Yo Horikawa Kagawa University Japan
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1 Exponential Transient Oscillations and Standing Pulses in Rings of Coupled Symmetric Bistable Maps Yo Horikawa Kagawa University Japan
description
Exponential Transient Oscillations and Standing Pulses in Rings of Coupled Symmetric Bistable Maps. Yo Horikawa Kagawa University Japan. 1. 1. Background Exponential transients Initial states → Transient states → Asymptotic states ↑ - PowerPoint PPT Presentation
Transcript of Yo Horikawa Kagawa University Japan
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Exponential Transient Oscillations and Standing Pulses in Rings of Coupled
Symmetric Bistable Maps
Yo HorikawaKagawa University
Japan
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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
)
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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)
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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
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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
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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)
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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(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
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N12
6
78
3
))}(())((ε/2{))(()ε1()1( 11 txftxftxftx nnnn
(1b), (2)
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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
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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
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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
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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)
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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)
