Synchronization of non-identical extended chaotic systems

This art icle was downloaded by: [ Pedro Garcia] On: 30 Novem ber 2011, At : 15: 26 Publisher: Taylor & Francis I nform a Lt d Regist ered in England and Wales Regist ered Num ber: 1072954 Regist ered office: Mort im er House, 37- 41 Mort im er St reet , London W1T 3JH, UK Applicable Analysis Publicat ion det ails, including inst ruct ions f or aut hors and subscript ion inf ormat ion: ht t p: / / www. t andf online. com/ loi/ gapa20 Synchronization of non-identical extended chaotic systems A. Acost a a , P. García b & H. Leiva c a Depart ament o de Mat emát ica Aplicada, Facult ad de Ingeniería, Universidad Cent ral de Venezuela, Caracas, Venezuela b Laborat orio de Sist emas Complej os, Depart ament o de Física Aplicada, Facult ad de Ingeniería, Universidad Cent ral de Venezuela, Caracas, Venezuela c Escuela de Mat emát ica, Facult ad de Ciencias, Universidad de Los Andes, Mérida, Venezuela Available online: 30 Nov 2011 To cite this article: A. Acost a, P. García & H. Leiva (2011): Synchronizat ion of non-ident ical ext ended chaot ic syst ems, Applicable Analysis, DOI: 10. 1080/ 00036811. 2011. 635654 To link to this article: ht t p: / / dx. doi. org/ 10. 1080/ 00036811. 2011. 635654 PLEASE SCROLL DOWN FOR ARTI CLE Full t erm s and condit ions of use: ht t p: / / www.t andfonline.com / page/ t erm s- andcondit ions This art icle m ay be used for research, t eaching, and privat e st udy purposes. Any subst ant ial or syst em at ic reproduct ion, redist ribut ion, reselling, loan, sub- licensing, syst em at ic supply, or dist ribut ion in any form t o anyone is expressly forbidden. The publisher does not give any warrant y express or im plied or m ake any represent at ion t hat t he cont ent s will be com plet e or accurat e or up t o dat e. The accuracy of any inst ruct ions, form ulae, and drug doses should be independent ly verified wit h prim ary sources. The publisher shall not be liable for any loss, act ions, claim s, proceedings, dem and, or cost s or dam ages what soever or howsoever caused arising direct ly or indirect ly in connect ion wit h or arising out of t he use of t his m at erial. Applicable Analysis 2011, 1–12, iFirst Synchronization of non-identical extended chaotic systems A. Acostaa, P. Garcı́ab* and H. Leivac a Departamento de Matemática Aplicada, Facultad de Ingenierı´a, Universidad Central de Venezuela, Caracas, Venezuela; bLaboratorio de Sistemas Complejos, Departamento de Fı´sica Aplicada, Facultad de Ingenierı´a, Universidad Central de Venezuela, Caracas, Venezuela; cEscuela de Matemática, Facultad de Ciencias, Universidad de Los Andes, Me´rida, Venezuela Downloaded by [Pedro Garcia] at 15:26 30 November 2011 Communicated by S. Leonardi (Received 18 May 2011; final version received 24 October 2011) In this article a technique to achieve synchronization in spatially extended systems is introduced. The basic idea behind this method is to map a system of partial differential equations (PDEs) into a high-dimensional space where the representation of this PDE is an ordinary differential equation. By using semi-group theory, we are able to find conditions that ensure the synchronization of two systems of non-identical reaction–diffusion equations with a master–slave coupling. Keywords: chaos synchronization; reaction-diffusion equations; coupled systems AMS Subject Classifications: 70G60; 74H65; 35K57 1. Introduction Synchronization in chaotic finite-dimensional systems is a subject that has been having great attention since 1983 with the seminal article of Fujisaka and Yamada [1]. However from 1990, with the work of Pecora and Carroll [2], the interest for the phenomenon has increased enormously and different techniques, based on some well establish theories (e.g. Lyapunov functions [3,4], exponential dichotomies theory [5], are some examples) have been applied. On the other hand, for spatially extended systems the synchronization is a very complicated problem and several techniques, in order to achieve this, have been implemented. Particularly, the scheme based on linear feedback control method has been very useful for synchronizing identical [6–11] and non-identical [12–16] infinite-dimensional chaotic systems. In order to study synchronization for non-identical spatially extended chaotic systems, we follow an approach based on semi-group theory. The idea behind our scheme is very simple, we consider an infinite-dimensional system and represent it in an abstract setting which corresponds to an ordinary differential equation in *Corresponding author. Email: pedro@fisica.ciens.ucv.ve ISSN 0003–6811 print/ISSN 1563–504X online ß 2011 Taylor & Francis http://dx.doi.org/10.1080/00036811.2011.635654 http://www.tandfonline.com 2 A. Acosta et al. a Hilbert space. Specifically, we consider a system of two vector reaction–diffusion equations linearly coupled using a master–slave scheme. Although, the setting is very formal, we explicitly exhibit the relation involving the parameters in the problem, and this allows us to estimate the intensity of the coupling as a function of the difference between the initial condition and the control parameters of the systems. 2. Setting of the problem We consider the system of two vector-coupled reaction–diffusion equations ð1Þ ð2Þ utð1Þ ¼ d1 uð1Þ xx þ f1 ð, u , u Þ Downloaded by [Pedro Garcia] at 15:26 30 November 2011 ð2Þ ð1Þ ð2Þ uð2Þ t ¼ d2 uxx þ f2 ð, u , u Þ, ð1Þ  vð1Þ , vð2Þ Þ þ ðuð1Þ  vð1Þ Þ vð1Þ t ¼ d1 vxx þ f1 ð, ð2Þ  vð1Þ , vð2Þ Þ þ ðuð2Þ  vð2Þ Þ, vð2Þ t ¼ d2 vxx þ f2 ð, ð1Þ ð2Þ with t > 0, 0 < x < l, homogeneous Dirichlet conditions: u(i)(0, t) ¼ v(i)(0, t) ¼ 0, u(i)(l, t) ¼ v(i)(l, t) ¼ 0, i ¼ 1, 2 and where , which is the coupling intensity, is a real positive number, di, i ¼ 1, 2, are positive constants and fi: Rm  R  R ! R, i ¼ 1, 2, are continuous functions. In a compact form, Equations (1) and (2), can be written as ut ¼ Duxx þ Fð, uÞ and  vÞ þ ðu  vÞ, vt ¼ Dvxx þ Fð, ð3Þ ð4Þ where u ¼ (u(1), u(2))T, v ¼ (v(1), v(2))T, D is a diagonal matrix with elements D11 ¼ d1 and D22 ¼ d2 and F(, u) ¼ ( f1(, u), f2(, u))T. The boundary conditions are expressed as u(0, t) ¼ v(0, t) ¼ 0, and u(l, t) ¼ v(l, t) ¼ 0. Notice that, without considering the coupling term, the systems (3)–(4) differs only in the parameter . The function F, because its components are continuous, is continuous. Also, we assume the following hypothesis: H1. Given a ball Br(0), with radius r and centre 0 in R2, there exists a constant L ¼ L(r, ) > 0 such that jFð, u1 Þ  Fð, u2 Þj  Lju1  u2 j for all u1 , u2 2 Br ð0Þ: H2. There exists K > 0, such that  uÞj  Kj  jjuj  jFð, uÞ  Fð, for all ,  2 Rm , u 2 R2 : In addition to the previous hypothesis it is important to remark that throughout this work we assume  and  as constant vectors. Next, we rewrite system (3)–(4) in terms of the difference u  v by using a solution of Equation (3) (master equation). Let u0(x, t) be a bounded solution of Equation (3) that satisfies u0(0, t) ¼ u0(l, t) ¼ 0, and consider it as an input in Equation (4) (slave equation). Applicable Analysis 3 Now consider the transformation z ¼ u  v, with u ¼ u0, i.e. z ¼ u0 ðx, tÞ  v: ð5Þ If v is a solution of the slave equation with input u0, then the transformation (5) applied to this equation yields the following equation:  u0  zÞ zt ¼ Dzxx  z þ Fð, u0 Þ  Fð, ð6Þ and z satisfies the boundary conditions zð0, tÞ ¼ zðl, tÞ ¼ 0 t 4 0: ð7Þ Downloaded by [Pedro Garcia] at 15:26 30 November 2011 We will obtain synchronization through the study of the problem (6)–(7). It means existence of solutions with v being arbitrarily close to u0 as t tends to þ1. 3. Abstract formulation of the problem In this section, by choosing an appropriate space, we will set our problem as an abstract ordinary differential equation. Let X ¼ L2((0, l ), R2) and consider the linear unbounded operator A: D(A)  X ! X defined by   d2 A ¼ D 2 þ I , ð8Þ dx where I is the identity operator, and DðAÞ ¼ H10 ðð0, l Þ, R2 Þ \ H2 ðð0, l Þ, R2 Þ: H2((0, l), R2) being the Sobolev space W2, 2((0, l), R2) and H10 ðð0, l Þ, R2 Þ the closure of the set C10 ðð0, l Þ, R2 Þ, the C2-functions with compact support, in the norm of W2,2((0, l), R2). The spectrum (A), of the operator A, consists of just eigenvalues n,i ¼ di(n/l)2 þ , n ¼ 1, 2, . . . , i ¼ 1, 2. We order the set of eigenvalues {n,i} according to the sequence 0 < 1 < 2 <   ; where 1 ¼ min{d1, d2}(/l)2 þ . For each n the corresponding eigenspace has dimension n, where n 2 {1, 2}. In this framework there exists a complete orthonormal set fg1 n¼1 of eigenvectors of A, where each element in this set has the form either ð2l Þ1=2 ðsinðjl xÞ, 0ÞT or ð2l Þ1=2 ð0, sinðjl xÞÞT with j 2 {1, 2, . . .}. Therefore for all  2 D(A) we have the representation A ¼ 1 X h, n in , n¼1 where h, i is the inner product in X. Also, A generates an analytic semigroup {eAt} given by eAt  ¼ 1 X n¼1 en t h, n in : 4 A. Acosta et al. In order to study the non-linear part of the abstract differential equation corresponding to the problem (3)–(4), we need to consider the fractional power space X ,  0 , associated with A. X is defined as X ¼ D(A ) and, because X is a Hilbert space, it has the structure of Hilbert space. Moreover, 1 X n h, n in , A ¼ n¼1 and the norm induced by the inner product on X is given by 1 X kk2 :¼ n jh, n ij2 : n¼1 ð9Þ Downloaded by [Pedro Garcia] at 15:26 30 November 2011 The next proposition contains estimates relating the semi-group {eAt} with the norms kk and kk. PROPOSITION 1 For each  2 X we have the following estimates: keAt k  e1 t kk , where M ¼ ð e Þ Proof =2 keAt k  Mt and t  0:  =2  21 t e kk, t 4 0, > 0. From the above notation, for  2 X we have * +2 1 1   X X     keAt k2 ¼ ej t , j j , i  i    j¼1 i¼1 ¼ 1 X i¼1  2 i ei t h, i i  e21 t 1 X i¼1 i jh, i ij2 ¼ e21 t kk2 : Therefore, keAt k  e1 t kk : The second inequality follows from the fact that, for > 0, the sequence of functions defined by fn ðtÞ ¼ ðn tÞ en t , n ¼ 1, 2, . . . ; t  0, satisfies the estimate fn(t)  ( /e) uniformly in n and t. In fact, keAt k2 ¼ 1 X i¼1 ¼ t i e2i t jh, i ij2 1 X i¼1 ei t fi ðtÞjh, i ij2  t e1 t 2  M t e 1 X i¼1 1 t fi ðtÞjh, i ij2 k k 2 : 5 Applicable Analysis Therefore, keAt k  Mt  =2  21 t e kk: g From Theorem 1.6.1 in [17] we obtain, for 1/2 < < 1, that there exists ~ 2 Cðð0, l Þ, R2 Þ such that  ¼  ~ almost everywhere and the operators i1:  X ! C((0, l), R2) and i2: X ! X defined by ~ i1 ðÞ ¼ , i2 ðÞ ¼  are continuous. Hence for the inclusions Downloaded by [Pedro Garcia] at 15:26 30 November 2011 X  Cðð0, l Þ, R2 Þ and X  X there are constants C > 1 and R > 0, such that sup jðxÞj  Ckk x2ð0,l Þ and kk  Rkk , 2X ð10Þ Now, we associate to the system (3)–(4) the following ordinary differential equation with initial condition on the space X: _ þ A ¼ Fe ð, ,  , tÞ  ðt0 Þ ¼ 0 , t  t0 4 0, ð11Þ where Fe: Rm  Rm  X  [0, þ1) ! X is given by  , tÞðxÞ :¼ Fð, u0 ðx, tÞÞ  Fð,  u0 ðx, tÞ  ðxÞÞ: Fe ð, , ð12Þ From now on, we will suppose that 1/2 < < 1. Let us assume that u0(, t) 2 X , t > 0, and it is uniformly bounded, i.e. there exists an N > 0 such that ku0 ð, tÞk  N for all t 4 0: ð13Þ The following lemma contains important estimates concerning to Fe, and shows that Equation (11) is well posed in X . LEMMA 1 (a) Given a ball with radius  and centre zero in X , there exists a constant L 4 0 such that for 1 , 2 2 B ð0Þ  1  2 k  1 , tÞ  Fe ð, ,  2 , tÞk  Lk kFe ð, , ð14Þ (b) There exists a constant K 4 0 such that   j,  0, tÞk  Kj  kFe ð, , for all ,  2 Rm , t > 0. Proof  4 0 such that (a) For H1, given r > 0 there exist a constant L ¼ Lðr, Þ  u2 Þj  Lju1  u2 j  u1 Þ  Fð, jFð, for all u1 , u2 2 B ð0Þ: ð15Þ 6 A. Acosta et al. Choosing r ¼ C(N þ ), with C as in (10) and N as in (13), we get  1 , tÞðxÞ  Fe ð, ,  2 , tÞðxÞj D ¼ jFe ð, ,  u0 ðx, tÞ  2 ðxÞÞ  Fð,  u0 ðx, tÞ  1 ðxÞÞj ¼ jFð,   LðCðN þ Þ, Þj 1 ðxÞ  2 ðxÞj: Therefore, if 1 , 2 2 B ð0Þ, then  1 , tÞ  Fe ð, ,  2 , tÞk  LðCðN þ Þ, Þk  kFe ð, , 1  2 k: Finally, the second estimate in (10) implies that  1  2 k  1 , tÞ  Fe ð, ,  2 , tÞk  Lk kFe ð, , Downloaded by [Pedro Garcia] at 15:26 30 November 2011  with L ¼ RLðCðN þ Þ, Þ: (b) For H2,  0, tÞðxÞj ¼ jFð, u0 ðx, tÞÞ  Fð,  u0 ðx, tÞÞj  Kj  jju  0 ðx, tÞj: jFe ð, , Now, by using (10) and (13), respectively, we get  0, tÞðxÞj  Kj  jCku   jFe ð, , 0 ð, tÞk  Kj  jCN: Therefore, Z l 0 and  0, tÞj2 dx  lK2 C2 N2 j  j  2 jFe ð, ,   j,  0, tÞk  Kj  kFe ð, , where K ¼ KCNl1=2 . g From Lemma 1 of this article and Lemma 3.3.2 of [17], for all T > t0 we have the following: a continuous function (): (t0, T ) ! X is solution of the integral equation Zt  ðsÞ, sÞds, t 2 ðt0 , TÞ, eAðtsÞ Fe ð, , ð16Þ ðtÞ ¼ eAðtt0 Þ 0 þ t0 if and only if () is a solution of (11). 4. Existence of bounded solutions Consider Cb([0, 1), X ), the space of the bounded and continuous functions defined on the interval [0, 1) and taking values in X . Cb([0, 1), X ) is a Banach space with the supremum norm kkb :¼ supfkðtÞk : t  0g,  2 Cb ð½0, 1Þ, X Þ: A ball of radius  > 0 and centre zero in this space is given by Bb ¼ f 2 Cb ð½0, 1Þ, X Þ : kkb  g: Now, we are ready to establish the main result of this article. 7 Applicable Analysis  K the constants in the Lemma 1. If the THEOREM 1 Let  be a positive number and L, following estimate holds:    1 =2  þ Kj   j  M L 1 , ð17Þ 4 ð1  =2Þ  2 where M ¼ ( /e) /2 and is the well-known gamma function, then Equation (11) admits one and only one bounded solution  2 Bb . Proof Let S be the set of all functions  2 Bb . With the norm that has been defined previously, S is a complete metric space. Next, inspired by (17), we define for  2 S Zt  ðsÞ, sÞds: eAðtsÞ Fe ð, , ð18Þ TðÞðtÞ ¼ Downloaded by [Pedro Garcia] at 15:26 30 November 2011 0 We have that T() is continuous. Now, the estimates giving in Lemma 1 imply that  ðsÞ,sÞ  Fe ð, ,  0,sÞ þ Fe ð, ,  0, sÞ  L kkb þ K j   j:  ðsÞ, sÞ ¼ Fe ð, , Fe ð, , Therefore, using the second estimate in Proposition 1 and a change of variables, its follows that Zt  ðsÞ, sÞ ds eAðtsÞ Fe ð, , TðÞðtÞ  0 Zt 1   Mðt  sÞ2 e 2 ðtsÞ L kkb þ K j   j ds  0   ¼ M L kkb þ K j   j Z 1 2t Z0  1   M L kkb þ K j   j es s2 ds es s2 ds 0     2 12   ¼ M Lkkb þ Kj   j 1 1 2  12    þ K j   j 2  M L 1 : 1 2 Hence, kT()(t)k   and T(S )  S. Now, we are going to establish that T is a contraction on S. For all 1, 2, t > 0, we have D :¼ kTð2 ÞðtÞ  Tð1 ÞðtÞk Zt  2 ðsÞ, sÞ  Fe ð, ,  1 ðsÞ, sÞÞk ds keAðtsÞ ðFe ð, ,  0 Zt 1  2 ðsÞ, sÞ  Fe ð, ,  1 ðsÞ, sÞk ds Mðt  sÞ2 e 2 ðtsÞ kFe ð, ,  0 Zt 1  ML ðt  sÞ2 e 2 ðtsÞ k2 ðsÞ  1 ðsÞkds 0  12 2  ML 1 1 2 k2  1 kb : 8 A. Acosta et al.  2 Þ12 ð1  Þ 5 1 and this implies that T is a Now, from (17) we have MLð 1 2 contraction. The above discussion allows us to conclude that T has a unique fixed ~ 2 S, point  Zt ~ ~  ðsÞ, ðtÞ ¼ eAðtsÞ Fe ð, , sÞds: 0 ~ is the solution defined on (0, 1), of the initial value problem with Moreover,  g t0 ¼ 0 and 0 ¼ 0, given for Equation (11). Downloaded by [Pedro Garcia] at 15:26 30 November 2011 When the initial condition in Equation (11) is not zero, a similar result could be obtained.  K the constants in the Lemma 1. If the THEOREM 2 Let  be a positive number and L, following estimate holds    1 =2  þ k0 kb þ Kj   j  M L 1 , ð19Þ 4 2 ð1  =2Þ  2 where M ¼ ( /e) /2, then the Equation (19) admits one and only one bounded solution ~  0 2 Bb .  2 Bb . Moreover, for all t > 0, ðtÞ  Notice that, if  ¼  and (0 ¼ 0, we have two identical systems with the same initial condition, in this case the system starts synchronizing and remains in that state. If  6¼  and 0 6¼ 0, then for each  > 0 there exists a critical value c such that the systems synchronize for each  > c. Also, it is important to mention that in the case of identical systems, inequalities (17) and (19) imply local synchronization, i.e. if the solutions for t ¼ 0 are sufficiently close, they converge to each other as t goes to infinity. However, the inclusion of the parameter  makes our result a bit more general than local synchronization. 5. Numerical results In order to show the performance of our technique, we use the Gray–Scott cubic autocatalysis model [18] @u1 @ 2 u1 ¼ d1 2  u1 u22 þ að1  u1 Þ @t @x @u2 @ 2 u2 ¼ d2 2 þ u1 u22  ða þ bÞu2 : @t @x This reaction–diffusion system correspond to two irreversible chemical reactions and exhibits mixed mode spatiotemporal chaos. Here, b, the dimensionless rate constant of the second reaction and a the dimensionless feed rate and di, i ¼ 1, 2, are the diffusion coefficients. As in reference [7], the constants associated with the system are chosen as: a ¼ 0.028, b ¼ 0.053, d2 ¼ 105, d1 ¼ 2d2 and l ¼ 2.5. Figure 1 shows the evolution of the component u1 of the Gray–Scott system. It displays, with the set of parameters given, the spatiotemporal chaotic behaviour associated to the system. 9 Applicable Analysis u1 10 000 8000 t 6000 4000 Downloaded by [Pedro Garcia] at 15:26 30 November 2011 2000 0 0.0 0.5 1.0 1.5 2.0 2.5 x Figure 1. Contour plot of component u1, of the Gray–Scott system. In our case, we have two of this reaction–diffusion systems coupled by mean of a master–slave scheme: @u1 @ 2 u1 ¼ d1 2  u1 u22 þ að1  u1 Þ @t @x @u2 @ 2 u2 ¼ d2 2 þ u1 u22  ða þ bÞu2 @t @x @v1 @ 2 v1   v1 Þ þ ðu1  v1 Þ ¼ d1 2  v1 v22 þ að1 @t @x @v2 @ 2 v2  2 þ ðu2  v2 Þ: ¼ d2 2 þ v1 v22  ða þ bÞv @t @x The vector field F(a, b, u1, u2) is given by ðu1 u22 þ að1  u1 Þ, u1 u22  ða þ bÞu2 Þ, and satisfies the condition H1 with L ¼ 2(4r2 þ a þ b), and satisfies a ¼ a condition H2 with K ¼ 1. On the other hand, the constants C and R appearing in (10) can be chosen as C¼ 1 1 =2 1=2 l , R¼ 1 1 =2 : Now, we realize a numerical implementation to illustrate the second theorem. We use the previous values for the constants a and b, a ¼ 0.028, b ¼ 0:049, ¼ 0.75 and ,  chosen according to (17) as 0.24 and 5  103, respectively. Also, according 10 A. Acosta et al. 0.1 e(t) 10–4 10–7 10–10 10–13 0 200 400 600 800 1000 Downloaded by [Pedro Garcia] at 15:26 30 November 2011 t Figure 2. Semi-logarithmic plot of the global error of synchronization versus the time. The solid line shows the before-mentioned error for different initial conditions and non-identical systems with given parameters and the dotted line shows the same results in the case of identical systems. to the expression for L and K given in the Lemma 1, we obtain ! pffiffiffi 1 3 2 N 2 ðN þ Þ þ a þ b , K ¼ L ¼ : 2 l  12 1 1 The initial conditions are given by  x , l  u2 ðx, 0Þ ¼ sin x , l u1 ðx, 0Þ ¼ sin 2 2 v1 ðx, 0Þ ¼ ðe10ðxl=3Þ þ e1000ðx2l=3Þ Þ sin  x , l 2 1  v2 ðx, 0Þ ¼ e10ðxl=2Þ sin x : 2 l Figure 2 shows a semi-logarithmic plot of the global error of synchronization, which is defined by ( Z )1=2 2 1 lX eðtÞ ¼ ð20Þ ðui ðx, tÞ  vi ðx, tÞÞ2 dx l 0 i¼1 versus time t. Here we show that the error (20) behaves in such a way that it cannot be greater than the value assign to . As can be seen, the synchronization error becomes very small in an exponential way. The solid line shows the above-mentioned error for different initial conditions and non-identical systems with given parameters and the dotted line shows same results in the case of identical systems. 6. Concluding remarks As a final comment, we want remark that in contrast with the works in [6–15], in this work the synchronization is achieved from an analytic result instead of numeric Applicable Analysis 11 search of the conditions for synchronization. Although the scheme is illustrated with a coupled reaction–diffusion equations, the methodology is quite general and seems to be useful in studying synchronization of other types of extended chaotic systems. Acknowledgements This work was partially supported by Consejo de Desarrollo Cientı́fico y Humanı́stico de la Universidad Central de Venezuela and CDCHT-ULA, Project C1667-09-05-AA. We also thank the second referee as we believe that their suggestions have improved the presentation and had suggested interesting lines of future research. Downloaded by [Pedro Garcia] at 15:26 30 November 2011 References [1] H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Prog. Theor. Phys. 69 (1983), pp. 32–47. [2] L.M. Pecora and T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990), pp. 821–824. [3] C.W. Wu and L.O. Chua, A unified framework for synchronization and control of dynamical systems, Int. J. Bifurcation Chaos 4(4) (1994), pp. 979–998. [4] H.M. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal. 62(3–4) (1996), pp. 263–296. [5] A. Acosta and P. 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