Yo Horikawa Kagawa University Japan

1

Exponential Transient Oscillations and Standing Pulses in Rings of Coupled

Symmetric Bistable Maps

Yo HorikawaKagawa University

Japan

2

1. Background Exponential transients

Initial states → Transient states → Asymptotic states 　　　　　　　　　　　 ↑Duration (Life time) of transients increases exponentially with system size.

T exp(∝ N) T: duration of transients N: system size

02468

10

0 2 4 6 8 10System size: N

log(T

)

3

1. Background

Systems never reach their asymptotic states in a practical time.

→ Transient states play important roles.

Two kinds of exponential transients I. Metastable dynamics in reaction-diffusion systems (Kawasaki and Ohta 1982) in ring neural networks (Horikawa and Kitajima 2008) II. Transient chaos in coupled map lattices (Crutchfield and Kaneko 1988) in neural networks (Bastolla and Parisi 1998)

4

1. Background Examples of exponential transients

1. Bistable reaction-diffusion equation

Transient kink, pulse patterns → Spatially homogeneous states: u = ±1

-1

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5

u

f

)2/2/()1()(

)(/2

222

LxLuuuf

ufxutu/

l

5

1. Background Examples of exponential transients

2. Ring neural network

Transient traveling waves and oscillations → Spatially homogeneous states

),1()tanh()()(/

0

1

N

nnn

xxNngxxfxfxdtdx

45

N12

6

78

3

l

6

1. Background Examples of exponential transients

3. Bistable ring of directly coupled maps

Traveling waves → Spatially homogeneous states

45

N12

6

78

3

l

)0,,1()1()(

)0()())(()1(

0

2

1

txxNnxxxf

ctcxtxftx

N

nnn

7

1. Background 1. Bistable reaction-diffusion equation 2. Ring neural network 3. Ring of directly coupled maps

Symmetric bistability Common kinematics dl/dt ～ – exp(–l) l: width of patterns

Purpose of this study Whether exponential transients exist in lattices of coupled circle maps.

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2. Unidirectionally coupled maps Ring of unidirectionally coupled bistable symmetric circle maps

n ： index of sites, N ： the number of sites t: discrete time xn(t): state of nth site at time t ε: coupling strength

Bistable steady states ： xn = ±1/2 (1 ≤ n ≤ N)

))((ε))(()ε1()1( 1 txftxftx nnn

201ε0,,1()π2/()π2sin()(

0

K,xxNnxKxxf

N

- 0.6-0.4-0.2

00.20.40.6

-0.5 0 0.5xf(x

) K = 0.1K = 0.5K = 1.0K = 1.5y = xfixed points

45

N12

6

78

3

(1a), (2)

9

2. Unidirectionally coupled maps Random initial states: xn(0) ~ N(0, 0.12)

→ Traveling pulse waves (xn(t): 1/2 ⇄ –1/2)

Fig. 1(a). Transient pulse waves (ε = 0.2, K = 0.5, N = 20) simulation

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2. Unidirectionally coupled maps Random initial states: xn(0) ~ N(0, 0.12)

→ Traveling pulse waves (xn(t): 1/2 ⇄ –1/2)

Fig. 1(b). Transient pulse waves (ε = 0.8, K = 0.5, N = 20) simulation

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2. Unidirectionally coupled maps N: even (N = 2M)Initial states ：

→ Unstable symmetric pulse wave → Saddle manifold in the state space

Stable in the subspace: xn = -xN/2+n (1≤ n ≤ N/2)

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

NnlxNlnx

hn

hn

lhlh

(3)

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3. Bidirectionally coupled maps Ring of bidirectionally coupled bistable symmetric circle maps

n ： index of sites, N ： the number of sites t: discrete time xn(t): state of nth site at time t ε: coupling strength

Bistable steady states ： xn = ±1/2 (1 ≤ n ≤ N)

201ε0,,1()π2/()π2sin()(

0

K,xxNnxKxxf

N

- 0.6-0.4-0.2

00.20.40.6

-0.5 0 0.5xf(x

) K = 0.1K = 0.5K = 1.0K = 1.5y = xfixed points

45

N12

6

78

3

))}(())((ε/2{))(()ε1()1( 11 txftxftxftx nnnn

(1b), (2)

13

3. Bidirectionally coupled maps Random initial states: xn(0) ~ N(0, 0.12)

→ Standing pulses (xn(t): 1/2 ⇄ –1/2)

Fig. 1(c). Standing pulse (ε = 0.5, K = 0.1, N = 40) simulation

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3. Bidirectionally coupled maps N: even (N = 2M)Initial states ：

→ Unstable symmetric standing pulses→ Saddles in the state space

Stable in the subspace: xn = -xN/2+n (1≤ n ≤ N/2)

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

NnlxNlnx

hn

hn

(3)

lhlh

4. Changes in pulse width

Locations of pulse fronts: n1, n2

Speeds of pulse fronts: v1 = Δn1/Δt, v2 = Δn2/Δt

Changes in pulse width l → Difference between the speeds of two pulse fronts

15

ln1 n2

N – l

x1 xN

v1 v2

dl/dt = Δ(n1 – n2)/Δt = v2 – v1

4. Changes in pulse width

Changes in pulse width: l

~ exponentially small with pulse width: l and N – l

16

ln1 n2

N – l

x1 xN

v1 v2

)]}(αexp[)αexp({β

/)(d/d

12

12

lNlvv

tnntl

(5)

α = 2.375, β = 1.304 in unidirectionally coupled maps α = 0.651, β = 0.487 in bidirectionally coupled maps

4. Changes in pulse width

Changes in pulse width: l

Initial pulse width: l(0) = l0 < N/2

→

l(T) = 0 →

T(l0; N): Duration of pulses with initial pulse width l0 17

)]}(αexp[)αexp({βd/d lNltl (5)

))]}2/(p(arctanh[ex)2/exp(tanh{)2/exp())(exp(

0 NltNNtl

)]}2/p(arctanh[ex

))]2/(p(arctanh[ex{)2/exp();( 00

N

NlNNlT

(6)

(7)

4. Changes in pulse width

Simple forms by letting N → ∞

T(l0) ～ exp(l0) ･･･ Duration increases exponentially with initial pulse width

18

010203040

0 2000 4000 6000 8000 10000t

l l(0) = 10l(0) = 15l(0) = 18l(0) = 22

)0)(()/(]1)[exp()()2/)0((])log[exp(/1)(

)exp(d/d

00

00

TlllTNlltltl

ltl(8)

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5. Duration of transient pulses 1. Duration of asymmetric pulses: T(l0)

T(l0) ～ exp(l0)

Fig. 4. Duration T vs initial pulse width l0 in unidirectionally coupled maps (ε = 0.2, K = 0.5, N = 21)

l0

0

2

4

6

8

10

0 2 4 6 8 10l0

log10

(T(l 0

)) simulation of Eq. (1)Eq. (8)

Fig. 7. Duration T vs initial pulse width l0 in bidirectionally coupled maps

0

2

4

6

8

10

0 10 20 30 40 50l0

log10

(T(l 0

))

ε = 0.5, K = 0.2ε = 0.5, K = 0.1ε = 0.9, K = 0.1

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5. Duration of transient pulses 2. Randomly generated pulses Random initial states: xn(0) ~ N(0, 0.12)

→ Pulses with initial pulse width obeying the uniform distribution: l0 ~ U(0, N/2)

Distribution h(T) of duration T of these pulses'd)'('d)2/,0(

00 00

Tl

TThlNU

NNTNN

NTNTl

NlNlTTh

/))]}2/(exp(arctanh)2/[exp(2{cosech)2/exp(4

2d

);(d2|d/);(d|

1)( 0

00

(9)

(10)

21

5. Duration of transient pulses 2. Distribution h(T) of duration T of randomly

generated pulses

Cut-off ： Tc = exp(αN/2)/(αβ) ≈ 3×106 (N = 20)

Prob{T > Tc} ≈ 4exp(-2)/(αN) ≈ 0.357/N ≈ 0.018 (N = 20)

)0(21

)( cTTNT

Th

)()2/exp(2)/()exp(4)(

cTTNNTTh

Fig. 5. Distribution of duration of random traveling pulses in unidirectionally coupled maps (ε = 0.2, K = 0.5, N = 20)

(11)

(12)

-11

-9

-7

-5

-3

-1

0 2 4 6 8log10(T )

log10

(h(T

))

simulation of Eq. (2)Eq. (10)Eq. (11)Eq. (12)

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5. Duration of tansient pulses2. Distribution h(T) of duration T of randomly

generated pulses

Cut-off ： Tc = exp(αN/2)/(αβ) ≈ 1.1×106 (N = 40)

Prob{T > Tc} ≈ 4exp(-2)/(αN) ≈ 0.832/N ≈ 0.021 (N = 40)

)0(21

)( cTTNT

Th

)()2/exp(2)/()exp(4)(

cTTNNTTh

Fig. 8. Distribution of duration of random standing pulses in bidirectionally coupled maps (ε = 0.5, K = 0.1, N = 40)

(11)

(12)

-11

-9

-7

-5

-3

-1

0 2 4 6 8log10(T )

log10

(h(T

))

simulation of Eq. (2)Eq. (10)Eq. (11)Eq. (12)

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6. Conclusion ･ Rings of unidirectionally and bidirectionally coupled maps

→ Transient traveling pulses and standing pulses

･ Duration T of transient pulses increases exponentially with initial pulse width l0. T ∝ exp(l0)

･ Duration T of transient pulses generated under random initial conditions is distributed in a power law form.

h(T) ~ 1/T

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2-2. Duration of pulse waves2. Duration of pulse waves occurring from random

initial states

2/)α()(/)(σ)(CV

)()βα/(

]α3)2/αexp(4)α[exp()(σ

)βα/(]2/α1)2/α[exp(2)(

2/1

223

2

2

NTmTT

TmN

NNNT

NNNTm

Fig. 6. Mean, SD and CV of duration of random pulse waves vs N

Mean: m(T) ～ exp(N)SD:σ(T) ～ exp(N)Coefficient of variation: CV(T) > 1

(14)

012345678

0 5 10 15Nlog

10(m

(T)),

log10

(σ(T

)), CV

(T) m(T) with Eq. (1)

m(T) in Eq. (13)σ (T) with Eq. (1)σ (T) in Eq. (13)CV(T) with Eq. (1)CV(T) in Eq. (13)