Synchronization in Networks of Coupled Harmonic Oscillators with Stochast

ic 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

1

sin( )N

ii j i

j

d K

dt N

: Number of oscillators

: Natural frequency of oscillator , 1, , .

: Phase of oscillator , 1, , .

: Coupling strength

i

i

N

i i N

i i N

K

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 info∈rmation to the agent i directly.

The in-degree neighborhood of the agent i :

Ni ={ j V : (j, i) E}.∈ ∈ A=(aij) R∈ n×n is the weighted adjacency matrix of G.

aij >0 if and only if j N∈ i. 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+∑j Ni∈ aij(xi’-xj’)=0 for i=1,…, n

where xi denotes the position of the ith oscillator,

k is a positive gain, and aij characterizes interact

ion 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)+∑j Ni∈ aij(xi’(t-s)-xj’(t-s))+bi(xi’(t-s)-x0’(t-s))+

[∑j Ni∈ pij(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 virt

ual 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 0 N∈ i.

W(t):=(w1(t),…,wn(t))T is an n dimensional stand

ard Brownian motion.

Let Ap=(pij) R∈ n×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.

Convergence analysis

Let ri=xi and vi=xi’ for i=0,1,…, n. Write r=(r1,…, r

n)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-B

0 00 -Lp-Bp

The theorem

Theorem: Suppose that vertex 0 is globally 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 positive definite matrices such that

P(E+F)+(E+F)TP=-Q,

then by using algorithms (1) and (2), we have

r(t)-r0(t)1→0, v(t)-v0(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, ∞) ×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.

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.

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 j N∈ i and aij=0 otherwise; bi=1 if 0 N∈ i

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*||.

Thanks

for your

Attention!

Email: shyl@sjtu.edu.cn