1

Self-organization Self-assembly

( )H. Haken : Synagetics, ( )Santa Fe group : Complex systems, ( )I. Prigogine : Dissipative structures

pattern formation

i)

ii)

iii

No 1-1.

-----

-----

----- -----

2

BZ

CSTR

BZ

CSTR BZ

BZ

CSTR BZ

3

( )NADH NADH NAD

--- ON OFFBZ D3

BZ

NHK TV

4

OR-NOT OR-NOT

OR OR

(circummutation)

20 1H

circadian rhythms 23

5

SCN

A+B A: B:

S.Yamaguchi, et al., Science 302(2003)1408

Per-I-luc mouse Lucifarase-transgenic SCN suprachiasmatic nucleus

Turing

A,B

CIMA system BZ

Q. Quyang & H. L. Swinney, Nature, 352, 610(1991)

6

Fig. 2. Rearrangement of the stripe pattern of Pomacanthus imperator (horizontal movement of branching points) and its computer simulation. a, An adult P. imperator ( ~ 10 months old ). b, Close-up of region I in a. c, d, Photographs of region I of the same fish taken two (c) and three (d) months later. e, Starting stripe conformation for the simulation (region I). f, g, Results of the calculation after 30,000 (f) and 50,000 (g) iterations. h, Close-up of region II in a. i-l, Photographs of region II of the same fish taken 30 (i), 50 (j), 75 (k) and 90 (l) days later, respectively. m, Stating stripe conformation for the simulation (region II). n-q, Results of the calculation after 20,000 (n), 30,000 (o), 40,000 (p) and 50,000 (q) iterations, respectively. Fish (Fish World Co. Ltd (Osaka)) were maintained in artificial sea water (Martin Art, Senju). Skin patterns were recorded with a Canon video camera and printed by a Polaroid Slide Printer. In the simulated patterns, darker color represents higher concentrations of the activator molecule. Equations and the values of the constants used, as Fig. 1.

7

39 -0.01 /min 39s

.

PbI2 PbNO3+2KI PbI2+K2NO3

8

Scientific American 279,No.4(1998)

DLA DLA Df=1.706

H.Fujikawa, Physica A189(1992)15

(20g/l ) (7g/l) (1g/l) (15g/l: ) DLA

9

Benard-Rayleigh

R-B

10

GP GP

11

Karman Reynolds Re

1.2.

12

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

1

13

2 2-1.

N0

E0

�

0

000

�

�����N

Nor

EE ��

�

(1)

energy

�� 0

00

��N

N��

T1 T2

.

energy E0 N0

14

(2) flow

�E1 �E2 , �N1 �2

�E1��E2 , �N1��2

�����00 N

Nor

EE ��

�

�� flow �

||

15

2-2.

�(S)�

�(S) = (flux ) ( ) =

J��: � X� : �

( : Fick’s law )

(1)

J� X�

Fick’s law

( )

Fourier’s law

�, = 1,2, ,n L� : L���: etc.

0�� ��

� XJ

xnDJ��

���

0)( �� � � dVXJSP ��

�

xnDJ��

�� �� �

� Jxt

n��

����

���

�

���

����

����

��

�

���

TgradLJ.etc

xT

J TT

1�

� � XLJ ��

16

i j

(i) .

J = LX

L X .

L=L0 + L1X .

L L0Xi + L1XiXj 2

Onsager

L� �= L � (��� )

detailed balance . cyclic balance .

2fluxflow .

JT , JN (1) T1>T2, 1

0<� 20

I II t = 0 .

(2) T1 , T2 . (3) 1

!< 2!��

I II JN t JN 0 . 1

!> 10�"� 2

!< 20�

XT : #T �XN $� /T �

��� �

17

DT 0 (t !) JN =0 JT

�(S) = JTXT + JNXN > 0 JT = L11XT + L12XN JN = L21XT + L22XN

t !"���JN 0 L21XT + L22XN = 0

Onsager L21 = L12 �(S) = L11XT2 + 2L21XTXN + L22XN2

X�

% &

% &0

2

22

2221

2221

�

'�

'���

�

NT

NTconstTXN

XLXL

XLXLSX

�

dVXJdVSP ��

��� � ��� )(

dVJdXdt

Pd

dVXdJdt

Pddt

Pddt

PddtdP

tJ

tx

Jx

��

�

��

�

� ��

� ��

'�

���

����

��

���

����

��

dtdJ

Jd

dtdX

Xd

t

t

��

��

0(dt

Pdx

18

Onsager

Soret

Soret

J1, J2 Jq

(1)

% &

dtPddVJdX

dVXLdX

dVXdXLdVXdJdt

Pd

Jt

t

ttx

�� ��

� ��

� �����

�

� ��

� �

� ��

�

dtPd

dtPd Jx �

02 (�'�dt

Pddt

Pddt

PddtdP xJx

)*

)+

,

''�

''�

''�

qqqqqq

qq

qq

XLXLXLJXLXLXLJ

XLXLXLJ

2211

22221212

12121111

19

Soret X1=0 X20�� �

J1=L12 - X2

Soret Doufor

Onsager

L12 = L21, L1q = Lq1 = -L2q = -Lq2 (2)

(3)

L1q (thermodiffusion Soret ) Lq1 Dufour (diffusion thermoeffect)

J1

(4)

L1q Lq1

)))

*

)))

+

,

�#���

��

�#���

T)(

LT

TLJ

JJT

)(L

TT

LJ

Tqqqq

Tq

2112

12

2111211

�

�

- .extbaropSoret CCCCDJ #'#'#'#�� /1

20

% &

% &

!�#

���

��� #�

���

��� �#��

#��

tTTTs

TTJ

TTJs

q

q

,0

2

2

2

��

0��

�

�T : �P :

D12 :

Soret entropy production

t ! , J1 = 0

( 0 : )

extB

Pbarop

TSoret

Uk

C

PP

C

TT

C

#12

�#

#�#

#�#

exp

�

�

1�

1��

���

����

�#'#��

logddC

logddC

TT

CDJ

T

T

21

121

�

�/

% & % &

20

03

1

TTJ

dttJJVTk

D

q

mqB

T

0�� �#��

� �!

% & % & dttJJVTk qq

B�!

�0

2 03

1�

12DDS TT �

% &dVsV� �

% & % & % &% &

% &% &t,rT

t,rJ

t,rT

t,rTt,rJs T

q �

�

�

��

�#

�#

�� 12

21

entropy production

1.Onsager

( ) #1� ( )

2.

Jm 0, Jq = const. (0)

3. 4. (1) Soret

Soret

Soret

0

t1 P

P = 0 � (s) = 0

P 0 � (s) 0

X P

22

(Onsager )

X1 X2

P X1 X2

X X+3X�

3P�

3P ��0

(3) L

L 4=JX=LX2

X X + 3X flux J(t)

J(t)=L0 X(t) L0 Onsager

flux

J(t)=L X(t) + R(t)

R(t)

LL0

L

Onsager

q(t)

% & % & % & dseqsqk

LL si

B

55 �!

�'�00 01

% & % & % & si

B

eRsRdsk

L 55 �!

��0

01

23

1 Onsager 2

EHD

G.Gallavoti PRL, 77 (1996), pp.4334-7, “Extension of Onsager’s Reciprocity.” W. I. Goldburg et al. PRL, 87(2001), pp.245502-1-4, “Fluctuation and Dissipation in Liquid Crystal

Convection.”

No

!

Nu = S :

= S0 + q(t) + 6(t)

q(t) 6(t)

( )

24

q(t) 1 2

Nu

2 (1) Onsager (2)

2 (1) (short time scale)

( ) (2) (long time scale) ( )

( )

2-3.

(1)

25

i) i) kB1 �kB17

ii) ii)

iii) 1 iii)

F=U-TS 1 S=0

F=U S F=U-TS 0 S

U TS

US

26

8 (1) U (2) S

i) (1),(2) (U S ) : S U

ii) S U

(1),(2) (U S ) (2)

Rayleigh-Benard

�T qc

27

RB

1)

2.7.1

(2.7.3)

2)

3Ekinetic = ( ) +( (z )) < 0

% & % &% &

% &- . 0

1

2 ��#�

�

���

�

���

�#'9�##�

�

�

dVw

C

dVpuCTwSP

v

v

: :�

/0�

3/

:::��

0)( ���� dVXJSP ��

�

% & % &%& &

% &- . 02& ��#%%�

�

��������

��������

#'#%#%

�

�

dVw

C

dVppuC

wS%%P

vCC

vCC

: :�--/0�

3/

:w::�

% & 00

2 ����

��� �#� � dzw

d: :�

0 ���0

28

< >

; (3/ ~ �: ) thermal expansion

3)RB

w w u, v, w

shear dissipation shear rate �xw , �zu

�xw qxw qx

qx �zu �zu

- . 00

2, <'�� dzwgu

d

ji :�;

wgwgxuu

j

iji /3:� �

��

�,

% &w,v,uu ��

du

dw

29

shear dissipation shear velocity

qx

shear velocity energy dissipation rate =�d

(qx )

ns equation

0��'� wu zx

wu zx ��

dqw

udw

uqx

x �8

2dqw

du

ux

z�

% &223

2

2

1xx

z

qqd

wuud �� /;=

vvvvvv

vv

vvv21

2

2

2

9#9��

��

�8

#���

���

����

���

���

;//

;

ttme

t

tm

dtdeme

�

30

qx shear dissipation rate qx ��zu �xw

shear velocity =d /;�(��x2w)w

w = qx d u) /;�(qx

4 d2 u2 ) qx

4

qx ��zu ( shear) qx ��xw ( shear)

=d qx qx-2 qx

qx4

Rayleigh

w

:� :� heat flux

= 0

TTtT 2v #�#9'�� �

�

wt

d :�:'#�

�2

wd

wq

�

�

:22�� 22

31

4)Rayleigh Ra Prandle Prqx d

energy rate =b

=b g3/w = g/�:w

energy rate =;

22

2

2 0

wdg

wd

wt

b �

/�=

�

:

:�:

9

�'#���

% &2

2

2

ud

uuz

/;/;=; ��

22

22

ud

wdgb

/;=

�

/�=

; �

� w z g

u

32

qx d -1 w u

Rayleigh

=

Prandle Pr

Pr

Pr

etc

2

24

u

wdgb

�;

� ==

;

�

�;� 4dg

�8

TdgRa ���;� 4

2

2

2

2

;>?

>�?

�;

??

;

�

;

�

d

d

Pr

�

�

��

33

2-4. 2-4-1.

L I a rod with the diameter L. 5 cm

1 R = 140

A.2

@LU

R9

�

� 140

34

U : L : ; :

U = 10 m/s, L 104 – 105 m ; 10-5 m2/s

Re 1010 !!

: reduction ( )

;U

Re �

3

22

10

10

e

T

R

smA

� ;;

sunshine U T

T+ T

35

Intermission

1750

40 30

2-4-2. (1)

F = m� = mv (4.1) �

U

F = -grad U

t -t

(4.2) : ( )

(2) i)

A + B C (4.3) [A] = CA

[B] = CB

(4.4) ii)

(4.5) t -t

(4.6)

Ugraddt

xd��2

2

Ugraddt

xd��2

2

)( ttCCkdt

dCCCkdt

dCBA

ABA

A ���B��

.)'(02 eqsFickDCDtC

C#���

)(02 C#���� DCD

tC

36

iii)

(4.7) (4.8)

t -t

k, D, �

D��

2-4-3.

Brown

:Langevin eq. (4.9)

: Newton

: (4.10)

t -t v -v f(t) f(-t)

2

i) f(t) ( 2 ) ii) v(t) ( 1 )

i) 2

ii) 1

Stoke s law (4.11) a : � :

i),ii) m M (

)

)(

.)'(0

2

2

1#���1�

��

C1#��1�

�

��

t

tt

eqsFouriert

FtfFext

�''E�� )(vv�

)(vv tf'E���

)(vv tf'E��

Ma�>6

�E

37

(M )

(m )

?M ?m

i) t ?M f (t) (4.12)

(4.13) : f(t) v(t) (M m ( ) ( ))

v(0) Langevin eq. (4.14)

(4.15)

Langevin eq.

(4.16) mpp( )

E�

1M?

'� Om?

0)0(v)(

)(2

)()(2

0

�

F��F

tf

ttM

Lktftf B 3

)0(v)()0(v)(v)0(v)(v tfttdtd

'E��

)(v)0(v)(v

)0(v)(v)0(v)(v

20 M

t tet

ttdtd

?��8

E��

E�

�t

tdt

0

��E��E� '�

t

t

sttt dssfetet0

)(0

)0( )()(v)(v

38

t0

(4.A1)

t 0

(4.A2)

v(0) v(0) <v(0)v(0)> = <v02>

(4.A3) (4.12)

(4.A4)

(4.A5) (4.A6) (4.A7)

(4.A8)

0)()0( ��� �!!'E��E� eee ttt

dssfet st )()v(t

-

)(�!�E��8

dssfe s )()0v(0

-�!E��

)()()0v(0

-

0

-

2 sfsfesdds ss FF� � �!

FE'E

!

2

0020

)(0 0

0

0

2

)(2

)(2)()(

M

LkedsLk

ssesddsLk

ssLksfsf

BsB

ssB

B

E��

F�F�

F��F

�� �

!�

E

F'E

!� !�3

3

T

Tv

Tv

T21

v21

0

2

020

20

20

ML

Mk

M

LkM

k

kME

BB

B

B

�E8

�E

�

�

��

2

0

02

2

)(2

)()()()(

M

Lk

dtttLM

ksdsfsftdtftf

B

B

�

F��FF�FF ���!

!�

!

!�

!

!�3

39

(4.A9)

L0 (4.A7)

(4.A10)

(4.A11)

E� f(s)

(4.16) v(t) 1

(4.17)

(4.18) 2

(4.19)

(4.20)

(4.21)

sdsfsfk

M

sdsfsfk

ML

B

B

FF�

FF�

�

�!

!

!�

0

2

20

)()(

)()(2

sdsfsfkM

sdsfsfk

MMM

L

B

B

FF�

FF91

�1

�E

�

�!

!

0

0

20

)()(T

)()(1

% & - .% & - .�

���E�E�

�E�E�

E'�

E��

E'�

E'��

tstt

tstt

t

dssfextx

dssfextxdtt

0

)(00

0

)(0

00

)(e-11

1v

)(

)(e-11

1v

)()(v

% & % &

���

����

��CC9�

���

����

��'�'��

��

����

��� ���

E?

E1

EEEE1

EEEE

12

14321

v 222

020

MB

ttBt

ttMk

eetMk

ex)t(x

% &t

xtxD

t 2

)(lim

20�

�!�

E�

E

1�

E

19�

!�

20v

221

limMk

tMk

tD BB

t

40

4.2

(4.A12)

: (4.22)

: (4.23)

t s ?M

!!

D kBT E� kBT T 0 : D 0 but E 0 ii) ?M t ?m Langevin eq.

tet E�� 20v)0(v)(v

E���

���� �

E��

���

���

E��

!E�

!

�202

0

0

20

0

vv0

1

v1

)0(v)(v tedtt

�!

�80

)0)v(v( dttD

�!

FF1

�E0

)()( sdsfsfkM

B

Ma

tfdtd

ak

DCDtC B

�>�>

6)(v

v6

2

�E�'E��

1��#�

�

�

)()v()(v

0tfstsds

dtd t

'��� � G

41

4.3 :

0 ( ?M t ?m )

t -t

s = -?�

? s �

1) 2)

( )

!

Onsager : Lij = Lji

Lij = Lij

0 + 3Lij

)(v

)0()()(

20

tftf

s �DD GG

)()(

vvv

v-v

tftf

dtd

dtd

dtd

��

���

����

�

�

���

�

GG

)(v)(

)(v)(

)(v)()(v)(

0

0

00

stsds

std

stsdsstsds

t

t

tt

��

��

'���

�

�

��

G

?G?

GG

42

4 Navier-Stokes random force

t -t v -v

Lagrangian picture r1 r2

r1 (Euler picture)

)(1

)(vvvv 2

0 tfPtFt

'#''#�#9'�

� ������

/;

)(vvvv 2

0 tft

'#�#9'�

� �����

;

vv���

9#9

DtDv�

t��v�

rd� v�d

vvvvvv

�����

���

gradgradtr

t

gradrdd

9�9��

���

8

9�

43

Fig. 11. Comparison of the trajectories of a Brownian or subdiffusive random walk (left) and a Lvy walk with index = 1.5 (right). Whereas both trajectories are statistically self-similar, the Lvy walk trajectory possesses a fractal dimension, characterising the island structure of clusters of smaller steps, connected by a long step. Both walks are drawn for the same number of steps (approx. 7000).

G. M. Viswanathan, H. E. Stanley, Nature, 381(1996)413

Fig.4. a, Possible flight path of a bird constructed from the longest time series, as described in the text. The time resolution of the data prevents us from considering changes in fright directions which occur more frequently than once per hour. b, Possible flight path given by the Lvy-walk model discussed in the text. Both flight paths have scale-invariant 'fractal' properties which may indicate that the distribution of food on the ocean surface is spatially scale invariant. FIG. 1 The longest of the 19 time series, with a length of 416 h. Each point in the time series gives the number of 15-s intervals in each hour for which the animal was wet for 9s or more.

44

=�= 0.1

correlation time

? 10s

total time for ex.

500 ?�

= =

Navier – Stokes equation Reynolds Re ;0#2v

;0#2v 0 Navier-Stokes equation

DtDv�

vvv ���

gradt

9'��

0vvv�#'

�� ����

t vv ���#

45

( :0.2cm 100m) 105

life time ?e ?e ?1?1�

Analogy ?m ?e E�7 ?1�

;eff = ;0 + 3; HI I �

EHD �eff = �0 + 34�������( ) BZ keff = k0 + 3k ( )

(1)

(2) (3) ( )

vv ���#

kdkver rki���� ��

)()(v � 9�

% &

% &� �

� �

��

�F��FF

FFFF�#

�9

FFF9FF'F

11

11

11vv

,

vvvv

kkdkei

kkkkk

kdkdkei

kkkrki

kkrkki

�����

�����

�������

�����

���

11vv kkk

�����

�

46

2 (1) Onsager (2)

2 (1) (short time scale)

( ) (2) (long time scale) ( )

( )

N

47

3 3-1.

d :�

-��(d ):

:

:�:: : sin2 gdMMd '��� ���

:��2Md : : ��� <<2Md

0�:�

���

����

�����

gMD

Ddd

gM �::�

: sinsin

% &Dd

dd

DDd

c

c

�

�J�

�

�*+,

KIL

'���

��� �

��

6

,0

0611

61sin

2

3

:

:

::

:::

48

49

3-2.

(1)

% &

% & % &

% & 42

3

41

21

xxxU

xUxd

d,xf

xx,xftdxd

'��

��

'���

=

=

==

=�� f (x, =�) = 0

=J�� Jss x,x 00

xsJ = = 0 xs0

(pitch-fork bifurcation) ( supercritical bifurcation)

'��'�� �� 202

2 xxdtdx

J��'� Jxx,x 02

50

�<0 xs (subcritical)

(2)

�= 0

x1 ��M�

(=�= 0, x2 = N2 , 0 )

N2 N2 > 0 transcritical pitchfork N2 < 0 hysterisis pitchfork

32

2 xxxdtdx

�'�� �=

032

2 ��'�� � xxxdtdx =

���

��� �J� �� =4

21

22

22x

51

(3)

(a)��N�> 0, ��> 0, A0 = 0 (normal bifurcation)

(b)��N�< 0, ��> 0, A0 = 0 (inverted bifurcation)

(c)��N�> 0, ��> 0, A0 0

( A0 < 0 )

( A0 > 0 )

(d)� �� � O�� 0 �� � ��� �O� 0

growth rate

O'�

'���� �i

Axxxdtdx

=�

� 053

=EEOE� 9�F'F� i

52

(4) a) Type I

stationary solution

( )

Im��= O�=0 Re� = 0 P Q qc = 0

oscillatory O��0 , qc = 0

Re� 0�=(=� + #2) =(=� q2)=0 -bx3

(1) (2)

Eckhaus

x x0 + 3x

=x = =x0 + =3x #2x = -q2x0 + #23x |x|2x = |x0|2x0 + 2|x0|23x + x0

23x* + 0(3x2, 3x3) |x|2x = (x0 + 3x) (x0 + 3x)* (x0 + 3x)

|x0|2x0 + 2|x0|23x + x023x*

% &% & 32

232

bxx

xxbbxxxx

�#'�

�#'�

=

=�

xxx ��� 3'� 0

% & xxbxx 22 �#'� =�

53

% & 02

002

0 xxbxqx ��� =�

% &220

1q

bx �� =

% & xbxxxbx 33=3 20

202 '���

3x growth rate ( )2

0x

3x = (=���2b|x0|2 + #2) 3x

��n

i q nn exx x0 + 3x

Type II

% &R S% &

% &R S0

322

22

322

�

�#'#��

T

���

�#'�#�

� � drbxxdrx

q

qqbxxx

m

=

=

=0

=

�

�

x

Type III (Swift-Hohenberg eq. )

% &R S 3221 bxxx �#'�� =�

i)Im� = O = 0 qc �0 ii) O �0, qc 0 (Oscillatory )

q 3x

54

Type IV (Kuramoto-Shivasinsky eq. Nikolaevskii eq. ) : Kuramoto-Shivasinsky eq. : Nikolaevskii eq. Goldstone Mode

3-3.

F1 = =

Fr = -mg sin :

F1 = Fr

3.3.5

( 3.3.6 ) : (3.3.7)

(1)

(2)

% & % &222 xbxax #'#�#��

% &R S % &2222 1 xbxx #�#'��#� =�

2

2

td

dml

:

022

2

2

2

�'

��

:5:

::

sintd

d

sinmgtd

dml

lg

T

lg

5>

5

2�

� 5 : T : g : m :

:

::::

�

�'�� �!5!3

53

sin

022

2

�' :5:

td

d

55

1)

C :

k2 3 i) k2 < 1 ii)k2=1 !

iii) k2 >1 :

022

2

�' :5:

sintd

d

0cos221 2

02

2�

���

�

���

����

�

����

� ::

:5::dd

dtd

dtd

dtd

% & Cdtd

dtd

dtd

dtd

'������

���

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���

:5:

:5:

cos121

0cos21

20

2

20

2

���

��� ��'���

��

���

2sin42

2sin4 222

022

0

2 :5:5: kCdtd

K : U :

% &:5: cos121 2

0

2

�'���

����

��'�

dtdUKC

20

2

25Ck �

56

i) C < 2502 ( k2 < 1 )

C 2502 250

2

:�0 k :�0 :

0 0 U�= 0

K(k) :

:�0 : << >>

:�M� :M� k2 1

���

����

�����

�

����

�2

sin4 2220

2:5: k

dtd

U: sin2

sin k� 2>

U �

�� ��

8

��

�

tdt

k

d

dtkdk

kd

00

0 22

022

sin1

cos2sin1

cos2

5U

U

U5UU

U:

U

4sin10

0 00 22

42 1��

�8 ��

1 55U

U>

dtkd

% & % &% &00

0

4

2

::

:

kKgl

T

sink

�

�

�

�

*+,

KIL ''''�

��

1

2

2

0

664422

0 220

sin165sin

83sin

211

sin11

4>

>

UUUU

UU5

�kkkd

kd

57

1� �

ii) C = 250

2 C

k2=1

t �� !� :��� >�

�

iii) C > 2502

k2 > 1

% &063.13

161

1612

00

20

0

20

1��

��

�

�

��

�

�'1�

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�

��

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>:

::>

gl

% &t

dt

0

2

2

0 20

tanh2

sin

sin1

sin1log

cos2

5:

:5

:

::

:

�8

�

'�� �

U:

:5:

sin2

sin

2sin14 2222

0

2

��

���

����

�����

�

����

� �

z

kkdtd

58

iv)

(+) (-) 2

(2)

Duffin equation

5�0

���

����

��

���

����

�V�

1

���

����

� 1����

�

����

���

�� � �

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kk

t

k

dkt

1,2

sin

12

20,

20

sin1

0

0

0 220

5:

5

>U

U

U5U

22

2

1

1

snkdn

sncn

��

��

mgA

a,mk

tcosAsinmglml

gl ��

�''

G

5::G: 0���

59

(3.3.14) Japanese attractor

(3)

5�0

Wdis( ) Ws ()

tcosasin

tt, lg

gl

0

0

2

5::G:5>

55

�''

��

���

6sin

cos6

3

0

3

2

2

:::

5:::G:

��

����

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lg

dtd

dtd

73.0,65.0,15

17.2,1,22.0

0

0

���

���

a

a

5G

5G

ta

tt

mgA

amk

tAmglml

lg

gl

gl

0

0

0

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,

cossin

5::G:5>

55

G

5::G:

�''

��

��

�''

���

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))K

))I

L

�

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10

ttcosasinx

x

�

��

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5::G

:

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limWdtt

limW ext

t

ts

t

tdis 9���� �� !�!� 000

2 115::G �

������

60

Wdis

% &::2 �,

= 2

21

; :G2:

2:G �

�� �

�

����

:G � disWQ �� S�

limit cycle

limit cycle (4) Van der Pol equation

Van der Pol equation ( :� )

(Kirchhoff’s law)

% & ���

���2����� :::G::G ��� ,22

dtdW dis

1�

1� disWQS

��

�

��

!�

!�

�

��

t

t

tt

distdis

dt

dtdtd

tdtW

tW

0

00

1lim

1lim1lim

::G

::G

�

�

Rdis iedW W����� � ::1G �

0

61

dtd

t,LC 0

20

15?5 �� ? t

% &

))))))

K

))))))

I

L

��

�

�

'�

�'''

dtvL

i

dtvd

Ci

Rv

i

Evfi

iiii

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C

R

LCR

1

0

% & 01

��'''' dtvLdt

vdC

Rv

Evf

% &

% &

% & 0v1v

v11v

0v1v

v1v

0v1vv1

vv

2

2

2

2

2

2

�')*

)+,

)K

)IL

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�')*

)+,

)K

)IL

'F''

�''''F9

LCdtd

EfCCRdt

d

Ldtd

EfRdt

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Ldt

dC

dtd

REf

dtd

0vv1v

0vv1v

2

2

2

2

�')*

)+,

)K

)IL

F''

�')*

)+,

)K

)IL

F''

dtd

fRC

L

dt

d

dd

fRC

L

d

d??

62

v v + E ( )

L g(x) :

( van der Pol )

% & % &02

2

3

v1

vv i

K

EEf '

)*

)+,

)K

)IL �

'��

�/

% &

)*

)+,

)K

)IL

���F

)*

)+,

)K

)IL

'��'

2

2

2

2

v1

1

v3

v1

1v

Kf

KEf

/

/

% &

% & 01

0111

22

2

2

2

2

2

�'�'

��

�')*

)+,

)K

)IL

����

�

����

���

xdtdx

xdt

xd

kv

RR

x

vdt

vdKR

vRRC

Ldt

vd

=

/

/

=

/�� ��� ��

63

3-4. 1.

||

5 50

entrainment Huygens : 17 synchronization

01

01

01

2

2

�'

�'

�

�' �

qLC

q

qCdt

qdL

iq

dtiCdt

diL

��

�

502

:NHK-TV

64

1 2 1

(3.5.1) 50 : 5 :

=�<< 1

(3.5.2)

A(t), U�(t) 50 , 5��

sin 5�t, cos 5�t 0

( ) sin 5�t + ( ) cos 5�t = 0 = 0 = 0 (3.5.3)

U= �� ,A, 2 0�UU== ������ ,A,,A

(3.5.3) 1 A(t), U�(t)

Dynamics

0��td

dtd

dA U5 (3) U�

% & tsinfxdtdx

xdt

xd

Poldervan

5=

5

0

1

22

2

0

1 �'��

����� ����� ��

% &% &tttAx U5 '� sin)(

))

K

))

I

L

�����

�

�

��

�

����

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���

U55

5U

U5

=

cosAf

dtd

sinf

AA

dtdA

221

241

2

02

02

65

A 5 f 0

0�td

dU

0�td

dU( -b < a < b)

(3) 2 0�td

dU

U cos U

��

�

�

��

�

� ��

��

�

�

��

�

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555

U

=5U

212

cos

41

22

sin

2

0

2

0

fA

AA

f

22

20

222

222

20

222 1

411sincos �

�

�

�

��

�

� �'�

�

�

�

��

�

����'8

555

=5

UUf

AA

A

f

2

222

2

2222

222

2

20

41

1

14

1

AA

AAAf

)*

)+

,

)K

)I

L

��

�

�

��

�

��'�

��

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� ��

��

�

�

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�

� �'

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�

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55

=5

555

5 Af

Af

221

20

20 �

�

55

5 21

2

20 �

Af

Af

Af

Af

Af

22

2)1(2)1)(1(1

221

2

00

2

02

0

����8

����'���

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�

5

55555

555

5

66

BZ

BZ

67

BZ

Definition of entrainment rate R

i, j : nearest neighbor

2

KIL

�

��

ij

n

j,iij

F

FN

R1

0 : non-entrainment 1 : entrainment

(2)

[Fig. 3.7] Entrainment in 2D Coupled Lattice Oscillators with distance d

d D [mm]

R

d D [mm]

R

��<

N

jiijr

NR

1

68

Ca

100

69

3-5. 3-5-1.

1982

2000

SN <6(t)> = 0 p t<6(t) 6(t ) > = 2QN23�(t-t’) SNR dB

N

s

PP

log1010� 3.6.1

Ps: PN: QN:

70

F.Moss etal:Int.J.Bif.Chaos, vol.4, No.6(1994)1383

(2)

Langevin eq. % & )(sin 0

3 ttAbxaxx 65 ����� ---(3.6.2)

)()()(sin

41

21

0

042

ttpttAc

cxUcxbxaxU

xUx

665 '�'�

'�''��

��

���

---(3.6.3)

6 (t) 0

c (t) c

---(3.6.4)

71

(1) c = 0 U = U0 (2) c = cth

c U0 Asin50 t A Cth

420 4

121

bxaxU '�� 03 �'�� bxaxcdxdU

ba

thc 274 3J�

72

( transient time)

U0 ( ) p (t) = Asin50 t A Cth

6 (t)

6 (t) QN

2 �U p(t) QN p(t) 6 (t)

73

E. Simonotto et al., Phys. Rev. Lett., Vol.78, No.6, pp.1186-1189(1997)

3-5-2.

(1) <;�> -B +B (-B no-output +B 1 )

threshold � pulse

74

6

��

6 (t) Gaussian �

�

0

21

0

0

2

0

3

1

)(

)(

;�

�

����

�

�

����

�

�

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��

c

cf

cf

f

dffP

dffPff

222

2

2exp)0()( NQgg �

���

�

���

� ���� �

�

���

�

���

� ���9

��

��

� 2

2

0 2exp

33)0()(

)0()(

N

cc

Q

ffgg

gg

;;

75

(2)

U�(t) 2 x1 x2 C (t)

<;>�

% & ?;?U <���

ttBt 121

)( 22

��

�

�

��

�

� ���

���

����

��

�

�

2

222

22

222

2exp

32121

sin21

)(

N

c

n

Q

fB

B

BfP

?

;?

55

;?

( 5 0 )

5���05�� ?

5; �

76

(3) modulation

< V >

NQ

A*

0 CC�

���

����

� �

���

�

���

�

���

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�� 2

2

02

2

020

2

2

22

441

3 NN

*

N

*

N

*c

Qexptcos

QA

tsinQ

AQ

AfB �55

�?

PS (f)

SNR | formal 5�0 Ps

tA 0*

0 sin 5'���

% & ��

���

�'�

��

���

����

� ����

�

20

*02

2

2

sin2

1exp3

2exp

3

tAQ

fB

QfBBV

N

c

N

c

5?

?;?

% &tcos 52121

�

��

�

�

��

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2

20

0

2

2

2*

0

2

2

*0

222

exp)2(4

)(3

)(NNN

cs QQ

A

Q

AfBfP 5353

?

���

�

���

� ��

��

�

�

��

�

� ��

�

2

20

4

2*20

10

10

2exp

3

2log10

log10

NN

c

n

sformal

QQ

Af

PP

SNR

ns PPSNR /�

77

S

C

�

�

R

% &0�)t(s

)t(x)t(sC � x t

)t(x)t(s)t(x)t(sC ��

R S 21

221

2

1

���

�

���

�

*+,

KIL �

�

)t(x)t(x)t(s

CC

5�5

0H�

H 50

22 )t(x)t(s

C�/

mT)Tvar(

R �T

first passage time var T Tm mean first passage time

<C><C1> Q [33] .

.

78

D. F. Russell, L. A. Wilkens, F. Moss, Nature, Vol. 402, pp. 291-294(1999)

79

Chaos, Vol. 8, No. 3, p. 599-603(1998)

Physical Review E, Vol. 56, No. 1, p. 923-926(1997)

A.Priplata et al., Phys.Rev.Lett.89, (2002)238101-1

80

81

BZ Fukuda, Nagano and Kai, JPSJ, 72,(2003) 487 BZ

82

83

K.Harada, S. Kai et al., IEICE trans., Vol.74, No.6, pp.1486-1491(1991)

84

P*(fa) : IN

* :

10

85

U (t)

p(x) : x

x2 : 1 | t | : x1 x2 ( )

50 = 1

212121

21

21*

),(

)()()(

dxdxxxpxx

xx

txtxt

� �!

!�

!

!�

�

�

�U

~~~~~~~

t1 x1 x1+dx1 t2 x2 x2+dx2

dtttxtx

ttxtx

)()(1

lim

)()(

*2

2

*

'1

�

'

�1

1�

!�1

dxtxpxt )1)(()(0

2 �� �!

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21

)exp(!

)( W�� xxx

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55

5 21

2

20 �

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% &220

20 1 555 ����8Af

% & 555 ��� 220

~~~~~~~

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Af

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0

0

00

21

0

21

0

21

21

5�5�

5�5�5�

86

f0

(3.5.3)

87

4 4-1.

(1)

vmmF ��� � (4.1) �

F = -grad U U

t t

A.2

(4.2)

Ugraddt

xd��2

2

Ugraddt

xd��2

2

88

4-2. van der Pol

(Kirchhoff’s law)

% & 01

�''�''Rv

dtvd

CdtvL

Evf (4.2.1)

0�''' RCL iiii

89

van der Pol

U x2 L x �x + ��x3

dU / dx

0)()(2

2

�'� xfdtdx

xgdt

xd(4.2.3)

x = x0

x = x0 + y (4.2.4)

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0111

011

22

2

2

2

�'*+,

KIL

'�'

�'*+,

KIL

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RCdtvd

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000 �F�F�F� 5

555

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Ra

CL

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20

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220

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555

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3

3212

2

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90

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2

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y2 = y3 = 0

% & 023 00

201

20 �''''� yxxyxyxyyεy ����� �� (4.2.7)

x0 = 0

0�'� yyεy ��� (4.2.8)

(4.2.8)

01,2

% & % &R S % & % & % & 03000

20102

2

�''''''��' yxyxyxdtd

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% & % & % &% &% &

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02

33

0201

30000

02

0201

320

20

30000

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constyeyty t :)( 000�

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00

421

0

22,1

2 ε

(4.2.9)

042 �� =

91

1 0�> 0 y : grow 0�< 0 y : decay

=2 4 < 0

= = 0

cii 5 0 J�J�2,1 (4.2.11) 5c : Hopf

= > 0 Hopf

= 20

2

45=

55 �� c 0(=�)

(i) 0� y(t) = A1e01t + A2e02t ( 4.2.12 )

(ii) 0�

y(t) = eRe(0)t (A ei5ct + A* e-i5ct ) (4.2.13) = = 0 ( )

y(t) = A ei5ct + A* e-i5ct (4.2.14) =�> 0 �A"�A* 5c �

�'''� 223

1021

yyyy === (4.2.15a)

t T

KIL

�'����

Ttt

t=

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���

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2,1 421

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% &

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120

230

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ti*ti

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cc

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55

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(20) (19) y0

30

20 yy �

2 (19) y2

tcitcitcitci eBeBeAeAy 5555 3*

23

2*222

�� '''� (4.2.21)

% &

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f0 : flash : 10 Hz Fig. 1 The international 10-20 electrode system. Top view of the scalp. The upper part is the front of the head. The abbreviations are as follows : Fp; frontal pole, F; frontal, C; central, T; temporal, P; parietal, O; occipital.

Fig. 2 The alpha spectra from O2 to Fp2: (i) at rest, (ii) under 10Hz photic stimulation. K. Harada, et al. IEICE trans., 74(1991)1486

166

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Fig. 5 The “on”-responses to the stimulation ; (a) the amplitude, (b) the phase and (c) the phase difference between posterior and anterior and regions.

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circadian rhythm Fig. 2 Time course of reentrainment of circadian rectal temperature rhythm to an 8-h phase-advanced schedule of sleep and social contacts. Top trace: with bright light; bottom trace: without bright light.

169

Fig. 3 Time courses of orthodromic reentrainment of circadian melatonin rhythm to an 8-h phase-advanced schedule of sleep and social contacts (subject YA). , With bright light; , without bright light. Shaded areas indicate the rest time, and horizontal open bars represent time of bright light exposure.

Fig. 6 Time courses of antidromic reentrainment of circadian melatonin rhythm to an 8-h phase-advanced schedule of sleep and social contacts (subject KA). See Fig. 3

Fig. 5 Time courses of antidromic reentrainment of circadian melatonin rhythm to an 8-h phase-advanced schedule of sleep and social contacts (subject AI). See Fig. 3 legend.

Fig. 4 Time courses of orthodromic reentrainment of circadian melatonin rhythm to an 8-h phase-advanced schedule of sleep and social contacts (subject IS). See Fig. 3 legend.

170

Circumnutation

Circummutation

Circadian Rhythm

171

Fig. 77 8:8 Canavalia ensiformis KLEINHOONTE 1929

502 - 512

Fig. 79 6:6 Canavalia ensiformis KLEINHOONTE 1929

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Fig. 80 18:18 Canavalia ensiformis KLEINHOONTE 1929

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A. G. Rossberg Physica D 115(1998)19.�

u SH

201

V

f > fc

Turbulence

(a)

(a)

(b)

(c)

200μ m

202

Nikolaevskii

22

2

2

2

2

)(1xuu

xu

xtu

��

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��

��

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�

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This was first proposed to describe the propagation of longitudinal seismic waves in viscoelastic media.

203

1D-Numerical study

M. I. Tribelsky and K. Tsuboi : PRL 76, 1631 (1996) Turbulence arises from the spatially uniform state as a result of one and the only one supercritical bifurcation.

cannot be scaled out from equations for the slowly varying amplitudes.

1~ �=t

The amplitudes of chaotic modes are scaled approximately as 1/2

.

Experimental study In electroconvection of LC S.Kai, et al J.Phys.Chem. (1996)

(Swift - Hohenberg Equation, 1977)

% & % & % &

% &% & parametercontrol:k

rategrowth:ikxtexpAuuuu,u

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204

EHD Goldstone mode A. G. Rossberg, et al., PRL 76, 4729 (1996)

Quasi-two-dimensional pattern-forming systems with spontaneously broken isotropy (experimentally accessible in electroconvection of homeotropically aligned liquid crystals)

)sin,(cos UU�C director q = (qc, pc) : critical wavenumber

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32

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223

211

212

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H : magnetic field (|| x) G1 : rotational viscosity K1 : splay elastic constant ^a : magnetic susceptibility K3 : bend elastic constant

EHD

A. G. Rossberg, et al., PRL 76, 4729 (1996) (1995 )

X6= AiAAAgAtA

'#'���

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Goldstone Mode coupling

(1) : OR( ) (2) : NR( )

Nikolaevskii-equation

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205

Coupling between Goldstone Modes and Convective Modes

M. Tribelsky & K. Tuboi PRL, 76(1996)1631 A. Rossberg, et al. PRL, 76(1996)4729 H. Sakaguchi Prog. Theor.. Phys., 96(1996)1037

q 0

206

1D Amplitude Equations of SMT

0122

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Nikolaevskii equation (1989).

% & % & % &T,Xf.c.cxiexpT,XAu 232

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Matthews & Cox PRE 62(2000)R1473

freely rotate resp. to akis. freely rotate resp. to

207

2D Nikolaevskii Equation

),()),((),( 2 tDtt rXrXFrX #'�� reaction-diffusion system

reduction

% & % &2122

32

212 X X;;;X #'##'#'#�� phase dynamics

1 2 < 0

% & % &212

212 X X;;X #'#�#�� Kuramoto-Sivashinsky eq.

1 < 0, 2 , 3 > 0

% & % &21

220

23

2 X X;=X #'���

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H. Fujisaka and T. Yamada: Prog. Theor. Phys. 106, 315 (2001)

2D-Numerical Study H. Fujisaka, et al.: Prog. Theor. Phys. 109, 911 (2003)

Deriving the amplitude equation from the two-dimensional Nikolaevskii eq. and carrying out the numerical integration of their amplitude equations

power spectrum for small k % &4.0e xp~)( CkkS �

3

22

13

20 4

,2 ;

;=;; vk '���

208

Goldstone Mode

Goldstone Mode

Schematic drawings of the director n and the angle �. The C-director represents a projection of the director n onto the xy-plane, and the angle � indicates the azimuthal rotation of the director n.

S.Kai, et al., J.Phys.Chem (1996) Rossberg 1995

leadership

H // q // x

H

( ) ( )

209

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210

Patterns EHD

Basic equation 1) orientational free energy ( elastic parts )

% & % & % &- .233

222

2112

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2) magnetic – dielectric free energy

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3) director torque balance equation

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21 GG33

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Patterns evolutions to STC in the route A and B. (a)-(c) shows each typical pattern observed for each =� in the route A (H=400 G) ((a) V = 7.498 V, = = 0.021, � = -0.017, (b) V = 7.704 V, = = 0.078, � = 0.040, (c) V = 8.017 V, = = 0.168, � = 0.13). (d)-(f) does in the route B (H=1000 G) ((d) V = 8.074 V, = = 0.074, � = -0.026, (e) V = 8.285 V, = = 0.13, � = 0.031, (f) V = 8.706 V, = = 0.249, � = 0.149). Here � ( = = -�=STC) represents a relative from a threshold =STC (H). When � is negative, the pattern ((a)or(d))shows a stationary state.

211

% &

% &

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j,ii,jj,i

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33

21

21 ���

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33

3 (6)

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reductive perturbation

Nonlinear Envelope Equation

212

6-2. (1)

i.

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21

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ii.

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% & % & % &*+,

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22

3

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121

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% & % &% & % &000 ,,HH,zcos,,zsinn x����

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213

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zyyzyxxyzxzyyzzyxxyx

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214

wall

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(1) wall ( real order parameter ) (2) dynamical wall ( complex order parameter )

(1)�f (2) transition

(wall ) 1 2

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: , sin:�cos:

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32

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232

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(phase transition of domain wall.)

215

A B (A)

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6

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31

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216

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3122

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217

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21

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218

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219

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DNA

ABA

DNA

228

(allelopathy)

C. A. Klausmeier, Science Vol.284, Jun 11 1999: 1826-1828 www.sciencemag.org

229

regular pattern Turing instability irregular pattern

W N

rate A rate L RG(W)F(N)N

G(W) : W

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F(N) = N

G(W) = W F(N) = N

J :

rate MN

v -x

230

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231

( )

232

1. 12.3.4.

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233

1000 80%

40% 100%

234

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235

(Turing type )

The Experimental Method Group Growth Dynamics Growth Stage Sensitivity for ABA

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236

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7: 2

(a) f(r)(b) L!(r)

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, dB = 2.5 9�10-3

(ii) dA = 2.5 9�10-3

, dB = 2.5 9�10-4

237

hormone sensitivity : x

10: 3(a) dA = dB = 1.0 9�10

-2, TA = TB = 2.0, ?A = ?b = 1.0 9�10

-1

(a) rA = rB = 0.0 ; ^a = ^

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D. F. Russell, L. A. Wilkens, F. Moss, Nature, Vol. 402, pp. 291-294(1999)

Chaos, Vol. 8, No. 3, p. 599-603(1998)

Physical Review E, Vol. 56, No. 1, p. 923-926(1997)

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E. Rodriguez et al., Nature, Vol. 397, pp.430-433(1999)

E. Rodriguez et al., Nature, Vol. 397, pp.430-433(1999)

305

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