , 1 , n n jj w k Lw f w y R y g z R n − = = ∆ ⊗ + ∈ = ∈∑ (2.4) where ( ) ( )1 1 2 1 0 0 ... 0 0 0 1 2 1 0 ... 0 0 0 0 1 2 1 ... 0 0 0 , : : : : ... : : : 0 0 0 0 ... 1 2 1 0 0 0 0 ... 0 1 2 n nR − × −    −   −   ∆ = − ∈      −   −  (2.5) 69 ADU A.M. WASIKE and the function ( ) ( ) ( )( )1 1, , ,......., , , T nF w y F w y F w y−= with ( ) ( ) ( )( )1, , 1j j jF w y g z g z j n+= − 1.≤ ≤ − Taking an initial condition in the manifold 0y 1,∏ we find that the solution of the second equation in (2.4) is for all t and the governing equations reduce to the decoupled system of the form 0y = 0≥ 20y R= ∈ , (2.6) ( )2 1 2 1 2 1 2 2 ,nj n j jw kl Lw g w− − −= − ⊗ + ∈ R )n (2.7) where The solution (1, 2, 3,......., / 2 .j = ( )2 1j −w of equation (2.7) corresponds to antisymmetric solutions of the system (2.1). To show the existence of we need consider the case for let only as the system equation (2.7) is decoupled and the result for n will follow easily. Let t 0,k 2n = 2> ( ) ( )2 2 , tan 2 1 2 . 1 2 1 k k k k α α φ= = = − − + We introduce a change of coordinates in equation (2.7) with n 2= given by 1 1 0 sin cos . cos sin0 1 U w V α φ φ φα  +    =      −     φ (2.8) The equation (2.7) becomes { ( ) ( ) ( ) ( ) ( ) ( ) ( ) } ' ' 2 2 2 2 22 1 1 2 2 sin 2 1 2 cos 2 1 1 1 2 cos 2 1 2 2 sin 2 1 . 1 k k k U V k k U V U V U I V k α φ φ α α φ φ α α α  +  − − − −   =    −    − + − + +   + − −   −   −    (2.9) in which the prime denotes differentiation with respect to time. In polar coordinates defines by ( ) ( ) ( ) ( )cos , sin ,U p t t V p t tσ σ= = equation (2.9) becomes ( ) ( ) 2 ' 2 ' 2 1 cos 2 1 2 1 1 1 2 sin 2 . p ,p k p k α σ α σ α α σ   = − − − − = − − + −  (2.10) Solving for 0p′ = p in the first equation in (2.10), one finds that ( )( )* 1 2 1p k 2α= − − (2.11) provided 1 and 2k> 1,α < is the radius of a unique invariant circle. ' 0σ < when 70 SYNCHRONIZATION AND OSCILLATOR DEATH 0 1k≤ < / 2 and * ' * 0, When ; 0, When . p P P p P < >  > < Therefore is the underlying point set if a limit cycle of (2.10) and attracts all of *P ( )2 \ 0, 0R as that exists when If we let t → ∞ 0 1/k≤ < 2. ( ) ( ) ( )( )1 cos , ,t t r tθ=w r then it can be shown that equation (2.7) with 0 1/ 2,k< < admits a unique and asymptotically stable periodic orbit with period T ( )α and radius ( )r α given by ( ) ( ) ( ) ( )( ) ( )( ) 2 2 1 2 12 , . 1 cos 21 2 T r t α απ α α α θ φα − − = − +− Clearly at i.e. there is no oscillation. Simple analysis shows that for the origin for system (2.7) is stable. 0 1/ 2, 0;k k r= = = 0 ,k k> 3. Two-dimensional lattice We consider an simple (or Bravais) square lattice. Let z in n× ,1 , ,ij j n≤ ≤ be coordinate of the site oscillator and the matrix Z z( ), thi j , 1 .: n i jij =  =   Let ( )n1, ,......Z Z Z Z= with Z R 2 1,nr ×∈ as its column. Let ve1 r≤ ≤ ,n rth c Z denote the vector valued function of ,Z and ⊗ be the Kronecker sum (See Graham 1981, for details on matrix tensor algebra). The system of differential equations describing the dynamics on the n n× lattice is given by ( ) ( ),H k G= +Z Z Z (3.1) where ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) .1 1 1 2, 1, and : , ,....,nij ni jH k k L G g z f Z f Z f Z= = ∆ ⊗∆ ⊗ = =    Z Equation (3.1) is symmetrically synchronization when 0ijz z= ≠ for all ij . This occurs when ( ) 0, 0H k Zµ= = ≠Z Z with 0µ = as an eigenvalue of ( ).H k Indeed, the eigenvalues of are ( )H k ( )( ) ( ){ }: : , 0 , 1s pH k s p nασ µ µ λ λ µ= = + ,≤ ≤ − with , , , 0,1,p s αλ λ µ α = as defined in section 2. Clearly 0µ = is an eigenvalue of and by equation (3.1), there is symmetric synchronization. In this case, the plane will be ramified with traveling-wave like pattern solutions to equation (3.1). ( )H k To facilities the understanding of other dynamics, we define some subspaces invariant to the flow defined by equation (3.1). Here, we shall follow a trend similar to that in Mallet-Paret and Chow (1995a, 1995b) where pattern formations are considered for LDE’s with a first order scalar ODE at each lattice site. Let { } ( ){ } ( ){ } { } 1 1 1 1 1 1 : , ,1 1 , : 1 ,1 1 , : 1 , ,1 : . j j j j j j j j j j T Z Z Z Z j n Z Z Z j n Z Z Z Z j n Z Z θθ πθ ππ πθ πθ + + + Θ = = ∈Θ ≤ ≤ − ∏ = = − ∈Θ ≤ ≤ − ∏ = = − ∈∏ ≤ ≤ − ∏ = ∈∏ 1 , We further make a coordinate transformation defined thus 71 ADU A.M. WASIKE 1 1 1 ,1 1, and . n j j j j W Z Z jn Y Z n+ = = − ≤ − = j∑ (3.2) Using this in equation (3.1) we get : ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 1 1 2 1 , 1 , n n n n n n jj W k I L W L W I G W Y R Y B k Y f Z R n − − − = = ⊗ ∆ + ∆ ⊗ + ∈   = + ∈∑ (3.3) where ( )1 1,..., ,nW W W −= ( ) ( ) ( )( )1 1, , ,....., , ,nG W Y G W Y G W Y−= with ( ) ( ) ( )1, ,1j jG W Y f Z f Z j n+= − 1.≤ ≤ − on the subspace θθΘ ( ) ( ) { } 2 1 2 0 , , 1, 2,......, . n n n W R Y Z f Z R nβ β β −= ∈ = = ∈ ∈ (3.4) The system in this case is symmetrically synchronized and the dynamics are described by equation (3.4). Proposition 2.1. Let be an even integer. Then there exists a constant for which the simple lattice equation (3.1) is ramified with horizontal and vertical bands of antisymmetric solutions for all Moreover, oscillator death occurs for n 1 0k > ( 10, .k k∈ ) 1.k k> Proof. On θπ∏ and πθ∏ , we have horizontal and vertical bands of stripes respectively and their flows are governed by the decoupled ODEs, ( )( ) (2 1 2 1 2 1 2 0, 2 ,j n n j j Y W I B k kI L W f W− − =   = ⊗ − ⊗ +    )− n (3.5) where W W Since all Z( )2 1 2 11 , with 1, 2, 3,...., / 2. j j j j+ −= − = 1,j ∈Θ equation (3.5) reduces to decoupled ODEs. Thus / 2n ( ) ( )2 1 2 1 2 12j n jW kI L W f W+ −= − ⊗ + ,j − (3.6) which upon the use of equation (2.7), Proposition 2.1 is proved with 1 1/ 2k = . Proposition 2.2. Let be an even integer. Then there is a constant for which the lattice equation (3.1) is ramified with checkerboard like solutions for all n 2 0k > ( )20, .kk ∈ Moreover, oscillator death occurs for . 2kk > Poof. The subspace ππ∏ represents vertical (horizontal) bands of symmetric and antisymmetric oscillators. This subspace constituted a checkerboard. A unit square on a checkerboard is constituted by four subsystems on a square lattice. Within each square, the subsystems are of alternate signs with their nearest neighbours. If 2 2× ( )z t represents an oscillator, then 72 SYNCHRONIZATION AND OSCILLATOR DEATH ( ) ( ) ( ) ( ) z t z t z t z t −    −  is the unit square. The entire checkerboard is defined by these squares that alternate is sign. The subspace ππ∏ is invariant under the flow defined by equation (3.1) and its flow governed by the solution of the equations / 2n ( ) ( )2 1 2 1 2 14j jW kI L W f W− −= − ⊗ + ,j − (3.7) We notice that equation (3.7) is similar to equation (2.7). The proof of the Proposition 2.2 follows easily if we replace k in equation (2.7) with . We find that 2k 2 1/ 4k = . 4. Three dimensional lattice We consider an simple cubic lattice. Let n n n× × ,1 , , ,pqrz p q r n≤ ≤ be the coordinate of the ( , , th)p q r site oscillator with its dynamics described by equation (1.1). We shall assume that the coupling is to the nearest neighbour and the coupling forces are equal and acting along the axes of the cube. Thus the oscillator can only interact with the oscillators z Z six in all. Let where pqrz 1 2, ,=Z Z Z 1 1, ,p q r p q r p q rZ± ± 1,± ( ): ....., TnZ , , 1: . n qr p q rr p z =   =Z The system of differential equations describing the dynamics of the entire lattice system is given by ( ) ( ),k= +Z H Z G Z (4.1) where ( ) ( ) ( )( ) ( ) ( ) ( )( )1 1 1 1, ,...., , T n nk k I L G G = ∆ ⊗∆ ⊗ ⊗∆ ⊗ = H G Z Z Z with G g( ) ( ) , , 1 . n r pqr p q r z =  =  Z To determine the invariant subspaces available to equation (4.1), we find the eigenvalues of and the corresponding eigenvectors. We have ( )kH ( )( ) ( )( ){ }: : , 0 , , 1p q rk pασ µ µ λ λ λ µ= = + + ,q r n≤ ≤ −H where , , , ,p q r αλ λ λ µ are as defined in section 2 with each rλ occurring with multiplicity .n { } { } { } ( ){ } ( ){ } ( ){ } { } 1 1 1 1 1 1 000 : , ,1 1 , : , ,1 1 , : , ,1 1 , : 1 , ,1 1 : 1 , ,1 : 1 , ,1 : 0 . j j j j j j j j j j j j j j j j j j j j j j n j n j n j n j n j n θθθ θθ ππθ ππ θπθ θπ θθπ θθ θππ θπ πππ ππ + + + + + + Θ = = ∈Θ ≤ ≤ − ∏ = = ∈∏ ≤ ≤ − ∏ = = ∈∏ ≤ ≤ − Θ = = − ∈Θ ≤ ≤ − ∏ = = − ∈∏ ≤ ≤ − , 1 , 1 ,∏ = = − ∈∏ ≤ ≤ ∏ = = Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z − The system of equations defining the dynamics on these manifolds follows readily with the coordinate transformation 1 1 : , 1 1 : . j j j n jj j n n + = − 1,≤ ≤ − = ∑ W Z Z Y Z (4.2) Using equation (4.2 in equation (4.1), we get 73 ADU A.M. WASIKE ( ) ( )( ) ( ) ( ) ( ) 1 1 1 : , 1 , n n jj k I L H k G n =  ∆ ⊗ ⊗ ∆ ⊗∆ ⊗ +  = + ∑ W W Y Y Z ,G W Y (4.3) where ( ) ( ) ( ) ( )( )1 1 1 1,...., , , , ,......, , , TT n nG G− −= =W W W G W Y W Y W Y with ( ) ( ) ( )1, , 1j jG G j n+= − 1,≤ ≤ −Y Z ZG W on , ,θθθΘ =W 0 and the flow is governed by ( ), 1,...., .Gβ β β= =Z Z n (4.4) We thus see that the system is synchronized Proposition 3.1. Let be an even integer. Then for all n ( )30,k k∈ the simple three-dimensional lattice equation (4.1) has layers of checkerboards. Besides, oscillator death occurs for . 3k k> Proof. On the subspace ,ππθ∏ the flow is defined by the solution of the decoupled equation ( ) ( )4 , , 1,...., ,j j jnk I L f j nβ β β β= − ⊗ + =ZZ Z ( 4.5) where jβ Z is the column of thj βZ k . Thus we shall have layers of checkerboards that persist for . Notice the similarity of equation (4.5) with equation (2.7). Thus replacing in equation (2.7) with we find that 0 k< < 1/ 4 ) k 2k 3 1/ 4.= Proposition 3.2. Let be an even integer. Then there exists a constant k such that for all the simple three-dimensional lattice equation (4.1), has sheets with vertical bands of stripes of checkerboards. Moreover, oscillator death occurs for . n 4 0> ( 40,k k∈ 4k k> Proof. On the subspace ,πθθ∏ the flow is defined by the solution of the decoupled equation ( ) ( )2 , , 1,..., ,j j jnk I L f j nβ β β β= − ⊗ + =.Z Z Z (4.6) where jβ Z is the column of thj βZ . Thus the entire system in the three-dimensional space is composed of sheets with vertical bands of stripes. Clearly, we see from equation (2.7) that this regime persists for . Proposition 3.2 follows with 0 1k< < / n 4 1/ 2.k = On ,θπθ∏ the flow is defined by the solution of the decoupled equations in (4.6). The solutions represent layers of sheets with horizontal band strips that persists for 0 1 3n /k 2.< < Proposition 3.3. Let be an even integer. Then there exists a constant k for which the simple three-dimensional lattice equation (4.1) has layers of horizontal sheets with horizontal bands that persist for all Moreover, oscillator death occurs for n 5 0> 5.k k>( 50, .k k∈ ) Proof. On the subspace πθθΘ the flow is defined by the solution of the decoupled equations ( ) ( )2 1 2 1 2 12 1, 2,...., / 2.j n j jk I L G j n− − −= − ⊗ + =W W W (4.7) Thus we shall have horizontal bands of synchronized sheets, like a “stack of sandwiches” that persists for 0 . Proposition 1.1 follows with k1k< < / 2 5 1/ 2= . On ,θππ∏ the flow is governed by equation (4.7) and we have sheets with horizontal bands. Proposition 3.4. Let be an even integer, then there exists a constant for which the simple three-dimensional lattice equation (4.1), has layers of horizontal sheets of checkerboards that persist for all n ( 0k σ > )0,k k σ∈ . Moreover, oscillator deaths occur for k k σ> . Proof. On the subspace πππ∏ , the flow is defined by the solution of the decoupled equations ( ) ( )2 1 2 1 2 -14 , 1, 2,...., / 2.j n j jk I L G j n− −= − ⊗ + =W W W (4.8) 74 SYNCHRONIZATION AND OSCILLATOR DEATH Thus the entire system in three-dimensional space is composed of layers of horizontal sheets of checkerboard. This dynamical regime persists for 0 1/k 4.< < Proposition 3.4 follows with . 6 1/ 4k = 5. Discussion In LDE’s on the Bravais lattices given in equations (2.1),(3.1) and (4.1) the coupling matrices ( ) ( ) { }( ), ,B k H k H k have 0 0λ = as an eigenvalue. Thus any solution to these equations that pass through a point in the generalized eigenspace corresponding respectively, shall be symmetrically synchronized. That is, they shall be in the set 0 , 1, 2, 3 v i vµ λ = 1, , , ,θθ θθθΘ Θ Θ n respectively. For an even number of oscillators in the bravais lattice, the coupling matrices have / 2 2λ = − as an eigenvalue. 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Mathematiques, 7: 375-421. Received 11 June 2002 Accepted 28 April 2003 75 Synchronization and Oscillator Death in Diffusively coupled lattice oscillators Adu A.M. Wasike Department of Mathematics, University of Nairobi, P.O.Box 30187, Nairobi Kenya, Email: aduwasike@yahoo.com.