The persistence of synchronization under environmental noise

The persistence of synchronization under environmental noise By Tom´ as Caraballo aand Peter E. Kloeden b aDepartamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain E-mail: carabal[email protected] bFachbereich Mathematik, Johann Wolfgang Goethe Universit¨at D-60054 Frankfurt am Main, Germany E-mail: kloe[email protected]ankfurt.de It is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise no matter how large the intensity of the noise provided asymptotically stable stationary stochastic solutions are used instead of asymptotically stable equilibria. Keywords: Synchronization, additive noise, random attractor, stationary stochastic process, one-sided Lipschitz dissipative condition 1. Introduction Synchronization of coupled systems is a very well known phenomenon in biology and physics, and also in the social sciences. A readable descriptive account of its diversity of occurence can be found in the recent book of Strogatz (2003), which contains an extensive list of references. The synchronization of coupled dissipative systems has been investigated mathematically in the case of autonomous systems by Afraimovich and Rodrigues (1998), Carvalho et al. (1998) and Rodrigues (1996), both for asymptotically stable equilibria and general attractors, such as chaotic attractors. Analogous results also hold for nonautonomous systems (Kloeden (2003)), but require a new concept of a nonautonomous attractor. In this note we investigate the effect of additive noise on the synchronization of coupled dissipative systems with asymptotically stable equilibria, which results in a coupled system of Ito stochastic differential equations. Such noise is often considered as modelling background environmental effects. We show that synchronization persists independently of noise intensity in terms of asymptotically stable stationary stochastic solutions rather than equilibria. Such asymptotically stable stationary stochastic solutions are a special case of random attractors (see Crauel & Flandoli (1994), Crauel et al. (1997), Schmalfuss (1992)) which are a probabilistic counterpart of the deterministic nonautonomous attractor used in Kloeden (2003). We present some required background material in Section 2 and then formulate the problem of synchronization of dissipative systems for both the deterministic and stochastic cases in Section 3. In Section 4 we show that the uncoupled systems with additive noise have random attractors consisting of single stationary stochastic Article submitted to Royal Society T EX Paper 2T. Caraballo & P.E. Kloeden processes and then establish the corresponding result for coupled systems in Section 5. Finally, the asymptotic behaviour of the coupled systems in the limit as the coupling parameter becomes arbitrarily large are presented in Section 6. 2. Background material We need some background notation and results. Let (Ω,F,P) be a probability space. Following Arnold (1998) a random dynamical system (RDS) (θ, φ) on Ω ×Rd consists of a metric dynamical system θon Ω and a cocycle mapping φ:R+×Ω×Rd →Rd. Essentially (and sufficient for our purposes here), θrepresents the driving noise process and φthe state space evolution of the system. For example, for a stochastic differential equation on Rdwith a two-sided scalar Wiener process Wt, i.e. defined for all t∈R,θis defined by θtω(·) = ω(t+·)−ω(·) on a canonical sample space Ω = C0(R,R) where, by definition, Wt(ω) := ω(t), t ∈R, and φis defined by φ(t, ω, x0) = Xx0 t(ω), the solution of the SDE starting at Xx0 0(ω) = x0. See the expositions in Arnold (1998) and Kloeden et al. (1999) for more details. A family b A={A(ω), ω ∈Ω}of nonempty measurable compact subsets A(ω) of Rdis called φ-invariant if φ(t, ω, A(ω)) = A(θtω) for all t≥0 and is called a random attractor if in addition it is pathwise pullback attracting in the sense that H∗ d(φ(t, θ−tω, D(θ−tω)), A(ω)) →0 as t→ −∞ for all suitable (i.e. in a given attracting universe as, for instance, in Kloeden et al. (1999)) families of b D={D(ω), ω ∈Ω}of nonempty measurable bounded subsets D(ω) of Rd. Here H∗ dis the Hausdorff semi-distance on Rd. The following result (cf. Arnold (1998), Kloeden et al. (1999), Schmalfuss (1992)) ensures the existence of a random attractor. Theorem 2.1. Let (θ, φ)be an RDS on Ω×Rd. If there exists a family b B= {B(ω), ω ∈Ω}of nonempty measurable compact subsets B(ω)of Rdand a Tb D,ω ≥ 0such that φ(t, θ−tω, D(θ−tω)) ⊂B(ω),∀t≥Tb D,ω for all families b D={D(ω), ω ∈Ω}in the given attracting universe, then the RDS (θ, φ)has a random attractor b A={A(ω), ω ∈Ω}with the component subsets defined for each ω∈Ωby A(ω) = \ s>0[ t≥s φ(t, θ−tω, B(θ−tω)). Note that if the random attractor consists of singleton sets, i.e A(ω) = {X∗(ω)} for some random variable X∗, then X∗ t(ω) := X∗(θtω) is a stationary stochastic process. We also need the following lemmata. Lemma 2.2. Let {xn}be a sequence in a complete metric space (X, d)such that every subsequence {xni}has a subsequence {xnij}converging to a common limit x∗. Then the sequence {xn}converges to x∗. Article submitted to Royal Society Persistence of synchronization under environmental noise 3 Proof. If not, then there would exist an η > 0 and a subsequence {xni}such that d(xni, x∗)≥ηfor all i, which is not possible because {xni}has a subsequence {xnij}converging to the x∗. Lemma 2.3. Let Wtbe a two-sided Wiener process, i.e. defined for all t∈R. Then the integrals νRt −∞ e−ν(t−s)dWt(ω)are pathwise uniformly bounded in ν > 0 on finite time intervals [T1, T2]of Rand the integrals Zt T1 e−ν(t−s)dWt(ω)→0as ν→ ∞ pathwise on finite time intervals [T1, T2]of R. Proof. Integrating by parts (see Mao (1997)), we have Zt −∞ e−ν(t−s)dWs(ω) = Wt(ω)−νZt −∞ e−ν(t−s)Ws(ω)ds. Then we use the sub-exponential growth of the paths of Wiener processes. For the second part we first notice that νZt T1 e−ν(t−s)Wt(ω)ds =Wt(ω)−e−ν(t−T1)Wt(ω) for any ω∈Ω. Integrating again by parts it follows Zt T1 e−ν(t−s)dWs(ω) = Wt(ω)−e−ν(t−T1)WT1(ω)−νZt T1 e−ν(t−s)Ws(ω)ds =e−ν(t−T1)(Wt(ω)−WT1(ω)) +νZt T1 e−ν(t−s)(Wt(ω)−Ws(ω)) ds from which the result follows. 3. Formulation of the problem Suppose we have two autonomous ordinary differential equations in Rd, dx dt =f(x),dy dt =g(y),(3.1) which are sufficiently regular to ensure the forwards existence and uniqueness of solutions and satisfy one-sided dissipative Lipschitz conditions hx1−x2, f(x1)−f(x2)i ≤ −L|x1−x2|2, hy1−y2, g(y1)−g(y2)i ≤ −L|y1−y2|2,(3.2) on Rdfor some L > 0, and thus have unique equilibria ¯xand ¯y, respectively, which are globally asymptotically stable (cf. Kloeden (2004)). Notice that the continuity of fand g, and the one-sided dissipative Lipschitz conditions (3.2) ensure the forwards existence and uniqueness of solutions to (3.1). Article submitted to Royal Society 4T. Caraballo & P.E. Kloeden Consider now the dissipatively coupled system dx dt =f(x) + ν(y−x),dy dt =g(y) + ν(x−y) (3.3) with ν > 0. It can be shown (see Afraimovich and Rodrigues (1998), and Carvalho et al. (1998)) that this also has a unique equilibrium (¯xν,¯yν), which is globally asymptotically stable. Moreover, (¯xν,¯yν)→(¯z, ¯z) as ν→ ∞, where ¯zis the unique globally asymptotically stable equilibrium of the “averaged” system dz dt =1 2(f(z) + g(z)) .(3.4) This phenomena is known as synchronization. Analogous results hold for more general autonomous attractors (cf. Afraimovich and Rodrigues (1998), Carvalho et al. (1998)) as well as for nonautonomous systems (Kloeden (2003)) with appropriately defined nonautonomous attractors. The aim of this note is to show that this synchronization effect is preserved under additive noise provided equilibria are replaced by stationary random solutions. Specifically, we consider two Ito stochastic differential equations in Rd, dXt=f(Xt)dt +α dW1 t, dYt=g(Yt)dt +β dW 2 t,(3.5) where α,β∈Rd +are constant vectors with no components equal to zero, W1 t, W2 tare independent two-sided scalar Wiener processes†, and f,gare as above, in particular, satisfying the one-sided dissipative Lipschitz conditions (3.2). We will show in the next section that each of these stochastic systems has a pathwise asymptotically stable random attractor consisting of a stationary random variable. For example, for linear drift terms, i.e the SDEs dXt=−Xtdt +α dW1 t, dYt=−Ytdt +β dW 2 t,(3.6) these random variables are given explicitly by ¯ Xt=αe−tZt −∞ esdW1 s,¯ Yt=βe−tZt −∞ esdW2 s.(3.7) The synchronized system corresponding to SDEs (3.5) reads dXt= (f(Xt) + ν(Yt−Xt)) dt +α dW1 t, dYt= (g(Yt) + ν(Xt−Yt)) dt +β dW 2 t.(3.8) It will be shown that this system is dissipative and has a unique stationary solution (¯ Xν t,¯ Yν t), which is pathwise globally asymptotically stable with (¯ Xν t,¯ Yν t)→(¯ Z∞ t,¯ Z∞ t) as ν→ ∞, pathwise on finite time intervals [T1, T2] of R, where ¯ Z∞ tis the unique pathwise globally asymptotically stable stationary solution of the “averaged” SDE dZt=1 2(f(Zt) + g(Zt)) dt +1 2α dW 1 t+1 2β dW2 t.(3.9) †alternatively, one could take α,βscalar valued and W1 t,W2 tvector valued Article submitted to Royal Society Persistence of synchronization under environmental noise 5 For example, for the linear SDEs (3.6) we have dZt=−Ztdt +1 2α dW1 t+1 2β dW2 t with ¯ Z∞ t=1 2e−tµαZt −∞ esdW1 s+βZt −∞ esdW2 s¶=1 2¡¯ Xt+¯ Yt¢, this averaged expression being due to the special linear structure. In addition to the one-sided Lipschitz dissipative condition on the functions f and gwe assume the following integrability condition: There exists m0>0 such that for any m∈(0, m0], and any continuous function u:R→Rdwith sub-exponential growth it follows Zt −∞ ems|f(u(s))|2ds < +∞,Zt −∞ ems|g(u(s))|2ds < +∞.(3.10) Without loss of generality, we can assume that L≤m0. 4. The uncoupled systems with additive noise We consider the first of the uncoupled equations in (3.5), dXt=f(Xt)dt +α dW 1 t.(4.1) Its solution paths are generally not differentiable, so in order to use the one-sided dissipative Lipschitz condition (3.2) we consider the difference Xt−¯ Xtwhere ¯ Xt is the Ornstein-Uhlenbeck stationary process (3.7) satisfying the first of the linear equations (3.6). This difference is pathwise differentiable since the paths Xtand ¯ Xt are continuous and satisfy the integral equation Xt−¯ Xt=X0−¯ X0+Zt 0¡f(Xs) + ¯ Xs¢ds, which, by the fundamental theorem of calculus, is thus equivalent to the differential expression d dt ¡Xt−¯ Xt¢=f(Xt) + ¯ Xt, from which it follows that d dt ¯¯Xt−¯ Xt¯¯2= 2 Xt−¯ Xt, f(Xt)−f(¯ Xt)®+ 2 Xt−¯ Xt, f(¯ Xt) + ¯ Xt)® ≤ −2L¯¯Xt−¯ Xt¯¯2+L¯¯Xt−¯ Xt¯¯2+1 L¯¯f(¯ Xt) + ¯ Xt)¯¯2 and thus ¯¯Xt−¯ Xt¯¯2≤¯¯Xt0−¯ Xt0¯¯2e−L(t−t0)+e−Lt LZt t0 eLs ¯¯f(¯ Xs) + ¯ Xs¯¯2ds. Article submitted to Royal Society 6T. Caraballo & P.E. Kloeden Pathwise pullback convergence (i.e. as t0→ −∞) gives pullback absorption (cf. Theorem 2.1) ¯¯Xt−¯ Xt¯¯2≤R2 X(θtω) := 1 + e−Lt LZt −∞ eLs ¯¯f(¯ X(θsω)) + ¯ X(θsω)¯¯2ds for all t≥Tb D(ω)for appropriate families b D(ω) of bounded sets {D(θtω), t ∈R}of initial conditions. The integrals here exist due to assumption (3.10) and the fact that OrnsteinUhlenbeck processes inherit the sub-exponential growth of their Wiener processes. Thus ¯¯Xt(ω)−¯ Xt(ω)¯¯≤RX(θtω),∀t≥Tb D(ω), and so |Xt(ω)| ≤ ¯¯¯ Xt(ω)¯¯+RX(θtω),∀t≥Tb D(ω), which means that this system has a random attractor b A={A(ω), ω ∈Ω}. But the difference of any two solutions satisfies the differential inequality d dt ¯¯X1 t−X2 t¯¯2≤ −2L¯¯X1 t−X2 t¯¯2, which means all solutions converge pathwise to each other and thus the random attractor sets are singleton sets A(ω) = {X∗(ω)}, i.e. the random attractor is formed by a stationary random process X∗ t(ω) = X∗(θtω) which pathwise attracts all other solutions. An analogous situation holds for the second equation of the uncoupled equations in (3.5). 5. The synchronized system with additive noise To show that the synchronized system (3.8) is strongly dissipative, we use the OU processes X∗,ν t=αe−νt Zt −∞ eνs dW 1 s, Y ∗,ν t=βe−νt Zt −∞ eνs dW 2 s,(5.1) which are the stationary solutions of the linear equations dXt=−νXtdt +α dW 1 t, dYt=−νYtdt +β dW 2 t.(5.2) The differences of the solutions of (3.8) and these stationary solutions are pathwise differentiable and satisfy the system of random differential equations d dt ¡Xt−X∗,ν t¢=f(Xt) + ν(Yt−Xt) + νX∗,ν t d dt ¡Yt−Y∗,ν t¢=g(Yt) + ν(Xt−Yt) + νY ∗,ν t, Article submitted to Royal Society Persistence of synchronization under environmental noise 7 which, with Uν t:= Xt−X∗,ν tand Vν t:= Yt−Y∗,ν t, is equivalent to d dtUν t=f(Uν t+X∗,ν t)+ν(Vν−Uν)+νY ∗,ν t,d dtVν=g(Yt)+ν(Uν−Vν)+νX∗,ν t. Thus 1 2 d dt ¡|Uν t|2+|Vν t|2¢=Uν t, f(Uν t+X∗,ν t)−f(X∗,ν t)®+Vν t, g(Vν t+Y∗,ν t)−g(Y∗,ν t)® +Uν t, f(X∗,ν t) + νY ∗,ν t®+Vν t, g(Y∗,ν t) + νX∗,ν t® −νhUν t−Vν t, Uν t−Vν ti ≤ −L¡|Uν t|2+|Vν t|2¢+L 2¡|Uν t|2+|Vν t|2¢ +2 L¯¯f(X∗,ν t) + νY ∗,ν t¯¯2+2 L¯¯g(Y∗,ν t) + νX∗,ν t¯¯2, and hence d dt ¡|Uν t|2+|Vν t|2¢≤ −L¡|Uν t|2+|Vν t|2¢+4 L¯¯f(X∗,ν t) + νY ∗,ν t¯¯2+4 L¯¯g(Y∗,ν t) + νX∗,ν t¯¯2. This means that |Uν t(ω)|2+|Vν t(ω)|2is pathwise absorbed by the family b Bν= {Bν(ω), ω ∈Ω}of closed balls in R2dcentered on the origin and of radius Rν(ω), where R2 ν(ω) is defined by 1+ 4 LZ0 −∞ eLs ³|f(X∗,ν (θsω) + νY ∗,ν (θsω)|2ds +|g(Y∗,ν (θsω) + νX∗,ν (θsω)|2´ds. Hence, by Theorem 2.1, the synchronized system has a random attractor b Aν= {Aν(ω), ω ∈Ω}with Aν(ω)⊂Bν(ω). But the difference (∆Xt,∆Yt) = (X1 t− X2 t, Y 1 t−Y2 t) of any pair of solutions satisfies the system of random differential equations d dt∆Xt=f(X1 t)−f(X2 t)+ν(∆Yt−∆Xt),d dt∆Yt=g(Y1 t)−g(Y2 t)+ν(∆Xt−∆Yt), so 1 2 d dt ¡|∆Xt|2+|∆Yt|2¢=∆Xt, f(X1 t)−f(X2 t)®+∆Yt, g(Y1 t)−g(Y2 t)® −νh∆Xt−∆Yt,∆Xt−∆Yti ≤ −L¡|∆Xt|2+|∆Yt|2¢, from which we obtain |∆Xt(ω)|2+|∆Yt(ω)|2≤¡|∆X0(ω)|2+|∆Y0(ω)|2¢e−2Lt, which means all solutions converge pathwise to each other as t→ ∞. Thus the random attractor consists of singleton sets formed by an ordered pair of stationary processes ( ¯ Xν t(ω),¯ Yν t(ω)). Article submitted to Royal Society 8T. Caraballo & P.E. Kloeden 6. The synchronized stationary solutions as ν→ ∞ Lemma 6.1. ¯ Xν t(ω)−¯ Yν t(ω)→0as ν→ ∞ pathwise on any bounded time interval [T1, T2]of R. Proof. Subtracting the second from the first equation of (3.8) gives d¡¯ Xν t−¯ Yν t¢=¡−2ν¡¯ Xν t−¯ Yν t¢+f(¯ Xν t)−g(¯ Yν t)¢dt +α dW1 t−β dW 2 t, or, with Dν t=¯ Xν t−¯ Yν t, d¡Dν te2νt¢=e2νt ¡f(¯ Xν t)−g(¯ Yν t)¢dt +αe2νt dW1 t−βe2νt dW 2 t, so pathwise |Dν t| ≤ e−2νT1|Dν T1|+Zt T1 e−2ν(t−s)¡¯¯f(¯ Xν s)¯¯+¯¯g(¯ Yν s)¯¯¢ds +|α|¯¯¯¯Zt T1 e−2ν(t−s)dW1 t¯¯¯¯+|β|¯¯¯¯Zt T1 e−2ν(t−s)dW2 t¯¯¯¯.(6.1) By Lemma 2.3 we see that the radius Rν(θtω) is pathwise uniformly bounded on each bounded time interval [T1, T2], so |Xν t(ω)|,|Yν t(ω))|and |Dν 0(ω))|are pathwise uniformly bounded on each bounded time interval [T1, T2]. Then by Lemma 2.3 and condition (3.10) we see that all of the integrals in (6.1) converge to zero as ν→ ∞ pathwise on the bounded time interval [T1, T2]. Theorem 6.2. (¯ Xνn t,¯ Yνn t)→(Z∞ t, Z∞ t)pathwise uniformly on bounded time intervals [T1, T2]for any sequence νn→ ∞, where Z∞ tis the stationary stochastic solution of the averaged SDE dZt=1 2(f(Zt) + g(Zt)) dt +1 2αdW 1 t+1 2βdW2 t.(6.2) Proof. Define Zν t:= 1 2¡¯ Xν t+¯ Yν t¢∀t∈R and observe that Zν tsatisfies the equation dZν t=1 2¡f(¯ Xν t) + g(¯ Yν t)¢dt +1 2α dW1 t+1 2β dW2 t. Also define ¯ Zt:= 1 2¡¯ Xt+¯ Yt¢∀t∈R where ¯ Xtand ¯ Ytare the OU processes satisfying the linear SDEs (3.6). The difference Zν t−¯ Ztis pathwise continuous and thus by Lemma 2.3 and the definitions of the respective absorbing sets, equi-bounded w.r.t. ν > 0 on the bounded time interval [T1, T2]. Moreover Zν t−¯ Ztis pathwise differentiable and satisfies the random differential expression d dt ¡Zν t−¯ Zt¢=1 2f(¯ Xν t) + 1 2g(¯ Yν t) + 1 2¯ Xt+1 2¯ Yt, Article submitted to Royal Society Persistence of synchronization under environmental noise 9 Since ¯¯¯¯ d dt ¡Zν t(ω)−¯ Zt(ω)¢¯¯¯¯≤1 2¯¯f(¯ Xν t(ω))¯¯+1 2¯¯g(¯ Yν t(ω))¯¯+1 2¯¯¯ Xt(ω)¯¯+1 2¯¯¯ Yt(ω)¯¯ ≤MT1,T2(ω)<∞ by Lemma 2.3, we can use the Ascoli Theorem to conclude that for any sequence νn → ∞, there is a (possibly) random subsequence νnj(ω)→ ∞ such that Zνnj t(ω)− ¯ Zt(ω)→Z∞ t(ω)−¯ Zt(ω) as j→ ∞, and thus that Zνnj t(ω)→Z∞ t(ω) as j→ ∞. Now Zνnj t(ω)−¯ Yνnj t(ω) = 1 2³¯ Xνnj t(ω)−¯ Yνnj t(ω)´→0, Zνnj t(ω)−¯ Xνnj t(ω) = 1 2³¯ Yνnj t(ω)−¯ Xνnj t(ω)´→0 as νnj→ ∞, so ¯ Xνnj t(ω)=2Zνnj t(ω)−¯ Yνnj t(ω)→Z∞ t(ω), ¯ Yνnj t(ω)=2Zνnj t(ω)−¯ Xνnj t(ω)→Z∞ t(ω) as νnj→ ∞. Moreover, Zν t−¯ Zt=Zν t−¯ Zt+1 2Zt T1 f(¯ Xν s)ds +1 2Zt T1 g(¯ Yν s)ds +1 2Zt T1 ¯ Xsds +1 2Zt T1 ¯ Ysds which converges pathwise to Z∞ t=Z∞ T1+1 2Zt T1 f(Z∞ s)ds +1 2Zt T1 g(Z∞ s)ds +¯ Zt−¯ ZT1+1 2Zt T1 ¯ Xsds +1 2Zt T1 ¯ Ysds =Z∞ T1+1 2Zt T1 f(Z∞ s)ds +1 2Zt T1 g(Z∞ s)ds +1 2αZt T1 dW1 s+1 2βZt T1 dW2 s, on the interval [T1, T2], so Z∞ tis a solution of the SDE (6.2) for all t∈R. The drift of this SDE satisfies the strongly dissipative one-sided Lipschitz condition (3.2), so it has a random attractor consisting of a singleton set formed by a stationary stochastic process which thus must be equal to Z∞ t. Finally, we note that pathwise all possible subsequences here have the same limit, so by Lemma 2.2 every full sequence Zνn tactually converges to Z∞ tas νn→ ∞. As a straightforward consequence of the arguments in the previous proof we have Article submitted to Royal Society