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Synchronization of Coupled Forced Oscillators
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
218, 97]116 Ž1998.
AY975749
Synchronization of Coupled Forced Oscillators Whei-Ching C. Chan* and Yi-Duo Chao Department of Mathematics, Tamkang Uni¨ ersity, Tamsui, Taiwan, 25137, Republic of China Submitted by William F. Ames Received June 10, 1997
1. INTRODUCTION Synchronization usually means the concurrence of all or some parts of motions Že.g., phase. with respect to time in coupled systems. For example, in some simple forced systems, one often observes phase-locking phenomena, or subharmonic motions Žsee Chan w4x, Chow and Hale w5x.. This is more often called phase synchronization. In recent years, Fujisaka and Yamada w9x observed the concurrence of motions in time for a coupled system but where each individual system is chaotic. They called this phenomena the stochastic synchronization. In w2x, Afraimovich, Verichev, and Rabinovich proved mathematically that stochastic synchronization exists in a system of globally coupled, periodically forced Duffin’s equations. Their method of proof was based on the theory of asymptotic stability. In w7x, Chua, Itoh, Kosarev, and Eckert showed experimentally, numerically, and mathematically that it is possible to synchronize Chua’s circuits even though each circuit is chaotic. Their theory was based on a detailed study of Liapunov exponents or asymptotic stability. Other types of mechanisms to produce synchronization in coupled systems have been shown by Carrol and Pecora w3x, Fabiny, Colet, and Roy w8x, and Heagy, Carrol, and Pecora w11x. Recently, Chow and Liu w6x showed how different notions of synchronization can be studied using the theory of invariant manifolds. In this paper, we consider an array of diffusively coupled Josephson junctions with periodic forcing. By using the theory of attractors and stability theory, we are able to prove the phenomena of synchronization *Partially supported by NSC Grant 86-2115-M-032-005. 97 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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CHAN AND CHAO
when the parameters in our system satisfy certain conditions. Our work in some way is similar to the recent work of Afraimovich, Chow, and Hale w1x. However, there are differences, especially, in studying the solvability of a matrix Riccati equation.
2. MODEL Josephson junctions are superconducting devices that are capable of generating voltage oscillations of high frequency. We now give a mathematical description. Let f be the difference of phases of the superconductors across the junction. The rate of slippage of f is governed by the Josephson voltage-phase relation Vs
F0 2p
f˙ ,
Ž 2.1.
where F 0 is a constant proportional to Planck’s constant. By Kirchhoff’s voltage and current laws, we have the equation CV˙ q
V R
q Ic sin f s I,
where C, R, Ic , and I are constants. By Ž2.1., we have F0 2p
f¨ q
F0 2p R
f˙ q Ic sin f s I.
Ž 2.2.
Equation Ž2.2. is refereed as a Josephson equation and has been studied by many authors Žsee, for example, Wiesenfeld, Benz, and Booi w12x.. If the devices are connected along a ring or line, we obtain the equation
¡¨ ¢
˙j
~f q af q sin f j
j
s G q k Ž f jq1 y 2 f j q f jy1 . q c f˙jq1 y 2 f˙j q f˙jy1 ,
ž
/
Ž 2.3.
where i is an integer index, a , k, and c are constants; and G is a time dependent function. k and c represent the strength of the interaction between nearest neighbors. In this paper, we consider the mathematical problem of synchronization of finitely many Josephson junctions. In w12x, the authors considered the synchronization of phases by using numerical methods and an approximation scheme. In Afraimovich, Chow, and Hale w1x, Duffin’s equations were considered.
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SYNCHRONIZATION
3. GLOBAL ATTRACTOR By rescaling the constants in Eq. Ž2.3., we consider an array of n ) 1 damped Josephson equations with diffusive coupling and Dirichlet boundary conditions at both ends,
¡˙x s y ~˙y s a y y sin x q g Ž t . ¢ qk Ž x y 2 x q x i
i
i
i i
i
iq1
Ž 3.1.
i
i
iy1
. q c Ž yiq1 y 2 yi q yiy1 . ,
where i s 1, . . . , n, x i s x i mod 2p g S 1, yi g R, , k, c, and the a i ’s are real constants, and the g i Ž t . are scalar T-periodic functions with T ) 0. The Dirichlet boundary conditions at both ends are given by x 0 s x nq1 s 0,
y 0 s ynq1 s 0.
Ž 3.2.
We note that
xs
x1 n ! # " n .. g S = ??? = S sT . xn
0
y1 .. n . gR . yn
0
ys
and
Notice that we have identical undamped oscillators at every lattice site. However, this is not a restriction and the theory presented in this paper works for non-identical ones. In vector notation, let ysin x 1 .. f Ž x. s , . ysin x n
0
gŽ t. s
g 1Ž t . .. , .
0 gnŽ t .
a1 0 0 As . ..
0 a2 0 .. .
0 0 a3 .. .
0 0 0 .. .
??? ??? ??? .. .
0 0
0 0
0 0
0 0
??? ???
0 0 0 .. .
a ny 1 0
0 0 0 .. . 0 an
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CHAN AND CHAO
and D be the following linear map which is the discrete Laplacian with Dirichlet boundary conditions Ž3.2.: y2 1 0 .. Ds . 0 0
1 y2 1 .. .
0 1 y2 .. .
0 0 1 .. .
??? ??? ??? .. .
0 0 0 .. .
0 0 0 .. .
0 0
0 0
0 0
??? ???
y2 1
1 y2
.
Then Eq. Ž3.1. and boundary condition Ž3.2. can be written as
½
˙x s y ˙y s Ay q f Ž x . q g Ž t . q kD x q cD y.
Ž 3.3.
The discrete Laplacian D is self-adjoint and the following is well known. LEMMA 3.1. The discrete Laplacian D has simple eigen¨ alues l s and its corresponding eigen¨ ectors j s, where s s 1, 2, . . . , n such that
l s s y2 y 2cos
ps nq1
s s 1, 2, . . . , n
,
l M s max l s s y4sin 2
ž
p 2 Ž n q 1.
/
-0
and the set of eigen¨ ectors B s j 1, . . . , j n4 forms an orthonormal basis for R n. Let ² ? , ? : be the usual inner product in R n. Since B is an orthonormal basis in R n, any x, y in R n have unique representations, n
xs
Ý
n
Xs j s ,
ys
ss1
Yssj s ,
Ý ss1
where scalars X s , Ys , s s 1, 2, . . . , n, are determined by x, y, and B. We also have n
² D x, y : s ² x, D y : s
Ý
l s X s Ys
ss1 n
² x, y : s
Ý is1
n
x i yi s
Ý ss1
X s Ys .
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In this section, we will show that under some conditions on the constants k, c and a i , systems Ž3.1. and Ž3.2. have a global attractor in T n = R n. To do so we will find an appropriate Liapunov function. First we define a new linear operator L and its adjoint L* in the usual Euclidean basis by 1 0 0 L s .. .
y1 1 0 .. .
0 y1 1 .. .
0 0 y1 .. .
??? ??? ??? .. .
0 0 0 .. .
0 0 0 .. .
0 0
0 0
0 0
??? ???
1 0
y1 1
0 0 and 1 y1 0 .. L* s . 0 0
0 1 y1 .. .
0 0 1 .. .
0 0 0 .. .
??? ??? ??? .. .
0 0 0 .. .
0 0
0 0
0 0
??? ???
1 y1
0 0 0 .. . . 0 1
Thus, along solutions of Eq. Ž3.1. we have d dt
² x, Lx : s ² y, Lx : q ² x, Ly : s ² y, Lx : q ² y, L*x : s ² y, Ž L q L* . x : s y² y, D x : .
Now, consider the following scalar-valued function, U Ž x, y . s 12 ² y, y : y F Ž x . q B² x, Lx : q v ² x, y : ,
Ž 3.4.
where F Ž x . s cos x i , B, v are positive constants to be determined latter. By taking the time derivative of U along solutions of Ž3.3., one obtains Ý nis1
dU dt
s ² y, ˙ y : y ² f Ž x . , y : y B² y, D x : q v ² y, y : q v ² x, ˙ y: s ² y, Ay : q v ² y, y : q ² y q v x, g Ž t . : q v ² x, Ay : q v ² x, f Ž x . : q Ž k y B . ² y, D x : q c² y, D y : q v k ² x, D x : q v c² x, D y : .
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CHAN AND CHAO
s ² y, Ay : q v ² y, y : q ² y q v x, g Ž t . : q v ² x, Ay : q v ² x, f Ž x . : q Ý Ž k y B q v c . l s X s Ys q c Ý l s Ys 2 q v k Ý l s X s2 . s
s
s
Thus, dU dt
s ² y, Ay : q v ² y, y : q ² y q v x, g Ž t . : q v ² x, Ay : q v ² x, f Ž x . : q
Ý l s Ž c y v k . Ys2 s
q Ý l s v k X s2 q
k y B q vc
s
PROPOSITION 3.1.
vk
X s Ys q Ys 2 .
Ž 3.5.
Let
v s 2,
B s 2Ž c y 2 k . q k
and c y 2k )
3 q aM
lM
,
where a M s max a i 4 .
Let UŽ x, y . be defined by Ž3.4.. Then along any solution Ž x Ž t ., y Ž t .. of Ž3.3., we ha¨ e dU Ž x, y . dt
- y< y < 2 q Ž 4pa M q < g < . < y < q 4 Ž < g < q 1 . p ,
where < g < is the sup norm of g Ž t . Proof. Since v s 2 and B s 2Ž c y 2 k . q k, we have k y B q vc
vk
s 2.
We also have < x < - 2p ,
c y 2k )
3 q aM yl M
) 0.
Thus
l M Ž c y 2 k . q a M q 2 - y1.
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It follows from Ž3.5. and the above inequality that dU Ž x, y . dt
F a m < y < 2 q 2 < y < 2 q Ž < y < q 2 < x <. < g < q 2 a m < x < < y < q 2 < x < 2
q Ý l s v k Ž X s q Ys . q
Ý l s Ž c y 2 k . Ys2
s
s
- lM Ž c y 2 k . q a M q 2 < y < 2 q Ž2 a M < x < q < g <. < y < q Ž 2 < g < q 2. < x < - y< y < 2 q Ž 4a M p q < g < . < y < q Ž 2 < g < q 2 . Ž 2p . s y< y < 2 q Ž 4a M p q < g < . < y < q 4p Ž < g < q 1 . . COROLLARY 3.1.
M0 s
Let
'
Ž 4pa M q < g < . q Ž 4pa M q < g < .
2
q 16p Ž < g < q 1 .
2
.
Then dU Ž x, y . dt
-0
for < y < G M0 . Proof. Since M0 is the largest root of the following second order equation in < y < y< y < 2 q Ž 4a M p q < g < . < y < q 4p Ž < g < q 1 . s 0, we have the result. We recall the following from Hale w10x. Let M be a smooth finite dimensional manifold and T Ž t . be a smooth flow on M, where t g R. T Ž t . is said to be bounded dissipative if there exists a bounded subset W ; M which attracts every bounded set, that is, for any bounded subset V ; M there exists t s t Ž V . F 0 such that T Ž t .Ž V . ; W for all t F t . Let A ; M be a compact invariant set. A is said to be maximal if every compact invariant set of T Ž t . belongs to A. A is said to be a global attractor if A is a maximal compact invariant set which attracts every bounded set. The following theorem follows directly from Theorems 3.4.7 and 3.4.8 in Hale w10x.
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CHAN AND CHAO
PROPOSITION 3.2. Let M be a smooth finite dimensional manifold and T Ž t . be a smooth flow on M. If T Ž t . is bounded dissipati¨ e, then T Ž t . has a global attractor. For our system, Proposition 3.1 and its corollary imply that system Ž3.3. is bounded dissipative. Hence, we have the following main result by Proposition 3.2 THEOREM 3.1.
Let
v s 2,
B s 2Ž c y 2 k . q k
and c y 2k )
3 q aM
lM
,
where a M s max a i 4 .
Then system Ž3.3. has a global attractor.
4. SYNCHRONIZATION In this section, we will show that if the coupling strengths are sufficiently large, then the global attractor of Ž3.3. is contained in an e neighborhood of the origin in T n = R n, where e ) 0 is a small constant. In other words, the system is synchronized in the e neighborhood of the origin. Let
as
1 n
n
Ý
ai ,
gŽ t. s
is1
1 n
n
Ý gi Ž t . , is1
d i s Ž a i y a . yi q Ž g i Ž t . y g Ž t . . and
d1 . d s .. , dn
0
GŽ t . s
gŽ t. .. . .
0 gŽ t.
Then Eq. Ž3.3. can be written as
½
˙x s y ˙y s a y q f Ž x . q G Ž t . q kD x q cD y q d .
Ž 4.1.
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By assuming for all t g R and i, j s 1, 2, . . . , n, < ai y a j < F e ,
< gi Ž t . y g j Ž t . < F e ,
we have that there exists a constant N ) 0 such that < d < - Ne .
Ž 4.2.
In the following, we will consider a series of changer of variables to transform Eq. Ž4.1. into a simpler system. First, consider the following change of variables on R n = R n, I x y s P
0 I
x v ,
where w g R n, I is the identity map and P: R n ª R n is a linear map that is to be determined. Note that the transformation is nonsingular for any P. When this transformation is applied to Ž4.1., one obtains
¡˙x s Px q v ¢ qŽ a P q kD q cD P y P . x.
~˙y s Ž cD v q a y v Pv . q f Ž x . q G Ž t . q d
Ž 4.3.
2
In order for the above equation to have a simpler form, we consider the following equation for P: P 2 s cD P q kD q a P.
Ž 4.4.
If P satisfies Ž4.4., then the coefficients of x in the second equation of Ž4.3. will vanish. By using the notation in Section 2, let P: R n ª R n be
Ý rs X s j s ,
Px s
r s g R,
s
where r s , s s 1, . . . , n, are to be determined later. Since Pj s s r s j s for any s s 1, . . . , n, we have r s g s Ž P . and j s is its corresponding eigenvector. By a simple computation, one obtains P 2 x s P Ž Px . s
Ý X s rs2j s s
cD Px s cD
ž Ý X r j / s cÝ X r l j s
s
s
s
kD x s k Ý X s l s j 2 s
a Px s a Ý X s r s j s . s
s
s
s
s
s
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CHAN AND CHAO
By substituting the above equation into Ž4.4., we have n
P2 x s
Ý
X s r s2j s
ss1
Ž cD P q kD q a P . x s Ý X s Ž ar s q k l s q c r s l s . j s . s
Thus, for arbitrary X s , s s 1, . . . , n, we have n
n
Ý
Ž X s rs2j s . s
ss1
Ý
X s Ž ar s q k l s q c r s l s . j s .
ss1
Hence, for any s s 1, . . . , n,
r s2 s a r s q k l s q c r s l s . In summary, we have the following: PROPOSITION 4.1. There exists a linear operator P: R n ª R n satisfying P 2 s cD P q kD q a P and P has simple eigen¨ alues r s gi¨ en by
rs s
a q cls q
'Ž a q c l .
2
s
q 4 k ls
2
,
s s 1, 2, . . . , n
with corresponding eigen¨ ector j s. By the change of variables I x y s P
0 I
x w
Ž4.1. takes the form
½
˙x s Px q w w ˙ s cDw q a w y Pw q f Ž x . q G Ž t . q d .
Ž 4.5.
Since we are interested in the synchronization of the oscillators, we consider the difference of any two arbitrary solutions of Ž4.5. Ž x, w ., Ž x*, w*., Let u s x y x* and ¨ s w y w*. Then,
½
u ˙ s Pu q ¨ ¨˙ s cD¨ q a ¨ y P¨ q g Ž x, x* . u q dˆ,
Ž 4.6.
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107
where
g1 . g s .. gn
0
and
gi s y
1 n
1
H0
cos Ž xUi q u Ž u i y x* . . du
for all i s 1, 2, . . . , n and
dˆ s d Ž x, w, t . y d Ž x*, w*, t . . Note that < dˆ < F 2 Ne . First observe that if Ž u, ¨ . s Ž0, 0. is asymptotically stable for Ž4, 6., then the system Ž4.1. is synchronized. On the other hand, if the global attractor is contained in a small ball around Ž u, ¨ . s Ž0, 0., then the system is close to synchronization. Next, consider the following nonsingular time dependent change of variables, u s
¨
I QŽ t .
0 I
u h ,
where Q is a time dependent bounded n = n matrix. If QŽ t . satisfies the matrix Riccati equation
˙ s RQ y Q 2 q g , Q
Ž 4.7.
where R: R n= n ª R n= n is a linear mapping defined by RQ s a Q y QP y PQ q cDQ,
Ž 4.8.
then by applying the above change of variables to Ž4.6., we obtain
½
u ˙ s Pu q Qu q h ¨˙ q cDh y Ph q ah y Qh q dˆ.
Ž 4.9.
Since Eq. Ž4.9. is lower triangular, its stability properties can be obtained directly from the coefficient matrices of u and h. Hence, we consider first the solvability of Eq. Ž4.7..
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CHAN AND CHAO
In order to simplify the notation, we will use the set B s j 1, . . . , j n4 of eigenvectors of D as a basis in R n. Then the linear operators D, P defined before have the matrix representation D s diag Ž l1 , . . . , l n . P s diag r 1 , . . . , rn 4 . The basis B can be extended to R n= n in which case the basis elements E i, j N i, j s 1, . . . , n4 have the representation a11 a21 Ei, j s . .. a n1
a12 a22 .. . a n2
??? ??? .. . ???
a1 n a2 n .. , . an n
where ak l s
½
if k s i , l s j otherwise.
1 0
Let L : R n= n ª R n= n be defined by L Ž E . s D E, where E g R n= n is a matrix. Then L Ž E i , j . s li E i , j ,
i s 1, . . . , n., j s 1, . . . , n.
Thus l s , s s 1, . . . , n, are eigenvalues of L with eigenvectors E s, j, j s 1, . . . , n. Next, let M : R n= n ª R n= n be defined by M Ž E . s EP q PE. Then M Ž E i , j . s Ž r j q ri . E i , j . Therefore r j q r i , i, j s 1, . . . , n, are eigenvalues of M with corresponding eigenvectors E i, j. Now R can be expressed by R E s a E y EP y PE q cD E s a E y M E q c L E. It is clear that R has eigenvalues def
def
ri , j s m i , j c s a y Ž r j q r i . q c l i , with eigenvectors E i, j.
i , j s 1, . . . , n
Ž 4.10.
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For the solvability of Eq. Ž4.7., we need to show that R is asymptotically stable. PROPOSITION 4.2. Let P and R be the linear operators gi¨ en in Proposition 1 and Ž4.8., respecti¨ ely. Then there exist constants l ) 0, K ) 0G ) 0, and k 1 such that if c G Gk G k 1 , then for all t G 0 we ha¨ e k
< P< F K
c
< e P t < F Key Ž l . t k c
,
and r s max ri , j - yKc. Proof. In the following we let K, l ) 0 be some large constants. Let a s < a < q c < l s <, b s k l s . Then a ) 0, b - 0, and 2 r s s ya q Ž a2 q 4 b .
1r2
.
Since a - 0 and l s - 0 are fixed, we have b
k < ls <
s
a
< a < q c < ls <
s
k < ls < c Ž < a
FK
k c
provided c is large enough. Next for any s s 1, 2, . . . , n, if c 4 0 then < rs < s s
1 2
ž ya q Ž a
1 2
-K
ž k c
2
q 4 b.
ya q a 1 q
ž
1r2
2b a2
/
q
2 b2 a4
q
4 b3
.
Let r s max< r s <4 ; we have 5 Px 5 5 x5
F r,
for x / 0.
Hence, < P< F K
k c
.
Similarly, we obtain
r s - yl
k c
.
a6
q ???
//
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CHAN AND CHAO
The last inequality can be obtained by observing that ri j s a y Ž r j q r i . q c l i s s
1 2 1
Ž a y 2 ri q c li q a y 2 r j q c li .
2
'Ž a q c l .
y
i
2
q 4 k li q c li q
ž
a c
y
2 rj c
/
.
This proves the proposition. It remains to solve the matrix Riccati differential equation Ž4.7. which is equivalent to the integral equation QŽ t . s
t
H0 e
R Ž tys.
2
yQ Ž s . q g Ž s . ds.
Ž 4.11.
PROPOSITION 4.3. There exist constants G ) 0 and k 1 such that if c G Gk G k 1 , then the matrix Riccati differential equation Ž4.7. has a bounded solution for t G 0. Proof. By Proposition 4.2, R is asymptotically stable. It follows from the contraction mapping theorem and Eq. Ž4.11. that Ž4.7. has a bounded solution. Since the arguments are standard, they are omitted. By Theorem 3.1 and the above arguments, we have the following main result on synchronization: Let Ž x, y . and Ž x*, y*. be any two solutions of the
THEOREM 4.1. equation
½
˙x s y ˙y s Ay q f Ž x . q g Ž t . q kD x q cD y.
For any e ) 0, there exist constants G ) 0 and k 1 , such that if c G Gk G k 1 , then the following holds: lim sup Ž < x Ž t . y x* Ž t . < q < y Ž t . y y* Ž t . < . F e . tª`
5. NUMERICAL EXPERIMENTS In this section, we use the Runge]Kutta method to simulate coupled Josephson equations when n s 9. The Josephson equation with different parameters is shown when they are not coupled. Figure 10 indicates the coupled Josephson equations after they are coupled. In Fig. 11, we show the maximal absolute value of the differences of oscillators. In Figs. 1]9, the x y y projection of attractors generated by the Josephson equations ¨ x q a˙ x q sin x s m cosŽŽpr3. t . are shown.
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FIG. 1.
FIG. 2.
a s 0.3, m s 3.65, x 0 s 3, y 0 s 0, t 0 s 0.
a s 0.31, m s 3.37, x 0 s 1, y 0 s y1, t 0 s 0.
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CHAN AND CHAO
FIG. 3.
FIG. 4.
a s 0.31, m s 3.52, x 0 s y.5, y 0 s y2, t 0 s 0.
a s 0.32, m s 3.51, x 0 s 2, y 0 s y2, t 0 s 0.
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FIG. 5.
FIG. 6.
a s 0.4, m s 3.65, x 0 s .2, y 0 s y1.2, t 0 s 0.
a s 0.42, m s 3.55, x 0 s y.5, y 0 s y2, t 0 s 0.
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CHAN AND CHAO
FIG. 7.
a s 0.7, m s 3.65, x 0 s y.2, y 0 s y1.2, t 0 s 0.
FIG. 8.
a s 0.72, m s 3.5, x 0 s y.5, y 0 s y1.5, t 0 s 0.
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FIG. 9.
115
a s 0.85, m s 3.2, x 0 s 1, y 0 s y.5, t 0 s 0.
FIG. 10. x y y projection of attractor generated by the coupled Josephson equations for c s 40, k s 10.
116
FIG. 11.
CHAN AND CHAO
t y max< x i y x j <4 projection generated by the coupled Josephson equations.
REFERENCES 1. V. S. Afraimovich, S. N. Chow, and J. Hale, Synchronizations in lattices of nonlinear Duffing’s oscillators, Phys. D, in press. 2. V. S. Afraimovich, N. N. Verichev, and M. I. Rabinovich, Stochastic synchronization of oscillations in dissipative system, Izy. Vyssh. Uchebn. Za¨ ed. Radiofiz. 29, No. 9 Ž1986., 1050]1060; So¨ iet Radiophys. 29 Ž1986., 795. 3. T. L. Carrol and L. M. Pecora, Synchronization nonautonomous chaotic circuits, IEEE Trans. Circuits and Systems 38 Ž1991. 453]456. 4. W. Chan, Stability of subharmonic solutions, SIAM Appl. Math. 47, No. 2 Ž1987., 244]253. 5. S. N. Chow and J. Hale, ‘‘Methods of Bifurcation Theory,’’ Springer-Verlag, New YorkrBerlin, 1982. 6. S. N. Chow and W. Liu, Synchronization, stability and normal hyperbolicity, Resenhas, in press. 7. L. O. Chua, M. Itoh, L. Kosarev, and K. Eckert, Chaos synchronization in Chua’s circuits, J. Circuits Systems Comput. 3, No. 1 Ž1993., 93]108. 8. L. Fabiny, P. Colet, and R. Roy, Coherence and phase dynamics of spatially coupled solid-state laser, Phys. Re¨ . A 47, No. 5 Ž1993.. 9. H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled oscillator systems, Progr. Theoret. Phys. 69, No. 1 Ž1983., 32]47. 10. J. Hale, Asymptotic behavior of dissipative systems, in Math. Surveys Monographs, Vol. 25, Amer. Math. Soc., Providence, Rhode Island, 1988. 11. J. F. Heagy, T. L. Carrol, and L. M. Pecora, Synchronous chaos in coupled oscillator systems, Phys. Re¨ . E 50, No. 3 Ž1994., 1874]1885. 12. K. Wiesenfeld, S. Benz, and P. Booi, Phase-locked oscillators optimization for arrays of Josephson junctions, J. Appl. Phys. 76 Ž1994., 3835]3846.
218, 97]116 Ž1998.
AY975749
Synchronization of Coupled Forced Oscillators Whei-Ching C. Chan* and Yi-Duo Chao Department of Mathematics, Tamkang Uni¨ ersity, Tamsui, Taiwan, 25137, Republic of China Submitted by William F. Ames Received June 10, 1997
1. INTRODUCTION Synchronization usually means the concurrence of all or some parts of motions Že.g., phase. with respect to time in coupled systems. For example, in some simple forced systems, one often observes phase-locking phenomena, or subharmonic motions Žsee Chan w4x, Chow and Hale w5x.. This is more often called phase synchronization. In recent years, Fujisaka and Yamada w9x observed the concurrence of motions in time for a coupled system but where each individual system is chaotic. They called this phenomena the stochastic synchronization. In w2x, Afraimovich, Verichev, and Rabinovich proved mathematically that stochastic synchronization exists in a system of globally coupled, periodically forced Duffin’s equations. Their method of proof was based on the theory of asymptotic stability. In w7x, Chua, Itoh, Kosarev, and Eckert showed experimentally, numerically, and mathematically that it is possible to synchronize Chua’s circuits even though each circuit is chaotic. Their theory was based on a detailed study of Liapunov exponents or asymptotic stability. Other types of mechanisms to produce synchronization in coupled systems have been shown by Carrol and Pecora w3x, Fabiny, Colet, and Roy w8x, and Heagy, Carrol, and Pecora w11x. Recently, Chow and Liu w6x showed how different notions of synchronization can be studied using the theory of invariant manifolds. In this paper, we consider an array of diffusively coupled Josephson junctions with periodic forcing. By using the theory of attractors and stability theory, we are able to prove the phenomena of synchronization *Partially supported by NSC Grant 86-2115-M-032-005. 97 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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CHAN AND CHAO
when the parameters in our system satisfy certain conditions. Our work in some way is similar to the recent work of Afraimovich, Chow, and Hale w1x. However, there are differences, especially, in studying the solvability of a matrix Riccati equation.
2. MODEL Josephson junctions are superconducting devices that are capable of generating voltage oscillations of high frequency. We now give a mathematical description. Let f be the difference of phases of the superconductors across the junction. The rate of slippage of f is governed by the Josephson voltage-phase relation Vs
F0 2p
f˙ ,
Ž 2.1.
where F 0 is a constant proportional to Planck’s constant. By Kirchhoff’s voltage and current laws, we have the equation CV˙ q
V R
q Ic sin f s I,
where C, R, Ic , and I are constants. By Ž2.1., we have F0 2p
f¨ q
F0 2p R
f˙ q Ic sin f s I.
Ž 2.2.
Equation Ž2.2. is refereed as a Josephson equation and has been studied by many authors Žsee, for example, Wiesenfeld, Benz, and Booi w12x.. If the devices are connected along a ring or line, we obtain the equation
¡¨ ¢
˙j
~f q af q sin f j
j
s G q k Ž f jq1 y 2 f j q f jy1 . q c f˙jq1 y 2 f˙j q f˙jy1 ,
ž
/
Ž 2.3.
where i is an integer index, a , k, and c are constants; and G is a time dependent function. k and c represent the strength of the interaction between nearest neighbors. In this paper, we consider the mathematical problem of synchronization of finitely many Josephson junctions. In w12x, the authors considered the synchronization of phases by using numerical methods and an approximation scheme. In Afraimovich, Chow, and Hale w1x, Duffin’s equations were considered.
99
SYNCHRONIZATION
3. GLOBAL ATTRACTOR By rescaling the constants in Eq. Ž2.3., we consider an array of n ) 1 damped Josephson equations with diffusive coupling and Dirichlet boundary conditions at both ends,
¡˙x s y ~˙y s a y y sin x q g Ž t . ¢ qk Ž x y 2 x q x i
i
i
i i
i
iq1
Ž 3.1.
i
i
iy1
. q c Ž yiq1 y 2 yi q yiy1 . ,
where i s 1, . . . , n, x i s x i mod 2p g S 1, yi g R, , k, c, and the a i ’s are real constants, and the g i Ž t . are scalar T-periodic functions with T ) 0. The Dirichlet boundary conditions at both ends are given by x 0 s x nq1 s 0,
y 0 s ynq1 s 0.
Ž 3.2.
We note that
xs
x1 n ! # " n .. g S = ??? = S sT . xn
0
y1 .. n . gR . yn
0
ys
and
Notice that we have identical undamped oscillators at every lattice site. However, this is not a restriction and the theory presented in this paper works for non-identical ones. In vector notation, let ysin x 1 .. f Ž x. s , . ysin x n
0
gŽ t. s
g 1Ž t . .. , .
0 gnŽ t .
a1 0 0 As . ..
0 a2 0 .. .
0 0 a3 .. .
0 0 0 .. .
??? ??? ??? .. .
0 0
0 0
0 0
0 0
??? ???
0 0 0 .. .
a ny 1 0
0 0 0 .. . 0 an
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CHAN AND CHAO
and D be the following linear map which is the discrete Laplacian with Dirichlet boundary conditions Ž3.2.: y2 1 0 .. Ds . 0 0
1 y2 1 .. .
0 1 y2 .. .
0 0 1 .. .
??? ??? ??? .. .
0 0 0 .. .
0 0 0 .. .
0 0
0 0
0 0
??? ???
y2 1
1 y2
.
Then Eq. Ž3.1. and boundary condition Ž3.2. can be written as
½
˙x s y ˙y s Ay q f Ž x . q g Ž t . q kD x q cD y.
Ž 3.3.
The discrete Laplacian D is self-adjoint and the following is well known. LEMMA 3.1. The discrete Laplacian D has simple eigen¨ alues l s and its corresponding eigen¨ ectors j s, where s s 1, 2, . . . , n such that
l s s y2 y 2cos
ps nq1
s s 1, 2, . . . , n
,
l M s max l s s y4sin 2
ž
p 2 Ž n q 1.
/
-0
and the set of eigen¨ ectors B s j 1, . . . , j n4 forms an orthonormal basis for R n. Let ² ? , ? : be the usual inner product in R n. Since B is an orthonormal basis in R n, any x, y in R n have unique representations, n
xs
Ý
n
Xs j s ,
ys
ss1
Yssj s ,
Ý ss1
where scalars X s , Ys , s s 1, 2, . . . , n, are determined by x, y, and B. We also have n
² D x, y : s ² x, D y : s
Ý
l s X s Ys
ss1 n
² x, y : s
Ý is1
n
x i yi s
Ý ss1
X s Ys .
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SYNCHRONIZATION
In this section, we will show that under some conditions on the constants k, c and a i , systems Ž3.1. and Ž3.2. have a global attractor in T n = R n. To do so we will find an appropriate Liapunov function. First we define a new linear operator L and its adjoint L* in the usual Euclidean basis by 1 0 0 L s .. .
y1 1 0 .. .
0 y1 1 .. .
0 0 y1 .. .
??? ??? ??? .. .
0 0 0 .. .
0 0 0 .. .
0 0
0 0
0 0
??? ???
1 0
y1 1
0 0 and 1 y1 0 .. L* s . 0 0
0 1 y1 .. .
0 0 1 .. .
0 0 0 .. .
??? ??? ??? .. .
0 0 0 .. .
0 0
0 0
0 0
??? ???
1 y1
0 0 0 .. . . 0 1
Thus, along solutions of Eq. Ž3.1. we have d dt
² x, Lx : s ² y, Lx : q ² x, Ly : s ² y, Lx : q ² y, L*x : s ² y, Ž L q L* . x : s y² y, D x : .
Now, consider the following scalar-valued function, U Ž x, y . s 12 ² y, y : y F Ž x . q B² x, Lx : q v ² x, y : ,
Ž 3.4.
where F Ž x . s cos x i , B, v are positive constants to be determined latter. By taking the time derivative of U along solutions of Ž3.3., one obtains Ý nis1
dU dt
s ² y, ˙ y : y ² f Ž x . , y : y B² y, D x : q v ² y, y : q v ² x, ˙ y: s ² y, Ay : q v ² y, y : q ² y q v x, g Ž t . : q v ² x, Ay : q v ² x, f Ž x . : q Ž k y B . ² y, D x : q c² y, D y : q v k ² x, D x : q v c² x, D y : .
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CHAN AND CHAO
s ² y, Ay : q v ² y, y : q ² y q v x, g Ž t . : q v ² x, Ay : q v ² x, f Ž x . : q Ý Ž k y B q v c . l s X s Ys q c Ý l s Ys 2 q v k Ý l s X s2 . s
s
s
Thus, dU dt
s ² y, Ay : q v ² y, y : q ² y q v x, g Ž t . : q v ² x, Ay : q v ² x, f Ž x . : q
Ý l s Ž c y v k . Ys2 s
q Ý l s v k X s2 q
k y B q vc
s
PROPOSITION 3.1.
vk
X s Ys q Ys 2 .
Ž 3.5.
Let
v s 2,
B s 2Ž c y 2 k . q k
and c y 2k )
3 q aM
lM
,
where a M s max a i 4 .
Let UŽ x, y . be defined by Ž3.4.. Then along any solution Ž x Ž t ., y Ž t .. of Ž3.3., we ha¨ e dU Ž x, y . dt
- y< y < 2 q Ž 4pa M q < g < . < y < q 4 Ž < g < q 1 . p ,
where < g < is the sup norm of g Ž t . Proof. Since v s 2 and B s 2Ž c y 2 k . q k, we have k y B q vc
vk
s 2.
We also have < x < - 2p ,
c y 2k )
3 q aM yl M
) 0.
Thus
l M Ž c y 2 k . q a M q 2 - y1.
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SYNCHRONIZATION
It follows from Ž3.5. and the above inequality that dU Ž x, y . dt
F a m < y < 2 q 2 < y < 2 q Ž < y < q 2 < x <. < g < q 2 a m < x < < y < q 2 < x < 2
q Ý l s v k Ž X s q Ys . q
Ý l s Ž c y 2 k . Ys2
s
s
- lM Ž c y 2 k . q a M q 2 < y < 2 q Ž2 a M < x < q < g <. < y < q Ž 2 < g < q 2. < x < - y< y < 2 q Ž 4a M p q < g < . < y < q Ž 2 < g < q 2 . Ž 2p . s y< y < 2 q Ž 4a M p q < g < . < y < q 4p Ž < g < q 1 . . COROLLARY 3.1.
M0 s
Let
'
Ž 4pa M q < g < . q Ž 4pa M q < g < .
2
q 16p Ž < g < q 1 .
2
.
Then dU Ž x, y . dt
-0
for < y < G M0 . Proof. Since M0 is the largest root of the following second order equation in < y < y< y < 2 q Ž 4a M p q < g < . < y < q 4p Ž < g < q 1 . s 0, we have the result. We recall the following from Hale w10x. Let M be a smooth finite dimensional manifold and T Ž t . be a smooth flow on M, where t g R. T Ž t . is said to be bounded dissipative if there exists a bounded subset W ; M which attracts every bounded set, that is, for any bounded subset V ; M there exists t s t Ž V . F 0 such that T Ž t .Ž V . ; W for all t F t . Let A ; M be a compact invariant set. A is said to be maximal if every compact invariant set of T Ž t . belongs to A. A is said to be a global attractor if A is a maximal compact invariant set which attracts every bounded set. The following theorem follows directly from Theorems 3.4.7 and 3.4.8 in Hale w10x.
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CHAN AND CHAO
PROPOSITION 3.2. Let M be a smooth finite dimensional manifold and T Ž t . be a smooth flow on M. If T Ž t . is bounded dissipati¨ e, then T Ž t . has a global attractor. For our system, Proposition 3.1 and its corollary imply that system Ž3.3. is bounded dissipative. Hence, we have the following main result by Proposition 3.2 THEOREM 3.1.
Let
v s 2,
B s 2Ž c y 2 k . q k
and c y 2k )
3 q aM
lM
,
where a M s max a i 4 .
Then system Ž3.3. has a global attractor.
4. SYNCHRONIZATION In this section, we will show that if the coupling strengths are sufficiently large, then the global attractor of Ž3.3. is contained in an e neighborhood of the origin in T n = R n, where e ) 0 is a small constant. In other words, the system is synchronized in the e neighborhood of the origin. Let
as
1 n
n
Ý
ai ,
gŽ t. s
is1
1 n
n
Ý gi Ž t . , is1
d i s Ž a i y a . yi q Ž g i Ž t . y g Ž t . . and
d1 . d s .. , dn
0
GŽ t . s
gŽ t. .. . .
0 gŽ t.
Then Eq. Ž3.3. can be written as
½
˙x s y ˙y s a y q f Ž x . q G Ž t . q kD x q cD y q d .
Ž 4.1.
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SYNCHRONIZATION
By assuming for all t g R and i, j s 1, 2, . . . , n, < ai y a j < F e ,
< gi Ž t . y g j Ž t . < F e ,
we have that there exists a constant N ) 0 such that < d < - Ne .
Ž 4.2.
In the following, we will consider a series of changer of variables to transform Eq. Ž4.1. into a simpler system. First, consider the following change of variables on R n = R n, I x y s P
0 I
x v ,
where w g R n, I is the identity map and P: R n ª R n is a linear map that is to be determined. Note that the transformation is nonsingular for any P. When this transformation is applied to Ž4.1., one obtains
¡˙x s Px q v ¢ qŽ a P q kD q cD P y P . x.
~˙y s Ž cD v q a y v Pv . q f Ž x . q G Ž t . q d
Ž 4.3.
2
In order for the above equation to have a simpler form, we consider the following equation for P: P 2 s cD P q kD q a P.
Ž 4.4.
If P satisfies Ž4.4., then the coefficients of x in the second equation of Ž4.3. will vanish. By using the notation in Section 2, let P: R n ª R n be
Ý rs X s j s ,
Px s
r s g R,
s
where r s , s s 1, . . . , n, are to be determined later. Since Pj s s r s j s for any s s 1, . . . , n, we have r s g s Ž P . and j s is its corresponding eigenvector. By a simple computation, one obtains P 2 x s P Ž Px . s
Ý X s rs2j s s
cD Px s cD
ž Ý X r j / s cÝ X r l j s
s
s
s
kD x s k Ý X s l s j 2 s
a Px s a Ý X s r s j s . s
s
s
s
s
s
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CHAN AND CHAO
By substituting the above equation into Ž4.4., we have n
P2 x s
Ý
X s r s2j s
ss1
Ž cD P q kD q a P . x s Ý X s Ž ar s q k l s q c r s l s . j s . s
Thus, for arbitrary X s , s s 1, . . . , n, we have n
n
Ý
Ž X s rs2j s . s
ss1
Ý
X s Ž ar s q k l s q c r s l s . j s .
ss1
Hence, for any s s 1, . . . , n,
r s2 s a r s q k l s q c r s l s . In summary, we have the following: PROPOSITION 4.1. There exists a linear operator P: R n ª R n satisfying P 2 s cD P q kD q a P and P has simple eigen¨ alues r s gi¨ en by
rs s
a q cls q
'Ž a q c l .
2
s
q 4 k ls
2
,
s s 1, 2, . . . , n
with corresponding eigen¨ ector j s. By the change of variables I x y s P
0 I
x w
Ž4.1. takes the form
½
˙x s Px q w w ˙ s cDw q a w y Pw q f Ž x . q G Ž t . q d .
Ž 4.5.
Since we are interested in the synchronization of the oscillators, we consider the difference of any two arbitrary solutions of Ž4.5. Ž x, w ., Ž x*, w*., Let u s x y x* and ¨ s w y w*. Then,
½
u ˙ s Pu q ¨ ¨˙ s cD¨ q a ¨ y P¨ q g Ž x, x* . u q dˆ,
Ž 4.6.
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107
where
g1 . g s .. gn
0
and
gi s y
1 n
1
H0
cos Ž xUi q u Ž u i y x* . . du
for all i s 1, 2, . . . , n and
dˆ s d Ž x, w, t . y d Ž x*, w*, t . . Note that < dˆ < F 2 Ne . First observe that if Ž u, ¨ . s Ž0, 0. is asymptotically stable for Ž4, 6., then the system Ž4.1. is synchronized. On the other hand, if the global attractor is contained in a small ball around Ž u, ¨ . s Ž0, 0., then the system is close to synchronization. Next, consider the following nonsingular time dependent change of variables, u s
¨
I QŽ t .
0 I
u h ,
where Q is a time dependent bounded n = n matrix. If QŽ t . satisfies the matrix Riccati equation
˙ s RQ y Q 2 q g , Q
Ž 4.7.
where R: R n= n ª R n= n is a linear mapping defined by RQ s a Q y QP y PQ q cDQ,
Ž 4.8.
then by applying the above change of variables to Ž4.6., we obtain
½
u ˙ s Pu q Qu q h ¨˙ q cDh y Ph q ah y Qh q dˆ.
Ž 4.9.
Since Eq. Ž4.9. is lower triangular, its stability properties can be obtained directly from the coefficient matrices of u and h. Hence, we consider first the solvability of Eq. Ž4.7..
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CHAN AND CHAO
In order to simplify the notation, we will use the set B s j 1, . . . , j n4 of eigenvectors of D as a basis in R n. Then the linear operators D, P defined before have the matrix representation D s diag Ž l1 , . . . , l n . P s diag r 1 , . . . , rn 4 . The basis B can be extended to R n= n in which case the basis elements E i, j N i, j s 1, . . . , n4 have the representation a11 a21 Ei, j s . .. a n1
a12 a22 .. . a n2
??? ??? .. . ???
a1 n a2 n .. , . an n
where ak l s
½
if k s i , l s j otherwise.
1 0
Let L : R n= n ª R n= n be defined by L Ž E . s D E, where E g R n= n is a matrix. Then L Ž E i , j . s li E i , j ,
i s 1, . . . , n., j s 1, . . . , n.
Thus l s , s s 1, . . . , n, are eigenvalues of L with eigenvectors E s, j, j s 1, . . . , n. Next, let M : R n= n ª R n= n be defined by M Ž E . s EP q PE. Then M Ž E i , j . s Ž r j q ri . E i , j . Therefore r j q r i , i, j s 1, . . . , n, are eigenvalues of M with corresponding eigenvectors E i, j. Now R can be expressed by R E s a E y EP y PE q cD E s a E y M E q c L E. It is clear that R has eigenvalues def
def
ri , j s m i , j c s a y Ž r j q r i . q c l i , with eigenvectors E i, j.
i , j s 1, . . . , n
Ž 4.10.
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SYNCHRONIZATION
For the solvability of Eq. Ž4.7., we need to show that R is asymptotically stable. PROPOSITION 4.2. Let P and R be the linear operators gi¨ en in Proposition 1 and Ž4.8., respecti¨ ely. Then there exist constants l ) 0, K ) 0G ) 0, and k 1 such that if c G Gk G k 1 , then for all t G 0 we ha¨ e k
< P< F K
c
< e P t < F Key Ž l . t k c
,
and r s max ri , j - yKc. Proof. In the following we let K, l ) 0 be some large constants. Let a s < a < q c < l s <, b s k l s . Then a ) 0, b - 0, and 2 r s s ya q Ž a2 q 4 b .
1r2
.
Since a - 0 and l s - 0 are fixed, we have b
k < ls <
s
a
< a < q c < ls <
s
k < ls < c Ž < a
FK
k c
provided c is large enough. Next for any s s 1, 2, . . . , n, if c 4 0 then < rs < s s
1 2
ž ya q Ž a
1 2
-K
ž k c
2
q 4 b.
ya q a 1 q
ž
1r2
2b a2
/
q
2 b2 a4
q
4 b3
.
Let r s max< r s <4 ; we have 5 Px 5 5 x5
F r,
for x / 0.
Hence, < P< F K
k c
.
Similarly, we obtain
r s - yl
k c
.
a6
q ???
//
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CHAN AND CHAO
The last inequality can be obtained by observing that ri j s a y Ž r j q r i . q c l i s s
1 2 1
Ž a y 2 ri q c li q a y 2 r j q c li .
2
'Ž a q c l .
y
i
2
q 4 k li q c li q
ž
a c
y
2 rj c
/
.
This proves the proposition. It remains to solve the matrix Riccati differential equation Ž4.7. which is equivalent to the integral equation QŽ t . s
t
H0 e
R Ž tys.
2
yQ Ž s . q g Ž s . ds.
Ž 4.11.
PROPOSITION 4.3. There exist constants G ) 0 and k 1 such that if c G Gk G k 1 , then the matrix Riccati differential equation Ž4.7. has a bounded solution for t G 0. Proof. By Proposition 4.2, R is asymptotically stable. It follows from the contraction mapping theorem and Eq. Ž4.11. that Ž4.7. has a bounded solution. Since the arguments are standard, they are omitted. By Theorem 3.1 and the above arguments, we have the following main result on synchronization: Let Ž x, y . and Ž x*, y*. be any two solutions of the
THEOREM 4.1. equation
½
˙x s y ˙y s Ay q f Ž x . q g Ž t . q kD x q cD y.
For any e ) 0, there exist constants G ) 0 and k 1 , such that if c G Gk G k 1 , then the following holds: lim sup Ž < x Ž t . y x* Ž t . < q < y Ž t . y y* Ž t . < . F e . tª`
5. NUMERICAL EXPERIMENTS In this section, we use the Runge]Kutta method to simulate coupled Josephson equations when n s 9. The Josephson equation with different parameters is shown when they are not coupled. Figure 10 indicates the coupled Josephson equations after they are coupled. In Fig. 11, we show the maximal absolute value of the differences of oscillators. In Figs. 1]9, the x y y projection of attractors generated by the Josephson equations ¨ x q a˙ x q sin x s m cosŽŽpr3. t . are shown.
SYNCHRONIZATION
FIG. 1.
FIG. 2.
a s 0.3, m s 3.65, x 0 s 3, y 0 s 0, t 0 s 0.
a s 0.31, m s 3.37, x 0 s 1, y 0 s y1, t 0 s 0.
111
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CHAN AND CHAO
FIG. 3.
FIG. 4.
a s 0.31, m s 3.52, x 0 s y.5, y 0 s y2, t 0 s 0.
a s 0.32, m s 3.51, x 0 s 2, y 0 s y2, t 0 s 0.
SYNCHRONIZATION
FIG. 5.
FIG. 6.
a s 0.4, m s 3.65, x 0 s .2, y 0 s y1.2, t 0 s 0.
a s 0.42, m s 3.55, x 0 s y.5, y 0 s y2, t 0 s 0.
113
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CHAN AND CHAO
FIG. 7.
a s 0.7, m s 3.65, x 0 s y.2, y 0 s y1.2, t 0 s 0.
FIG. 8.
a s 0.72, m s 3.5, x 0 s y.5, y 0 s y1.5, t 0 s 0.
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FIG. 9.
115
a s 0.85, m s 3.2, x 0 s 1, y 0 s y.5, t 0 s 0.
FIG. 10. x y y projection of attractor generated by the coupled Josephson equations for c s 40, k s 10.
116
FIG. 11.
CHAN AND CHAO
t y max< x i y x j <4 projection generated by the coupled Josephson equations.
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