Introduction to pattern-formation

목표 : 맛보이기 , 개념설명

정성옥 (Seong-Ok Jeong)

* 이 version 은 약식 version 으로 정식 version에 있는 수식들과 그림들이 생략되고 , 개요만 나와 있습니다 . (2 장부터 5 장까지 )

1. What is Pattern Formation?

2. Basic Concepts

2.1 dissipative structure

2.2 symmetry breaking

2.3 primary instability

2.4 mode interaction

2.5 triad interaction

2.6 quasi-crystal

2.7 slaving principle

2.8 parametric resonance

2.9 self-organization

3. Classification of systems (bistable, oscillatory, excitable)

4. Amplitude Equation

5. Some Theoretical Aspects

5.1 amplitude eq. 의 application

5.2 secondary instabilities

5.3 defects

5.4 front dynamics

6. Simple models (CO,CML,CA)

7. Future Aspects (ex. Biology)

8. Summary ( 중요 개념 15 가지 정리 )

레일리 버나드 대류

마랑고니 패턴

( 식당국 , 라면국물 )

패러데이 패턴

super-lattice

Spatiotemporal patterns arise in numerous physical, chemical, and biological systems.

[F. Melo, P. B. Umbanhowar, H. L Swinney PRL 1995] Oscillon[Nature 1996]

Decoration of kinks

P. Umbanhowar, Physica A (1998)

-peculiar to granular material

-phase front instability

Fibrillation of Heart

2. Basic concepts

2.1 dissipative structure - 외부에서 공급되는 에너지가 구조 유지에 쓰임

2.2 symmetry breaking

2.3 primary instability - 특정한 모드가 자람 . - bifurcation (supercritical, subcritical) : subcritical 의 경우는 hysterisis

2.4 Mode interaction

2.5 triad interaction - up-down symmetry 가 깨진 system 에서 k1+k2+k3=0 인 세 모드가 공명 .

2.5 quasi-crystal - 예를 들어 동시에 8 개의 모드가 생겨나면 ?

Penrose tiling - Quasi-Crystal

2.7 slaving principle - 빠르게 decay 하는 모드는 느리게 성장하는 mode 에 빠르게 적응한다 .

- ex) mode projection method

2.8 parametric resonance

- 외부 driving 과 internal mode 의 resonance.

2.9 self-organization

3. Classification of systems

3.1 bistable - trigger wave (front propagation)

3.2 oscillatory - plane waves

- center or target

- archimedian spirals

3.3 excitable - ex) neuron, heart, BZ reaction

- trigger wave, spiral formation

- “drift”, “resonance”, “meandering”

- target wave (impurity 때문 ), vortex filament

- “activator-inhibitor model”

- cf) “turing instability” : long-range inhibitor diffusion

-> “quick long-range negative feedback”

-> 그러면 bistable, excitable, oscillatory 별 관계 없이

stable stationary dissipative patterns such as spikes or broad strata.

4. Amplitude equation - slow, fast variable 분리 후 perturbation 전개하면 secular term 없애면서 나온

- universality 는 강력한 tool 이다 .

- ex) RGL, CGL (Benjamin-Feir instability)

- 그러나 threshold 근처에서만 성립함을 기억하자 .

5. Some theoretical aspects

5.1 amplitude eq. 의 application

5.2 secondary instability - “phase equation” (cf. Kuramoto-Sivashinsky eq.)

- ex) zigzag & Eckhaus instability

- “Busse baloon”

5.3 Defects - codimension of defect = physical space dimension - defect dimension

- Codim(D) = nonattracting set 의 unstable manifold 의 dim.

- 즉 2 차원 상의 scalar filed 에서는 point defect 가 불가능하다 !

- defect 들도 dynamics 가 있다 !

5.4 Front dynamics - “Ising wall” (immobile), “Bloch wall” (chirality 에 비례한 속도 )

- “Neel point”

- “labyrinth formation”

- snow flake : “Mullins-Sekerka instability”, latent heat 관여

- flame

5.5 Simple models - Maps are proven to be powerful tools! (O.D. 의 경우 특히 )

state time-step space

CA: d d d

CML: c d d

CO: c c d (d:discrete, c:continuous)

SCTM: c d d

PDE: c c c

Patterns in granular systems are formed by discrete events – collisions between sand and container.

t=(n-1)T

t=(n+1)T

t=nT

),(),(: 11 iiii XXXXF

]),[ ],,[ (]),[ ],,[ (: TtrhTtrhtrhtrhF

Patterns in granular system can be described by models

discrete in time.

t=(n-1)T

t=(n+1)T

t=nT

),(),(: 11 iiii XXXXF

]),[ ],,[ (]),[ ],,[ (: TtrhTtrhtrhtrhF

The variable ],[ trh

is not important in the formation of patterns in

vibrated granular system

What kind of model systems can we use to capture the essential features of complex systems?

0 0

0

0 0

for

|| for

for

)(

hhh

hhh

hhh

hF

02

2tan)( hhhG

rdtrhtrhWS

trh rr

2),(),(1

),(

))(()()(1 rhGrhFrh nnn

S.-O.Jeong, H.-T.Moon, Phys. Rev. E vol.59 p.850 (1999)S.-O.Jeong,H.-T.Moon,T.-W.Ko, Phys. Rev. E vol.62 p.7778 (200

0)S.-O.Jeong,T.-W.Ko,H.-T.Moon, Physica D vol.164 p.71 (2002)

Bifurcation diagram of the model

Nucleation of oscillons

A localized structure ‘oscillon’ of the model

(a) The peak phase

(b) The crater phase

2.13

Skew-varicose Crossroll

Kink Solutions

Dynamics of kinks : We can use simple models to study some specific features of complex systems. (Universality)

7. Future Aspects

7.1 Various New Viewpoints

7.2 information processing

7.3 pattern control

7.4 biology-oriented studies

- noise

- time delay

- network

F. Melo, PRL(1993)

Localized excitations form

patterns!

Nucleation of oscillons

Oscillon ‘molecules’ and ‘crystal’[P.B.Umbanhowar, Nature 1996]

(b) The nucleation of the oscillon(a) Oscillon

Nucleation of oscillons

2.13 123.14

Stripes and oscillons

0.15 0.15

3.13

Global stripes Growing pattern

< In the hysteresis region > Oscillon chains Oscillon lattice

Time-Delay

• Interaction is not instantaneous.• In some systems such as biological systems, time-delay is co

mparable to the characteristic time of systems.• Recently, noting that an inclusion of time-delay is more natu

ral in realistic systems, several authors have investigated the effects of time delay.

• In case of coupled oscillators, previous results: - Uniform delay induces multistability, desynchronization, and

amplitude death.

- A delay proportional to a distance between elements could produce a propagating structure in 1-dim. coupled oscillators.

Time Delay Effects

M.K.S.Yeung and S.H.Strogatz, Phys. Rev. Lett. Vol.82, 648 (1999)

j

iji ttN

Kt )()( sin)(

S.-O.Jeong, T.-W.Ko, and H.-T.Moon, Phys. Rev. Lett. Vol. 89, 154104 (2002)

00

,0

)()( sin1

)()(

rr

lkijklij t

v

rt

rrN

Kt

0

2

00

,0

22 1)(N and )()(|),(),(|

rr

lk rrljkilkjir

References

I. E. Niebur, PRL (1991)

II. S. Kim, PRL (1997)

III. D.V.R.Reddy, PRL (1998)

IV. S.R.Campbell, PhysicaD (1998)

V. M.K.S.Yeung, PRL (1999)

Coupled Phase Oscillators with Time Delay

the phase of a oscillator located on in a 2-dim. discrete space

Time DelayIntrinsic frequency Coupling term

We start with the following coupled oscillator model equation:

Synchronization Frequency vs Time Delay

Patterns

Planar Oscillations

Only planar solutions occur.

Planar solutions as well as

patterns are possible.The planar solution is not

available but patterns occur.

Phase diagram of the model

Typical Patterns

The space discreteness of the system admits topological defects, so other phase configurations are also possible.

Lattice plane

H. G. Schuster and P. Wagner, Prog. Theo. Phys. Vol. 81, 939 (1989)

M.K.S. Yeung and S. H. Strogatz, Phys. Rev. Lett. Vol. 82, 648 (1999)

S.-O. Jeong, T.-W. Ko, and H.-T. Moon, Phys. Rev. Lett. Vol. 89, 154104 (2002)

vr /0

In the systems of two-coupled oscillators or globally coupled oscillators with uniform time-delays, the dynamics are determined by the value of .

in-phase locking

anti-phase locking

b t

( )a t

( )a t

( )b t

( )b t

( )b t( )a t

a t

a t

a t b t

b t

8. Summary (중요개념정리 )(1) pattern formation (2) dissipative structure

(3) primary instability (4) secondary instability

(5) mode interaction (6) triad interaction (ud sym)

(7) quasi-crystal (8) slaving principle

(9) parametric resonance

(10) bistable, oscillatory, excitable

(11) turing pattern (12) Am Eq. Ph. Eq. Op. Eq.

(13) SH, CGL, BF-instability

(14) CA, CML, CO, Time-map

(15) Time-delay induced pattern formation