Dynamical Order in Systems of Coupled Noisy

Oscillators

Dedicated to Professor Pavol Brunovsky on the occasion of his 70th birthday

Shui-Nee Chow

School of MathematicsGeorgia Institute of Technology

Atlanta, GA 30332

Wenxian Shen ∗

Department of Mathematics and StatisticsAuburn University

Auburn University, AL 36849

and

Haomin Zhou †

School of MathematicsGeorgia Institute of Technology

Atlanta, GA 30332

May 2, 2007

∗Partially supported by NSF grant DMS-0504166†Partially supported by NSF grant DMS-0410062

1

Abstract. We investigate a dynamical order induced by coupling and/or noise in systemsof coupled oscillators. The dynamical order is referred to a one-dimensional topologicalstructure of the global attractor of the system in the context of random skew-productflows. We show that if the coupling is sufficiently strong, then the system exhibits onedimensional dynamics regardless of the strength of noise. If the coupling is weak, thenit is shown numerically that the system also exhibits one dimensional dynamics providedthe noise is sufficiently strong. We also show that for any coupling and any noise, thesystem has a unique rotation number and hence all the oscillators tend to oscillate withthe same frequency eventually (frequency locking).

Key words. Coupled oscillators, dynamical order, white noise, random attractor, invari-ant measure, invariant manifold, invariant foliation, one dimensional dynamics, horizontalcurve, rotation number, frequency locking

Mathematics subject classification. 34C15, 34F05, 37H10, 37H30, 60H10

2

1 Introduction

In this paper , we study systems of coupled oscillators with random perturbations. Forinstance, the following first order random system models a system of coupled oscillatorsperturbed by white noise,

dψj = (αj − sinψj +K(ψj−1 − 2ψj + ψj+1))dt+ εjdWj(t), j = 1, · · · , N. (1.1)

with boundary conditionsψ0 = ψ1, ψN+1 = ψN

where ψi is a scalar angular variable, Wj(t) is a Brownian motion, K, εj are couplingconstant and perturbation coefficient respectively, αj are given constants.

System (1.1) and its generalizations (see (2.1)), appear in a variety of applied problemsincluding Josephson junction arrays ([12], [16], [29]), chemical reactions ([15]), neuralnetwork patterns ([19], [25], [30]). Due to the stochastic perturbations in the system, thetrajectories are random in time and space, specially when the initial states are randomas well. Recently, a large amount of research has been carried out toward the influenceof noises on the dynamics of various noisy systems (see [2], [11], [18], [20], etc.). Theobjective of the current paper is to investigate the influence of the coupling and/or noiseon the dynamics of (1.1), in particular, the order in (1.1) induced by coupling and/ornoise.

In the literature on properties of systems of coupled noisy oscillators, one of the mainfocuses is the global attractor and frequency locking in some special cases. For example,in [24], the authors studied the autonomous case of the system and showed the existenceof a global attractor and the existence of a rotation number, that is, the following limitsare identical and independent of the initial values

limt→∞

ψi(t)

t= lim

t→∞

ψj(t)

t

for all i, j = 1, 2, · · · , N . This implies that all the oscillators tend to oscillate in the samefrequency after a long period of time and hence frequency locking is successful in theautonomous case. In [22], the authors studied (1.1) and proved that the rotation numberof (1.1) exists. Hence, frequency locking is also successful under white noise. The readeris referred to [14], [23], etc. for other related works.

Despite of these interesting results, more understanding of the systems of coupled noisyoscillators, especially the topological or geometrical structures of the induced order, aredesirable. Using standard approaches, such as probability density functions or numericaltrajectories, it is difficult to observe any dynamical orders in noisy and coupled systems.For example, in Figure 1, we show the distribution of 500 randomly selected initial statesfor two (N = 2) coupled oscillators with α1 = 2 and α2 = 3 on the left. In the middle,the paths between t = 150 and t = 200 of all trajectories starting from these randominitial states but with the same Brownian motions are displayed. The system is stronglycoupled (K = 2) with high level of noise (ε1 = ε2 = ε = 2). For all numerical simulations

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1

2

3

4

5

6

7

2.159 2.16 2.161 2.162 2.163 2.164 2.165 2.166 2.167 2.1681.919

1.92

1.921

1.922

1.923

1.924

1.925

1.926

1.927

1.928

Figure 1: Left: The distribution of 500 randomly selected initial states (modulated by 2πin spaces in all the plots in this paper. Middle: The paths between t = 150 and t = 200of all trajectories starting from the random initial states. It is hard to observe specialstructures of the oscillators. Right: The skew-product view of the dynamics, all the pathsfrom different random initial states attract to a 1 − D (one dimensional) structure at agiven large time, t = 200.

in this paper, we modulate the state variables by 2π because 2π is the period for theun-perturbed systems. We note that it is hard to obverse any structures of the dynamicsin the middle plot in Figure 1.

However, if one plots the snap shots (not the histories) of the trajectories at any givenlarge time (we take t = 200 in all examples showing in this paper), a clear 1−D structuresurprisingly appears as shown on the right of Figure 1, despite the random noise andcoupling effects between the oscillators. This comparison suggests that random dynamicalsystems or skew product flows will provide a way to study the dynamical properties ofthe oscillators. Furthermore, from the plot, one can easily observe a 1 − D topologicalstructure in its global attractor if the oscillators have strong coupling and high level ofnoise. In fact, this 1−D topological structure is related to the notion of horizontal curves.Thus, it is natural to ask:

(a) do coupling and/or noise induce dynamical order in a system of oscillators?

(b) what is the topological structure of the induced order?

In order to answer some of these questions, we first perform some numerical experimentswhich are shown in Figure 2. On the left, we show the positions of 500 trajectories withrandom initial states at time t = 200 for a system of two (N = 2) coupled oscillators withweak coupling (K = 0.001) and low level of noise (ε = 0.1). This shows that dynamicalorder is not present if the coupling is weak. On the right, we have a similar plot of thepositions at time t = 200 without noise but with strong coupling, It is clear that there isa global attractor with a 1 −D structure.

Our main purpose in this work is to show, by using general random attractor theory[9] and the invariant foliations theory [4], that for a more general class of oscillators (see

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0 1 2 3 4 5 6 70

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2

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0 1 2 3 4 5 6 70

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Figure 2: States of trajectories with random initial states at t = 200 with weak coupling(K = 0.001) and low level of noise (ε = 0.1) (left); and strong coupling (K = 2) but nonoise (right). Clearly, a 1 −D structure is displayed in one case and not the other.

(2.1)) than the class given by the system of coupled stochastic oscillators (1.1), we havethe following under certain condition:

(1) the existence of a ‘compact’ global attractor;

(2) 1 −D topological structure for the global attractor;

(3) the existence of a rotation number

We remark that the above results apply to systems of coupled periodic or almostperiodic oscillators by embedding them into the framework of skew-product flows. Theresult (3) extends those in [22] for systems of coupled stochastic oscillators to generalsystems of coupled random oscillators.

The rest of this paper is organized as follows. In Section 2, we collect some results forgeneral random dynamical systems to be used in later sections. We also present somebasic properties about (1.1) and its generalization in this section. The main results (1)-(3)above of the paper are proved in Sections 3 to 5. We present some numerical simulationin Section 6 to see how the dynamics change when the “noise” and coupling propertiesare changed.

2 Preliminary

In this section, we first collect some results for general random dynamical systems to beused later. Then we present basic properties for a general class of systems of coupledrandom oscillators (see (2.1) below) and show that it includes systems of coupled periodicand almost periodic oscillators and systems of coupled oscillators perturbed by whitenoises as special cases in proper sense.

First, Let (Ω,F ,P) be a probability space, θtt∈R be a family of P-preserving trans-formations (i.e. P(θ−1

t (F )) = P(F ) for any F ∈ F and t ∈ R) such that (t, ω) 7→ θtω

5

is measurable, θ0 = id, and θt+s = θt θs for all t, s ∈ R. Thus θtt∈R is a flow on Ωand ((Ω,F ,P), θtt∈R) is called a metric dynamical system. ((Ω,F ,P), (θt)t∈R) is said tobe ergodic if for any F ∈ F satisfying P(θ−1

t (F )∆F ) = 0 for any t ∈ R, P(F ) = 1 orP(F ) = 0.

Let E be a complete and separable metric space with metric d.

Definition 2.1. Πt : E × Ω → E × Ω (t ≥ 0) is called a continuous random dynamicalsystem over ((Ω,F ,P), (θt)t∈R) if it is of form Πt(x, ω) = (πt(x, ω), θtω) for any t ≥ 0,x ∈ E, and ω ∈ Ω, and πt : E × Ω → E (t ≥ 0) satisfies

(i) π0(x, ω) = x for any x ∈ E;

(ii) πt+s(x, ω) = πt(πs(x, ω), θs(ω)) for any t, s ≥ 0 and x ∈ E ;

(iii) πt(x, ω) is measurable in ω and continuous in t and x.

In the following, let Πt : E × Ω → E × Ω be as in Definition 2.1.

Definition 2.2. (1) A set valued map φ : Ω → 2E is call a (compact) random set if foreach ω ∈ Ω, φ(ω) is a nonempty closed (compact) subset of E and for each x ∈ E,d(x, φ(ω) (the distance between x and φ(ω)) is measurable in ω.

(2) Given a random set ω 7→ K(ω), denoted by K = (K(ω), ω)|ω ∈ Ω, the set valuedmap ΩK : Ω → 2E,

ΩK(ω) = ∩τ>0

(

closure(∪t≥τπt(K(θ−tω), θ−tω)))

is called the Ω-limit set of K.

(3) A random set ω 7→ K(ω) is called an absorbing set if for every bounded set B ⊂ E,

d(πt(B, θ−tω), K(ω)) → 0

as t→ ∞.

(4) A random set ω 7→ A(ω) is called a global attractor if πt(A(ω), ω) = A(θtω) fort ≥ 0 and it is an absorbing set.

Theorem 2.1. If there is a compact set ω 7→ K(ω) which is absorbing, then the randomset ω 7→ A(ω) is a global attractor, where

A(ω) = closure(

∪ΩB(ω) : B ⊂ E bounded set)

Proof. See [9].

6

Let Pr(E) be the space of probability measure on E, Pr(E×Ω) the space of probabilitymeasure on X × Ω, and

PrP(E × Ω) = µ ∈ Pr(E × Ω)|P = the marginal of µ

A map µ : Ω → Pr(E), ω 7→ µω, is called a random probability measure on E if for anyB ∈ B(E), ω 7→ µω(B) is measurable and for P-almost ω ∈ Ω, B 7→ µω(B) is a Borelprobability measure.

If µ : Ω → Pr(E) is a random probability measure, then µ ∈ PrP(E × Ω), where

µ(A) =

∫

Ω

∫

E

1A(x, ω)dµω(x)dP(ω)

Conversely, if µ ∈ PrP(E × Ω), then there is a random probability measure ω 7→ µω suchthat

∫

E×Ω

h(x, ω)dµ(x, ω) =

∫

Ω

∫

E

h(x, ω)dµω(x)dP(ω)

for every bounded measurable function h : E × Ω → R. Denote PrΩ(E) as the spaceof random probability measure. There is one-to-one correspondence between PrΩ(E)and PrP(E × Ω). For given µ ∈ PrP(E × Ω), the corresponding ω 7→ µω is called thedisintegration of µ. A random probability measure is usually called random measure. Itis invariant if its associated measure in PrP(E × Ω) is invariant.

Theorem 2.2. (1) If there is a compact random set which is forward invariant, thenthere is a (random) invariant measure.

(2) Any ergodic invariant measure is an extremal point of the convex set

µ ∈ PrΩ(E)|µ is invariant

(3) If P is ergodic, then any extremal point of the set µ ∈ PrΩ(E)|µ is invariant isergodic.

Proof. See [7].

Next, let ((Ω,F ,P), θtt∈R) is an ergodic metric dynamical system. We consider thefollowing general class of systems of coupled oscillators

dφj

dt= αj − sin(φj + hj(θtω)) +K(φj−1 − 2φj + φj+1) +Hj(θtω) (2.1)

with boundary conditionsφ0 = φ1, φN+1 = φN

where j = 1, 2, · · · , N , φj is the phase angle of the jth oscillator, ω ∈ Ω, hj, Hj : Ω → R

(j = 1, 2, · · · , N) are F ,BR-measurable, and for each ω ∈ Ω, hj(θtω) and Hj(θtω) are

7

continuous in t, and limt→±∞Hj(θtω)

t= 0 for ω ∈ Ω and j = 1, 2, · · · , N . We first present

some basic properties of (2.1) and then show that coupled systems of oscillators withalmost periodic and/or white noise perturbations can be viewed as special cases of (2.1).

Let φ(t;φ0, ω) be the solution of (2.1) with initial φ0 ∈ RN . Then Πt : R

N × Ω →R

N × Ω,Πt(φ0, ω) = (φ(t;φ0, ω), θtω) (2.2)

is a continuous random dynamical system (see [1]). Note that Πt(φ0, ω) is (2π, 2π, · · · , 2π)-periodic in φ0 in the sense that

φ(t;φ0 + (2π, 2π, · · · , 2π), ω) = φ(t;φ0, ω) + (2π, 2π, · · · , 2π)

for any t ≥ 0, φ0 ∈ RN , and ω ∈ Ω.

Let ≤ be the usual order in RN , that is,

(φ11, φ

12, · · · , φ

1N) ≤ (φ2

1, φ22, · · · , φ

2N) if φ1

j ≤ φ2j for j = 1, 2, · · · , N.

For (φ1, ω), (φ2, ω) ∈ Rn × Ω, we define

(φ1, ω) ≤ (φ2, ω) if φ1 ≤ φ2.

Note that for K > 0, (2.1) is a cooperative system. The following monotonicity propo-sition for Πt then follows from the general theory for cooperative systems of differentialequations (see [13]).

Theorem 2.3. For any ω ∈ Ω and φ10, φ

20 ∈ R

N with φ10 ≤ φ2

0, there holds

Πt(φ10, ω) ≤ Πt(φ

20, ω)

for t ≥ 0.

In the following, we present the definition of horizontal curves. The notion of horizontalcurves was first introduced in [21] in the study of global dynamics of Josephson junctiontype second order oscillators (this idea was also used in [17] independently) and wasadopted later in [14], [24], [26], [27], etc.

Definition 2.3. Let β1 = (N − 1)(1 + 2K

), βj = 1 + (N − j)(1 + 2K

), j = 2, · · · , N − 1,and γi = 1

βN−i, i = 1, 2, · · · , N − 1. A curve l given by parametric equations

φ1 = η1(s) ≡ s, φi = ηi(s), i = 2, · · · , N

satisfyingηi(s+ 2π) = ηi(s) + 2π, i = 2, · · · , N

andγi[ηi(s2) − ηi(s1)] ≤ ηi+1(s2) − ηi+1(s1) ≤ βi[ηi(s2) − ηi(s1)]

s1 ≤ s2, i = 1, 2, · · · , N1, is called a horizontal curve.

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Theorem 2.4. If l is a horizontal curve, then for any t ≥ 0, ω ∈ Ω, φ(t; l, ω) is also ahorizontal curve.

Proof. It follows from similar arguments as those in [24, Lemma 2.2].

We now show (2.1) includes systems of coupled time periodic and almost periodicoscillators as special cases. To see this, consider

dφj

dt= αj − sin φj +K(φj−1 − 2φj + φj+1) + fj(t), j = 1, · · · , N (2.3)

with boundary conditionsφ0 = φ1, φN+1 = φN

where f(t) = (f1(t), f2(t), · · · , fN(t)) is almost periodic. Let Ω = H(f) = clf(t + ·)|t ∈R be the hull of f , F = B be the σ-algebra of H(f), P be the Harr measure of H(f),and θt : Ω → Ω be the time translation flow, θtω(·) = ω(t+ ·). Then ((Ω,F ,P), θtt∈R) isan ergodic metric dynamical system and (2.3) induces (2.1) with hj(ω) = 0 and Hj(ω) =ωj(0) (j = 1, 2, · · · , N).

We show that coupled first order oscillators with white noises (1.1) can be transferredto coupled random oscillators of form (2.1). Let

Ωj = Ω0 = C0(R,R) = ω0|ω0 : R → R is continuous and ω0(0) = 0

equipped with open compact topology, Fj = B(Ω0) (the Borel σ-algebra), Pj the Winnermeasure. Let

Ω = Ω1 × ΩN × · · · × ΩN

F be the induced product σ-algebra of Ω, P the induced product Winner measure, andθt : Ω → Ω, θtω(·) = ω(· + t) − ω(t). Then ((Ω, F ,P), (θt)t∈R) is an ergodic metricdynamical system. System (1.1) can then be changed to an equivalent system of randomequations as follows. First let ψ∗

i be the unique stationary process of

dψi + ψidt = didWi (2.4)

Following from [10], we have

Theorem 2.5. There is a θt-invariant subset Ω ⊂ Ω (i.e. θtΩ = Ω for t ∈ R) withP(Ω) = 1 such that for any ω ∈ Ω, the following holds.

1) ψ∗,ω(t) = ψ∗(θtω) is continuous in t ∈ R.

2) limt→±∞ψ∗,ω(t)

t= 0.

3) limt→±∞1

t

∫ t

0

ψ∗,ω(s)ds = 0.

9

Next let Ω ⊂ Ω be as in Theorem 2.5 and F = F ∩ Ω ≡ F ∩ Ω|F ∈ F. Then((Ω,F ,P), (θt)t∈R) is also an ergodic metric dynamical system.

Let nowφi(t, ω) = ψi(t, ω) − ψ∗

i (θtω). (2.5)

for ω ∈ Ω. Then φi (i = 1, 2, · · · , N) satisfy

dφj

dt= αj − sin(φj + ψ∗

j (θtω)) +K(φj−1 − 2φj + φj+1) +Hj(θjω) (2.6)

with boundary conditionsφ0 = φ1, φN+1 = φN

where ω ∈ Ω,Hj(θtω) = K(ψ∗

j−1(θtω) − 2ψ∗j (θtω) + φ∗

j+1(θtω))

j = 1, 2, · · · , N , and ψ∗0(θtω) = ψ∗

1(θtω) and ψ∗N (θtω) = ψ∗

N+1(θtω).Therefore (2.6) is a special case of (2.1) and the results we prove for (2.1) can be

applied to (1.1).Finally, consider the following system of coupled oscillators with both white noises and

almost periodic perturbations,

dψj = (αj − sinψj +K(ψj−1 − 2ψj + ψj+1) + fj(t))dt+ εjdWj(t), j = 1, · · · , N (2.7)

with boundary conditionsψ0 = ψ1, ψN+1 = ψN

where fj(t) is almost periodic. Combining the ideas of imbedding (2.3) into (2.1) andtransforming (1.1) into (2.1), we can also transform (2.7) into (2.1).

3 Existence of Global Attractor

In this section, we study the existence of global attractor of (2.1). Let Πt : RN × Ω →

RN × Ω be the random dynamical system (2.2) generated by (2.1). By the periodicity of

Πt(φ0, ω) in φ0, we say that a set K ⊂ RN (B ⊂ R

N) is compact (bounded) if K mod(2π, 2π, · · · , 2π) (B mod (2π, 2π, · · · , 2π)) is compact (bounded). The following is themain result of this section.

Theorem 3.1. There is A : Ω → 2RN

such that A(ω) (mod) (2π, 2π, · · · , 2π) is compactand A is a global random attractor of (2.1).

Proof. First of all, we change (2.1) to another system of random equations as follows.

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Let

M =

1 −1 0 · · · 0 0 0

−1 2 −1 · · · 0 0 0

· · ·

0 0 0 · · · −1 2 −1

0 0 0 · · · 0 −1 1

(3.1)

There is an orthonormal matrix P = (pij) such that P>MP is diagonal,

D := P>MP =

0 0 0 · · · 0 0

0 λ2 0 · · · 0 0

· · ·

0 0 0 · · · λN−1 0

0 0 0 · · · 0 λN

(3.2)

where λj = 4 sin2(

(j−1)π2N

)

, j = 1, 2, · · · , N , are eigenvalues of M , and pj1 = 1√N

.

Let Φ = P>φ. Then Φ satisfies

Φ = P>

α1 − sin(φ1 + h1(θtω)) +H1(θtω)

α2 − sin(φ2 + h2(θtω)) +H2(θtω)

· · ·

αN − sin(φN + hN (θtω)) +HN(θtω)

−KDΦ (3.3)

where φ = PΦ.By [1], (3.3) also generated a random dynamical system Σt : R

N × Ω → RN → Ω,

Σt(Φ0, ω) = (Φ(t; Φ0, ω), θtω) (3.4)

where Φ(t; Φ0, ω) is the solution of (3.3) with initial condition Φ(0; Φ0, ω) = Φ0. We notethat Σt(Φ0, ω) is (2π, 0, · · · , 0) in Φ0 in the sense that

Φ(t; Φ0 + (2π, 0, · · · , 0), ω) = Φ(t; Φ0, ω) + (2π, 0, · · · , 0)

We next show that the random dynamical system Σt admits a compact absorbing set.More precisely, we claim that there is r : Ω → R such that for each ω ∈ Ω and any subset

11

B = R ×K ⊂ RN , where K ⊂ R

N−1 is bounded,

Σt(B, θ−tω) ⊂ R × U(r(ω)) × ω for t 1 (3.5)

where U(r(ω)) = x ∈ RN−1 : ‖x‖ ≤ r(ω). Moreover, for any τ ∈ R,

|r(θτω)| ≤ r(ω)(1 + |τ |). (3.6)

In fact, for given ω ∈ Ω, since limt→±∞Hj(θtω)

t= 0, there is M1(ω) > 0 such that

|Hj(θtω)| ≤M1(ω)(1 + |t|), j = 1, 2, · · · , N, t ∈ R. (3.7)

Let r(ω) be given by

r(ω) = 1 + max2≤j≤N

αj + 1

Kλj

+ 4M1(ω)

∫ 0

−∞eKλjs(1 + |s|)ds.

Note that for 2 ≤ j ≤ N ,

Φj(t; Φ0, θ−tω) = Φj(0)e−Kλjt +

∫ t

0

e−Kλj(t−s)(αj − sin(φj(s; Φ0, θ−tω) + hj(θs−tω)))ds

+

∫ t

0

e−Kλj(t−s)Hj(θs−tω)ds.

It then follows from (3.7) that

|Φj(t; Φ0, θ−tω)| ≤ r(ω) for t 1.

HenceΣt(B, θ−tω) ⊂ R × U(r(ω)) × ω for t 1

for any set B ⊂ R ×K, where K is a bounded subset of RN−1.

Moreover, we haver(θτω) ≤ r(ω)(1 + |τ |)

for any τ ∈ R. The claim is thus proved.We now show that there is a global random attractor of (3.3).Note that Σt : R

n × Ω → RN × Ω induces a random dynamical system Σt : (S1 ×

RN−1) × Ω → (S1 × R

N−1) × Ω on S1 × RN−1,

Σt(Φ0, ω) = (Φ(t; Φ0, ω) (mod) (2π, 0, · · · , 0), θtω). (3.8)

By (3.5), the set ω 7→ K(ω) = S1 × U(r(ω)) is a compact absorbing set of Σt. It thenfollows from Theorem 2.1 that Σt has a compact random attractor A : Ω → 2S1×R

N−1

.Let A(ω) = Φ ∈ R

N : Φ (mod) (2π, 0, · · · , 0) ∈ A(ω). Then A : Ω → 2RN

is a globalcompact random attractor of Σt : R

N × Ω → RN × Ω.

Finally, we show that (2.1) has a global attractor. In fact, let A(ω) = φ ∈ RN :

P>φ ∈ A(ω). Then A : Ω → 2RN

is a global random attractor of (2.1). The theorem isproved.

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4 Dimension of Global Attractor

In this section, we investigate the dimension of the global attractor of (2.1) and show thatwhen K 1, the global attractor is one dimensional. Our main result of this sectionstates as follows.

Theorem 4.1. There is K0 > 0 (independent of the noises) such that when K ≥ K0, foreach ω ∈ Ω, A(ω) is one-dimensional and is a horizontal curve, where A : Ω → 2R

N

isthe global random attractor of (2.1).

Proof. First, consider (3.3). We prove that there is K0 > 0 (independent of the noises)such that when K ≥ K0, there is a one-dimensional center-unstable manifold W cu(ω) of(3.3) near R × 0.

In fact, let K0 > 0 be such that 1γ

+ 2β−γ

< 1 for some 0 < γ < β, where β = Kλ2. Let

P : RN → R

N and Q : RN → R

N be the projections P (ψ1, ψ2, · · · , ψN ) = (ψ1, 0, · · · , 0)and Q(ψ1, ψ2, · · · , ψN ) = (0, ψ2, · · · , ψN ). Let

T (t)Φ0 = e−KDtΦ0

and

F (Φ, θtω) = P>

α1 − sin(φ1 + h1(θtω)) +H1(θtω)

α2 − sin(φ2 + h2(θtω)) +H2(θtω)

· · ·

αN − sin(φN + hN (θtω)) +HN(θtω)

where φ = PΦ and P = (pij) is as in (3.2).Then by Theorem 3.3 in [4], for any K ≥ K0, there is a center-unstable manifold of

(3.3) near R × 0,W cp(ω) = ξ + h(ξ, ω)|ξ ∈ PR

N,

where h : PRN → QR

N , and for Φ0 = ξ + h(ξ, ω),

Φ(t; Φ0, ω) = T (t)ξ+

∫ t

0

T (t−s)PF (Φ(s; Φ0, ω), θsω)ds+

∫ t

−∞T (t−s)QF (Φ(s; Φ0, ω), θsω)ds.

Hence

h(ξ, ω) =

∫ 0

−∞T (−s)QF (Φ(s; Φ0, ω), θsω)ds.

Therefore there is M2(ω) > 0 such that

|h(ξ, θτω)| ≤ M2(ω)(1 + |τ |). (4.1)

Next, by Theorem 3.4 in [4], for anyK ≥ K0, there is a stable foliation W cus (Φ0, ω)|Φ0 ∈

W cu(ω) of the center-stable manifold W cu(ω) of (3.3),

W cus (ξ + h(ξ, ω), ω) = ζ + h(ξ, ω) + h(ξ, ζ, ω)|ζ ∈ QR

N , h(ξ, ζ, ω) ∈ PRN,

13

where h(ξ, 0, ω) = ξ. Moreover there is M > 0 such that for any Φ0 = ζ + h(ξ, θ−tω) +h(ξ, ζ, ω−t),

|Φ(t; Φ0, θ−tω) − Φ(t; ξ + h(ξ, θ−tω), θ−tω)| ≤Me−γt|ζ| (4.2)

for t > 0.Now we claim that A(ω) = M cu(ω), where A : Ω → 2R

N

is the global random attractorof (3.3). In fact, let B ⊂ R

N be such that B (mod) (2π, 0, · · · , 0) is bounded. For anyΦ0 ∈ B and t > 0, there is ξ0 ∈ PR

N such that Φ0 ∈ W cus (ξ0 + h(ξ0, θ−tω), θ−tω). Let

ζ0 = h(ξ0, θ−tω) − QΦ0. Then by (4.2),

|Φ(t; Φ0, θ−tω) − Φ(t; ξ0 + h(ξ0, θ−tω), θ−tω)| ≤Me−γt|ζ0|

for t > 0. By (4.1),|ζ0| ≤M2(ω)(1 + t) + |Φ0|

for t > 0. Therefored(Φ(t;B, θ−tω),W cu(ω)) → 0

as t→ ∞. This implies thatA(ω) = M cu(ω).

Therefore, A(ω) is one-dimensional. This implies that A(ω) = PA(ω) is also one-dimensional.

Finally, we show that A(ω) is a horizontal curve. Note that for a given horizontal curvel, l (mod) (2π, 2π, · · · , 2π) is bounded. Hence

d(ψ(t; l, θ−tω), A(ω)) → 0

as t→ ∞. By Theorem 2.4,, for any t > 0, ψ(t; l, θ−tω) is a horizontal curve. By the abovearguments, A(ω) is one-dimensional. We then must have that limt→∞ ψ(t; l, θ−tω) = A(ω)and A(ω) is a horizontal curve.

Suppose that A(ω) = l(ω) and l(ω) is given by parametric equations: ψ1 = s, ψ2 =η2(s, ω), · · · , ψN = ηN(s, ω). Then restricted to the global attractor, (2.1) reduces to thefollowing scalar ODE on S1,

φ1 = α1 − sin(φ1 + h∗1(θtω)) +K(η2(φ1, θtω) − φ1) +H1(θtω).

We refer to [3] for some interesting dynamical scenario of quasi-periodic ODEs on S1.It should be pointed out that in [28], the authors studied the rotation number of a classof scalar ODEs on S1 perturbed by white noise, i.e.

dφ(t) = (I − sin φ(t))dt+DdW (t)

where I ≥ 0, W (t) is a Brownian motion on R, and D is a constant representing thestrength of white noise perturbation. They showed the following interesting result: Therotation number ρ(D) > 0 when I > 0 and D 6= 0. This is different to the definite case(i.e. D = 0 case), where ρ = 0 when 0 ≤ I ≤ 1 and ρ > 0 when I > 1.

14

5 Rotation Number

In this section, we study the existence of rotation number of (2.1), i.e,

dφj

dt= αj − sin(φj + hj(θtω)) +K(φj−1 − 2φj + φj+1) +Hj(θtω)

with boundary conditionsφ0 = φ1, φN+1 = φN

where ω ∈ Ω, ((Ω,F ,P), (θt)t∈R) is an erdogic metric dynamical system, and hj, Hj : Ω →R (j = 1, 2, · · · , N) are F ,BR-measurable, and for each ω ∈ Ω, hj(θtω) and Hj(θtω) are

continuous in t, and limt→±∞Hj(θtω)

t= 0 for ω ∈ Ω and j = 1, 2, · · · , N .

Recall that φ(t;φ0, ω) denotes the solution of (2.1) with initial condition φ(0;φ0, ω) =φ0.

First, we introduce the notion of the rotation number.

Definition 5.1. (2.1) is said to have a rotation number ρ if there is Ω0 ⊂ Ω with P(Ω0) =

1 such that for any ω ∈ Ω0 and φ0 ∈ RN , the limit limt→∞

φj(t;φ0,ω)

texists and

limt→∞

φj(t;φ0, ω)

t= ρ

for j = 1, 2, · · · , N .

Observe that if the rotation number of (2.1) exists, then all the oscillators tend tooscillate in the same frequency after a long period of time almost surely, that is to say,the frequency locking is successful.

Theorem 5.1. For any positive coupling K > 0 and any given constants αj, the rotationnumber of (2.1) exists.

Proof. First of all, let Σt : RN × Ω → R

N × Ω be the random dynamical system (3.4)generated by (3.3) and Σt : (S1×R

N−1)×Ω → (S1×RN−1)×Ω be the random dynamical

system (3.8) induced from Σt : RN × Ω → R

N × Ω.By the arguments of Theorem 3.1, the random dynamical system Σt has a compact

global random attractor A. Then by Theorem 2.2, there is an ergodic invariant measureµ of Σt supported on A. Note that

Φj(t; Φ0, ω) = Fj(Σt(Φ0, ω)),

where Fj(Φ, ω) is the jth component of the right hand side of (3.3). Hence

Φj(t; Φ0, ω) − Φ0j

t=

1

t

∫ t

0

Fj(Σs(Φ0, ω))ds.

By Birkhoff’s Ergodic Theorem, there is A0 ⊂ A with µ(A0) = µ(A) and such that for(Φ0, ω) ∈ A0,

limt→∞

Φj(t; Φ0, ω)

t=

∫

AFj(Φ, ω)dµ, j = 1, 2, · · · , N.

15

Let

ρj =

∫

AFj(Φ, ω)dµ

for j = 1, 2, · · · , N and

ρ1

ρ2

···ρN

= P

ρ1

ρ2

···ρN

where P is as in (3.2). Note that

Φj(t; Φ0, ω) = Φj(0)e−Kλjt +

∫ t

0

e−Kλj(t−s)(αj − sin(φj(s; Φ0, ω) + hj(θsω)))ds

+

∫ t

0

e−Kλj(t−s)Hj(θsω)ds.

Since limt→∞Hj(θtω)

t= 0, we have limt→∞

Φj(t;Φ0,ω)

t= 0 for j = 2, 3, · · · , N . This implies

thatρ2 = ρ3 = · · · = ρN = 0,

and thenρ1 = ρ2 = · · · = ρN ,

limt→∞

φ1(t;φ0, ω)

t= lim

t→∞

φ2(t;φ0, ω)

t= · · · = lim

t→∞

φN(t;φ0, ω)

t= ρ1,

where φ0 = PΦ0.Let ρ = ρ1. Since the marginal of µ is P and ((Ω,F ,P), θtt∈R) is ergodic, there is

Ω0 ⊂ Ω with P(Ω0) = 1 such that for any ω ∈ Ω, there is Φ∗0 ∈ R

N such that (ω,Φ∗0) ∈ A0.

Hence for any ω ∈ Ω0, there is φ∗0 ∈ R

N such that

limt→∞

φ1(t;φ∗0, ω)

t= lim

t→∞

φ2(t;φ∗0, ω)

t= · · · = lim

t→∞

φN(t;φ∗0, ω)

t= ρ.

By the periodicity of Πt, for any n ∈ N and 1 ≤ j ≤ N ,

limt→∞

φj(t;φ∗0, ω)

t= lim

t→∞

φj(t;φ∗0 ± (2nπ, 2nπ, · · · , 2nπ), ω)

t. (5.1)

Now for given ω ∈ Ω0, for any φ0 ∈ RN , there is n ∈ N such that

φ∗0 − (2nπ, 2nπ, · · · , 2nπ) ≤ φ0 ≤ φ∗

0 + (2nπ, 2nπ, · · · , 2nπ)

This together with (5.1) and Theorem 2.3 implies that

limt→∞

φj(t;φ0, ω)

t= lim

t→∞

φj(t;φ∗0, ω)

t= ρ

for j = 1, 2, · · · , N , ω ∈ Ω0, φ0 ∈ RN . The theorem is thus proved.

16

Figure 3: Trajectory plots (between t = 150 and t = 200 of two strongly coupled (K = 2)oscillators with 500 random initial states with different level of noise: no noise (ε = 0,left), and low level noise (ε = 0.1, right).

6 Numerical Simulation and Remarks

In this section, we show some numerical simulations to see whether in general A(ω) is onedimensional for K > 0 not so large. We also do some numerical simulations to see howthe dynamics changes when the noise level and coupling coefficients change.

In the first set of examples, we make a complete comparison between trajectory plotsand skew product plots for two coupled oscillators (N = 2) with different coupling strengthand noise level. We take the same noise level for both oscillators so that ε1 = ε2 = ε. Weset α1 = 2 and α2 = 3 to aviod equilibrium solutions in the noise free (ε = 0) oscillators.In Figures 3-4, we present the paths between t = 150 and t = 200 for all trajectoriesstarting from 500 random initial states as shown in Figure 1. In our experiments, we alsonotice that the trajectories originated from 500 randomly distributed initial states almostcover the entire modulated phase space and no structure is observable when there areonly weak coupling (K = 0.001) and low level noise (ε = 0, or ε = 0.1) in the systems.

In Figures 5-7, we show the skew product plots of the attractors with different strengthof coupling and noise level. It is pretty clear that skew product plots can better demon-strate the 1 −D global attractor structures if order is induced by noise or coupling.

In the last set of examples, we illustrate three coupled oscillators (N = 3). We takeα1 = 2, α2 = 3 and α3 = 0.5 to avoid the equilibrium solutions in the un-perturbedsystem. This implies that if the initial states are random, the solutions remain randomeven if there is no noise. In the experiments, again we take the same noise level for eachoscillator, i.e. ε1 = ε2 = ε3 = ε. On the left of Figure 8, we show the random initialstates in 3 − D. The states with a weak coupling (K = 0.001) and without stochasticperturbation (ε = 0) at t = 200 are shown on the right of Figure 8. It is obvious that thetrajectories are still random, and no apparent order is induced by the weak coupling.

With the same random initial states, we simulate the states for the weak couplingand low level stochastically perturbed system (ε = 0.1) on the left of Figure 9. Theperturbation is too small to create any order in the system. In the middle of Figure 9, we

17

Figure 4: Trajectory plots between t = 150 and t = 200 with 500 random initial stateswith high level noise (ε = 2) but different strength of coupling: weak coupling (K = 0.001,left), and strong coupling (K = 2, right).

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Figure 5: Skew product plots at t = 200 with 500 random initial states without noisebut different strength of coupling: weak coupling (K = 0.001, left), and strong coupling(K = 2, right).

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

0 1 2 3 4 5 60

1

2

3

4

5

6

Figure 6: Skew product plots at t = 200 with 500 random initial states with low levelnoise (ε = 0.1) but different strength of coupling: weak coupling (K = 0.001, left), andstrong coupling (K = 2, right).

18

0.25 0.255 0.26 0.265 0.27 0.275 0.28 0.285 0.29 0.2952.48

2.49

2.5

2.51

2.52

2.53

2.54

2.55

2.159 2.16 2.161 2.162 2.163 2.164 2.165 2.166 2.167 2.1681.919

1.92

1.921

1.922

1.923

1.924

1.925

1.926

1.927

1.928

Figure 7: Skew product plots at t = 200 with 500 random initial states with high levelnoise (ε = 2) but different strength of coupling: weak coupling (K = 0.001, left), andstrong coupling (K = 2, right).

−3−2

−10

12

34

−4

−2

0

2

4−3

−2

−1

0

1

2

3

−3−2

−10

12

3−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

Figure 8: Left: The randomly distributed initial states of the oscillators. Right: Thestates for the oscillators at t = 200. No noise is presented in the dynamics. The couplingconstant is K = 0.001. The weak coupling does not induce any order in the dynamics.

19

−3−2

−10

12

3−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

−3−2

−10

12

3−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

−0.9495

−0.949

−0.9485−0.08

−0.079

−0.078

−0.077

−0.076

−0.075

−2.2515

−2.251

−2.2505

−2.25

Figure 9: Left: The states at t = 200 with low level noise noise ε = 0.1 and weak couplingK = 0.001. Weak coupling and small noise do not restore the order in the randomoscillators. Middle: The states at t = 200 with very noisy force ε = 2 but weak couplingK = 0.001. Right: A zoom in of the states shown in the middle picture. A clear 1 − D

structure is evident.

−3−2

−10

12

3−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

−3−2

−10

12

3−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

0.6610.662

0.6630.664

0.6650.666

0.6670.668

1.033

1.034

1.035

1.036

1.037

1.038

−0.208

−0.207

−0.206

−0.205

−0.204

−0.203

−0.202

−0.201

Figure 10: Left: The states at t = 200 without noise ε = 0 and strong coupling K = 2.Middle: The states at t = 200 with large noise ε = 2 and strong coupling K = 2. Right:A zoom in of the middle picture.1−D structures are easily observed in all of the pictures

20

increase the noise level to ε = 2. 1 −D structures are restored in the system after beingseverely perturbed. To better view the structure, we zoom in of the middle plot and showthe structure on the right of Figure 9.

In Figure 10, we vary the noise level and coupling strength to see the structure of theglobal attractors. On the left, we take strong coupling with K = 2 and no perturbation.The strong coupling alone creates an order in the states even they are randomly distributedinitially. In the middle of Figure 10, we demonstrate the effect if both coupling and noiseare strong. Certainly, order re-appears. A zoom in of the states (on the right of Figure10 ) demonstrate the clear 1 −D structure in the global attractor.

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