Synchronization in Networks of Coupled Harmonic Oscillators with Stochastic Perturbation and Time Delays

OutlineIntroduction Backgrounds Problem formulationMain result Synchronization of coupled harmonic oscillators Methods of proofNumerical examples

Synchronized oscillatorsCellular clocks in the brainPacemaker cells in the heartPedestrians on a bridgeElectric circuitsLaser arraysOscillating chemical reactionsBubbly fluidsNeutrino oscillationsSynchronous firings of male fireflies

Kuramoto modelAll-to-all interactionIntroduced by Kuramoto in 1975 as a toy model of synchronization

We want to study synchronization conditions for coupled harmonic oscillators over general directed topologies with noise perturbation and communication time delays.

Basic definitions For a matrix A, let ||A||=sup{ ||Ax||: ||x||=1}. ||.|| is the Euclidean norm. Let G=(V, E, A) be a weighted digraph with vertex set V={1, 2,..., n} and edge set E. An edge (j, i) E if and only if the agent j can send information to the agent i directly. The in-degree neighborhood of the agent i : Ni ={ j V : (j, i) E}. A=(aij) Rnn is the weighted adjacency matrix of G. aij >0 if and only if j Ni. D=diag(d1,..., dn) with di=|Ni|. The Laplacian matrix L=(lij) =D-A.

Our model Consider n coupled harmonic oscillators connected by dampers and each attached to fixed supports by identical springs with spring constant k. The dynamical system is described as xi+kxi+jNiaij(xi-xj)=0 for i=1,, n where xi denotes the position of the ith oscillator, k is a positive gain, and aij characterizes interaction between oscillators i and j.

Here, we study a leader-follower version of the above system. Communication time delay and stochastic noises during the propagation of information from oscillator to oscillator are introduced. xi(t)+kxi(t)+jNiaij(xi(t-s)-xj(t-s))+bi(xi(t-s)-x0(t-s))+ [jNipij(xi(t-s)-xj(t-s))+qi(xi(t-s)-x0(t-s))]wi(t) = 0 for i=1,, n (1) x0(t)+kx0(t)=0, (2) where s is the time delay and x0 is the position of the virtual leader, labeled as oscillator 0.

Let B=diag(b1,, bn) be a diagonal matrix with nonnegative diagonal elements and bi>0 if and only if 0Ni. W(t):=(w1(t),,wn(t))T is an n dimensional standard Brownian motion. Let Ap=(pij) Rnn and Bp=diag(q1,, qn) be two matrices representing the intensity of noise. Let pi=jpij, Dp=diag(p1,, pn), and Lp=Dp-Ap.

Convergence analysis Let ri=xi and vi=xi for i=0,1,, n. Write r=(r1,, rn)T and v=(v1,,vn)T. Let r0(t)=cos(kt)r0(0)+(1/k)sin(kt)v0(0) v0(t)=-ksin(kt)r0(0)+cos(kt)v0(0) Then r0(t) and v0(t) solve Equation (2) x0(t)+kx0(t)=0

Let r*=r-r01 and v*=v-v01. we can obtain an error dynamics of (1) and (2) as follows dz(t)=[Ez(t)+Fz(t-s)]dt+Hz(t-s)dW(t) where, z= (r*, v*)T,

E= , F= , H=

and W(t) is an 2n dimensional standard Brownian motion.

0 In-kIn 00 00 -L-B0 00 -Lp-Bp

The theorem Theorem: Suppose that vertex 0 is globally reachable in G. If ||H||2||P||+2||PF|| {8s2[(k1)2+||F||2]+2s||H||2}

Method of proof Consider an n dimensional stochastic differential delay equation dx(t)=[Ex(t)+Fx(t-s)]dt+g(t,x(t),x(t-s))dW(t) (3) where E and F are nn matrices, g : [0, ) RnRnRnm is locally Lipschitz continuous and satisfies the linear growth condition with g(t,0,0) 0. W(t) is an m dimensional standard Brownian motion.

Lemma (X. Mao): Assume that there exists a pair of symmetric positive definite nn matrices P and Q such that P(E+F)+(E+F)TP=-Q. Assume also that there exist non-negative constants a and b such that Trace[gT(t,x,y)g(t,x,y)] a||x||2+b||y||2 for all (t,x,y). If (a+b)||P||+2||PF||{2s(4s(||E||2+||F||2)+a+b)}

Simulations We consider a network G consisting of five coupled harmonic oscillators including one leader indexed by 0 and four followers.

Let aij=1 if jNi and aij=0 otherwise; bi=1 if 0Ni and bi=0 otherwise. Take the noise intensity matrices Lp=0.1L and Bp=0.1B. Take Q=I8 with Eigenvaluemin(Q)=1. Calculate to get ||H||=0.2466 and ||F||=2.4656. In what follows, we will consider two different gains k.

Firstly, take k=0.6 such that ||E||=1>k. We solve P from the equation P(E+F)+(E+F)TP=-Q and get ||P||=8.0944 and ||PF||=4.1688. Conditions in the Theorem are satisfied by taking time delay s=0.002. Take initial value z(0)=(-5, 1,4, -3, -8, 2, -1.5, 3)T.

||r*||0

||v*||0

Secondly, take k=2 such that ||E||=k>1. In this case, we get ||P||=8.3720 and ||PF||=7.5996. Conditions in the Theorem are satisfied by taking time delay s=0.001. Take the same initial value z(0).

||r*||0

||v*||0

The value of k not only has an effect on the magnitude and frequency of the synchronized states (as implied in the Theorem), but also affects the shapes of synchronization error curves ||r*|| and ||v*||.

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Email: shyl@sjtu.edu.cn