A Model of Diffusive Coupling

CJAO

CHAPTER 7 CHAPTER 7

5. A Model of Diffusive Coupling

We perform a simple analysis to determine the form of coupling function K(d) that corresponds to diffusion. Suppose the oscillators are distributed uniformly in space, and that an oscillator at spatial position q has xi= x(q, t) and y = y(q, t). In this case, we replace (1.1) by the continuum approximation

ax at

at

I f

K(p)

^\12

o(p),

F(x, y) +

J K(~ ^{q) x(p,}

G(x, y) +

^£

J K(p - q) y(p, then (5.1) becomes

ax

- - = F(x, y) +

£ \1

x at

~ ^G(x,

^y)

⁺

^£

¹¹ ²

t)dp

(5.1)

(5,2)

This is the reaction diffusion system studied in Chapter 5. Hence K(d)

v ² o(d) provides a model of diffusive coupling.

6. A Diffusively Coupled System

We study synchronization in a system of diffusively coupled oscillators. Initially, the phases ~ of the oscillators are distributed uniformly throughout [-

~. ~J.

except for those oscillators in an infini- tesimal neighborhood of spatial position q

=

0, which are constrained to have ~

=

0 for all slow time T > 0. That is,

n(~, q, T) + o(~)

as q +

0

for fixed T >

0.

While

n(~, q, 0) n (~, q)

(6.1)

(6.2)

for q

f:

0. The oscillators at q

=

0 can be regarded as pacemakers that initiate a synchronization process in the rest of the medium.

To see how this synchronization process proceeds, we must determine the solution for B(q,

r).

Comparing

(6.2)

with the initial condition

(3.16), we find that

p {q) 1 , u{q) 0

for q

+

^0. With these values for P and u, and K(q) equation (3.15) for B{q, T) reads

as

v

² (tanh

~)

(6.3)

v

² o(q), the

(6.4)

The initial condition is B(q, 0)

=

⁰ ^for ^q

#

^0. To determine the condition on B at q

=

0, we recall that the distribution of phases becomes sharply peaked about ~ 0 as B becomes large and negative.

Since n(~, q, T) o(~) as q ~ 0, we require B(q, T) ~ - m as q ~ 0 for fixed T > 0.

It is convenient to introduce the variable U tanh

2·

^B ^The

problem for U is

au

2 2

=

^~(1- U ) V U , T > 0

(6.5)

u (

q' 0)

0 , U(O,

T) - 1 for T > 0

(6.6)

We consider the one dimensional case, with q replaced by a scalar q

"2 a2

and v replaced by - - -

2. This one dimensional problem admits a simi-

aq '-'

larity solution

U =

X(s), where s

=

~ and X(s) satisfies

IT

X"

X(O)

--=s--=-

X' on 0

1 - x ²

- 1 , X(m)

< X < m

(6. 7)

(6.8)

In Section 7, we prove that this singular boundary value problem has a monotone increasing solution X(s).

With this solution for X(s) one can determine B(q, T) from

1+X hl

B(q, T) ~ log

;-;

(6.9)

1-X liL

IT

As T

~ ⁺

^oo ^{with q} ^fixed, ^X

(~I)~-

¹ ^and ^B

~-

^oo Hence, for each q, the picture of the characteristics, in the (~, T) plane looks like Figure

7.4

with all the characteristics converging to ~

= 0.

The rate of synchronization is determined by q, decreasing as lql ~ oo This is illustrated in Figure

7.6,

where n(~, q, T) is plotted vs.

~ for various q at a fixed instant of time T.

7. An

Existence Theorem

We prove that the singular boundary value problem

X"

X(O)

___ s __ 7 X' on 0 < s < oo

1 -

x

- 1 , X{oo) 0

(7.1)

has a monotone increasing solution. The proof employs an iteration scheme on an equivalent integral equation. This scheme is an alternating pincer movement, analogous to the one employed in

D.S.

Cohen's treatment of a nonlinear two point boundary value problem[?].

For our constructive proof, it is convenient to consider

Y =- ^X(s).

We seek a monotone decreasing solution of

Y" --.::.⁸--~ Y' on 0 <

1 - y2

(7. 2)

It is easily verified that the iterates for

^{n >} ¹

are well defined, positive, decreasing functions with

(0) = 1,

y (~)

= 0.

n n

Using the

facts that and in 0

^< s < ~

' and the Lemma 7.1, we decue the ordering of the

Lemma 7.2. In 0 < s <

we have

y .

The result is given in

0 =YO< Y2

Y4

... Y2n

••• Y2n+1

.•. Y3

Y1 < 1 . We see that the even iterates converge to a lower limit function YL(s) and that the odd iterates converge to an upper limit function YR(s). Clearly,

YL(s)

YR(s). To show that the pincer movement closes, so that the sequence

converges to a solution of the boundary value problem (7.2), we prove the reverse inequality.

Theorem. YR(s)

YL(s).

Proof. YL(s) and YR(s) satisfy the equations

[y

^(s)

YL (s)

F[YR] (s) 1 -

lyR

^(~)

(7. 11)

YL (s) YR(s) F[YL] (s) 1 -

YL

^(~)

(7.12)

Differentiating equations (7.11) and (7 .12) with respect to s gives

rs

^\)

₂

Y~(s) Y~(O)

e

⁰

1 - YR(v) dv (7 .13)

fs

^\)

₂

Y~(s) Y~(O)

e

⁰ ^{1 -}

YL(v)

_dv

(7.14)

Y(O)

1 , Y(oo) 0 (7. 2)

The boundary value problem (7.2) is equivalent to the integral equation

Y(s) F[y J ^(s) ¹ ^{G[Y] (s)}

c[Y]

^(oo)

(7. 3)

where

{u 1

^vdv

c[ ^Y J ^{s) Is ^e ^{- Y} ²

^(v)

^du

(7. 4)

The operator F has a property that is crucial to all our results.

Lemma 7

.1. I f

0 Y1 (s)

Y

2 (s)

1 for 0

s

< ⁰⁰

' ^then

=

1 F[ Y 1 ] ( s) ~ F [YJ (s)

0 for 0

s

< ⁰⁰

Proof. We first note that F [Y ]<s) is well defined for Y(s) with

0 Y(s)

1 in 0

s

< 00

and that 0

F[Y ]<s)

^< ^1.

These follow

immediately from the definitions (7.3) and (7.4). Since the integrand in u2

(7 .4)

is positive and bounded above by - 2 , c[: J

^(oo)

exists, and

[ ] [ ] [ ] e

c[y]

(s) [ ]

G Y (s)

G Y

^(oo).

Hence, F Y (s) = 1 - G[YJ

^(oo)

has 0

F Y (s)

1 in 0

s

< oo •

To prove that F[Y 1 ] (s)

~ F[Y2

J ^(s), it is sufficient to show that

Using the definition of G, (7.5) can be written as

J

⁰⁰ ⁽ ^s

o Jo

- <P2(u)- <f>1(v)

- <f>1(u)- <P2(v)

- e }dudv

(7. 5)

(7. 6)

where

1,2

(7. 7)

The integrand of (7.6) is antisymmetric about the line u = v, hence, the contribution to the integral from the square 0

u

s, 0

< v ~

s is zero. This allows us to change the region of integration from

0 u

s, 0

< v < ~

to 0

u

s

< v < ~.

With this change in the

region of integration, and some simple algebraic manipulations, (7.6) becomes

= I

s ^~

^J 0

^{s e-} ^4>¹^{( u) -} ^4>²^{( v)}

^{e ^{r (}

^{v) -}

^{r (}

^u)

^{- 1}dudv}

^{(7 .8)}

where

r(u) = 4>2(u)- 4>1(u) = f

⁰

^u{

^{1 -}¹

_y; -

^{1 -}¹

^y~

^}dv ^(7.⁹⁾

Since 0

Y

1 Y

2 1 in s

0, the integrand in (7.9) is non-negative.

Hence, r(u) is a monotone increasing function. The monotone increasing nature of f(u) implies that the integrand of (7.8) is non-negative in the region of integration, which has 0

u

s

< v < oo

M > 0.

Define the sequence

Y (s)

by

We conclude that

(7 .10)

where

Y' (0) L

= -

-G...,[...-y..::,~.--(::-co~) 1 , Yi ( 0)

Using the facts that 0

YL (s)

YR(s)

1 in

=

we argue from (7.13) that

l s vdv

Y' (s) L = Y' (0) e L 0 1 - YR(v)

= Y{(O) e

From (7.14), we have

Hence, (7.16) becomes

Y{(s)

From (7.15), we see that

G G

(7 .15)

s

0, and Xj_(O)

0,

~s ¹ ^vdv

- YL(v) (7.16)

(7. 17)

(7 .18)

(7 .19)

From the definition

(7. 4)

of c[ ^Y J , and the fact

0 <

YL (s)

YR (s)

< 1

in

s > 0,

we find

(7.20)

Hence,

(7.19)

implies that

Y{ (0)

0 < < 1

Yi(O) and

(7.18)

together with YL(s), Yi(s) < 0, implies

Hence

=

+ ~s

^Y{(v)dv ^> ¹

⁺ J

^s ^Yi(v)dv

(7. 21)

(7. 22)

This is the required result. The pincer movement in

(7.10)

closes, and the sequence

{Y }

n converges to a single limit Y(s), which satisfies

Y(s) (7.23)

Y(s) provides the solution to the boundary value problem (7.2). Since, the iterates Y (s) are all monotone decreasing, the solution Y(s) is

n monotone decreasing.

--A

3 2

FIGURE 1.1

.... [

I I

, I

! r

-·!. : -~.

y

FIGURE 1~. 2

t·

=

⁰ ^-+³⁷^'

' j

' ;

,

' I

I

'··

' I

' '

I '

I I i.

I I

-'--,

...

^..,:.

' '

1 1 r

; I

I •

····- _J.

I I

,'1 I.,·- ..

r

J -·

I I

·i _I

' I

I '

-79-! . ~

I y

~ ... 0

FIGURE

l.

i

- r 1--

1=' '37 -t 73

:-·--

^{. .} ^- ,-·

I I l

,..

.. -- I

·j

l I

' I

. 1

_L.::_·-·-

^.^L.

-·-·----l

I I

I I '

191

0 ^Li"' 0

0 ^N ^1.1"\

FIGURE 1.3

1.1"\ 0

r--. 0

-

0 00

0 1.1"\

-

51T 2

FIGURE 2.1

I I I r-~---~~

~~~---~---L---~T

41T ⁰ 47T

FIGURE 2.2 I 2

w - 1

I

w - 1

FIGURE 2.3

R(T)

41T ⁰ 41T

.; 2

w - 1

.; 2

w - 1

FIGURE 2.4

= oA

A~£~

FIGURE 3

;;

a b

*

FIGURE 4. 2

'

I I

~ N

' I , ' !

-85-

r-~--~ . <~ ~ ~ . :--- . ~~---~---·~

--•-:.

I '

V"'

"" ^: ^o

^V"' ⁰ ^V"' ^Cl> ^~^·

... :~~"~

.

_._c'iN ~ ^... ^Ll!'l

N ,~ ^I_l ^(')I

- - -

I FIGURE

4.3 ...

• ... .J.._

I ' ^! ^I

I I ;

I Î i• Î Î

\ ' '

' ^I

t I ¹

r.

;- ' ^j

I_, r ~ ~ ~

.,-

- -^.-~

I '

FIGURE 5.1

S e:Ut

FIGURE 5.2

(x,y) (x,y) + £A(y - x )

¹ ¹

L

TUBE REACTOR

Dalam dokumen nonlinear oscillations in discrete and continuous systems (Halaman 76-95)

CJAO

CHAPTER 7 CHAPTER 7

5. A Model of Diffusive Coupling

ax at

at

K(p)

o(p),

F(x, y) +

J K(~ q) x(p,

G(x, y) +

J K(p - q) y(p, then (5.1) becomes

ax

- - = F(x, y) +

x at

~ G(x,

+

11 2

t)dp

t)dp

This is the reaction diffusion system studied in Chapter 5. Hence K(d)

v 2 o(d) provides a model of diffusive coupling.

~. ~J.

=

=

0

0.

(6.2)

f:

=

r).

(6.2)

+

as

v

~)

v

(6.4)

=

#

=

2·

au

=

(6.5)

u (

0 , U(O,

(6.6)

aq '-'

U =

=

IT

X"

X(O)

--=s--=-

1 - x 2

(6.8)

1+X hl

;-;

1-X liL

IT

~ +

(~I)~-

~-

7.4

= 0.

7.6,

7. An

X"

x

(7.1)

D.S.

Y =- X(s).

It is easily verified that the iterates for

are well defined, positive, decreasing functions with

(0) = 1,

= 0.

Using the

facts that and in 0

' and the Lemma 7.1, we decue the ordering of the

J K(~ ^{q) x(p,}

~ ^G(x,

⁺

¹¹ ²

v ² o(d) provides a model of diffusive coupling.

1 - x ²

~ ⁺

Y =- ^X(s).

₂

₂

Y(s) F[y J ^(s) ¹ ^{G[Y] (s)}

c[ ^Y J ^{s) Is ^e ^{- Y} ²

^du

' ^then