尚轶伦 上海交通大学 数学系

1. Synchronization in Networks of Coupled Harmonic Oscillators with Stochastic Perturbation and Time Delays 尚轶伦 上海交通大学 数学系

2. Outline • Introduction ● Backgrounds ● Problem formulation • Main result ● Synchronization of coupled harmonic oscillators • Methods of proof • Numerical examples

3. Synchronized oscillators • Cellular clocks in the brain • Pacemaker cells in the heart • Pedestrians on a bridge • Electric circuits • Laser arrays • Oscillating chemical reactions • Bubbly fluids • Neutrino oscillations • Synchronous firings of male fireflies

4. Kuramoto model All-to-all interaction Introduced by Kuramoto in 1975 as a toy model of synchronization

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

6. 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) ∈Rn×n 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.

7. 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+∑j∈Niaij(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.

8. 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)+∑j∈Niaij(xi’(t-s)-xj’(t-s))+bi(xi’(t-s)-x0’(t-s))+ [∑j∈Nipij(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.

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

10. 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

11. Let r*=r-r01 and v*=v-v01. we can obtain anerror 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 2ndimensional standard Brownianmotion. 0 In -kIn 0 0 0 0 -L-B 0 0 0 -Lp-Bp

12. The theorem Theorem:Suppose that vertex 0 isglobally reachable in G. If ||H||2||P||+2||PF||√ {8s2[(k∨1)2+||F||2]+2s||H||2} <Eigenvaluemin(Q), where P and Q are two symmetric positivedefinite matrices such that P(E+F)+(E+F)TP=-Q, then by usingalgorithms (1) and (2), we have r(t)-r0(t)1→0, v(t)-v0(t)1→0 almost surely, as t→∞.

13. 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 n×n matrices, g : [0, ∞) ×Rn×Rn→Rn×m 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.

14. Lemma (X. Mao): Assume that there exists a pair of symmetric positive definite n×n 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)} <Eigenvaluemin(Q), then the trivial solution of Equation (3) is almost surely exponentially stable.

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

16. Let aij=1 if j∈Ni and aij=0 otherwise; bi=1 if 0∈Ni 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.

17. 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.

18. ||r*||→0

19. ||v*||→0

20. 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).

21. ||r*||→0

22. ||v*||→0

23. 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*||.

24. Thanks for your Attention! Email: shyl@sjtu.edu.cn