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

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

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

Kuramoto model All-to-all interaction Introduced 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 : N i ={ j ∈ V : (j, i) ∈ E}. A=(a ij ) ∈ R n×n is the weighted adjacency matrix of G. a ij >0 if and only if j ∈ N i. D=diag(d 1,..., d n ) with d i =|N i |. The Laplacian matrix L=(l ij ) =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 x i ’’+kx i +∑ j ∈ Ni a ij (x i ’-x j ’)=0 for i=1,…, n where x i denotes the position of the ith oscillator, k is a positive gain, and a ij 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. x i ’’(t)+kx i (t)+∑ j ∈ Ni a ij (x i ’(t-s)-x j ’(t-s))+b i (x i ’(t-s)-x 0 ’(t-s))+ [∑ j ∈ Ni p ij (x i ’(t-s)-x j ’(t-s))+q i (x i ’(t-s)-x 0 ’(t-s))]w i ’(t) = 0 for i=1,…, n (1) x 0 ’’(t)+kx 0 (t)=0, (2) where s is the time delay and x 0 is the position of the virtual leader, labeled as oscillator 0.

Let B=diag(b 1,…, b n ) be a diagonal matrix with nonnegative diagonal elements and b i >0 if and only if 0 ∈ N i. W(t):=(w 1 (t),…,w n (t)) T is an n dimensional standard Brownian motion. Let A p =(p ij ) ∈ R n×n and B p =diag(q 1,…, q n ) be two matrices representing the intensity of noise. Let p i =∑ j p ij, D p =diag(p 1,…, p n ), and L p =D p -A p.

Convergence analysis Let r i =x i and v i =x i ’ for i=0,1,…, n. Write r=(r 1,…, r n ) T and v=(v 1,…,v n ) T. Let r 0 (t)=cos(√kt)r 0 (0)+(1/k)sin(√kt)v 0 (0) v 0 (t)=-√ksin(√kt)r 0 (0)+cos(√kt)v 0 (0) Then r 0 (t) and v 0 (t) solve Equation (2) ： x 0 ’’(t)+kx 0 (t)=0

Let r*=r-r 0 1 and v*=v-v 0 1. 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 I n -kI n L-B 0 0 -L p -B p

The theorem Theorem: Suppose that vertex 0 is globally reachable in G. If ||H|| 2 ||P||+2||PF||√ {8s 2 [(k ∨ 1) 2 +||F|| 2 ]+2s||H|| 2 } <Eigenvalue min (Q), where P and Q are two symmetric positive definite matrices such that P(E+F)+(E+F) T P=-Q, then by using algorithms (1) and (2), we have r(t)-r 0 (t)1→0, v(t)-v 0 (t)1→0 almost surely, as t→∞.

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, ∞) ×R n ×R n →R n×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.

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) T P=-Q. Assume also that there exist non-negative constants a and b such that Trace[g T (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)} <Eigenvalue min (Q), then the trivial solution of Equation (3) is almost surely exponentially stable.

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

Let a ij =1 if j ∈ N i and a ij =0 otherwise; b i =1 if 0 ∈ N i and b i =0 otherwise. Take the noise intensity matrices L p =0.1L and B p =0.1B. Take Q=I 8 with Eigenvalue min (Q)=1. Calculate to get ||H||= and ||F||= 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) T P=-Q and get ||P||= and ||PF||= Conditions in the Theorem are satisfied by taking time delay s= 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||= and ||PF||= Conditions in the Theorem are satisfied by taking time delay s= 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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