Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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A
Ω/π = γ
πr
K
D
eff
=
1
2
d
d t
hφ
2
i − hφi
2
=
π
2
2
d
d t
hk
2
i − hki
2
.
φ
t
n
D
DMP
eff
=
π
2
2
γ +
hω
stat
out
i
π
− (2γ − (a
1
+ a
2
)) hσ
stat
i
2
−
1
2
(a
2
− a
1
)hσ
stat
i(1 + hσ
stat
i
2
)
!
,
D
DPP
eff
=
π
2
2
hω
stat
out
i
π
−
2
Ω
π
− (a
1
+ a
2
)
hσ
stat
i
2
−
1
2
(a
2
− a
1
)hσ
stat
i(1 + hσ
stat
i
2
) +
Ω
π
hσ
stat
i
!
.
D
eff
= D
in
+ D
out
− D
coh
hσ
stat
i
n
n = 1, 2, 3
hT i
hφ
2
i =
dφ
dt
2
hT
i
2
+ 2D
eff
hT i = π
2
.
hT i = D
eff
dφ
dt
−2
v
u
u
t
1 +
π
D
eff
dφ
dt
2
!
−1
.
hT i
D
0.00 0.02 0.04 0.06 0.08
D
10
2
10
3
10
4
T
lock
0.00 0.02 0.04 0.06 0.08
D
10
2
10
3
10
4
T
lock
D
T
E
A
1
= 0 A
2
= 0.1 A
3
= 0.2
A → 0+
hσ
stat
i
∆U = 150
f
c
= 100 f
0
= 100
hfi = lim
M→∞
1
M
P
M
k=1
π
t
k+1
−t
k
A
th
0
50
100
150
200
250
30 40 50 60 70 80 90 100 110
<f> (Hz)
Vn (mV)
<f>,çÃ
V
n
, Í÷
<f>, Hz
V
N
, mV
0.02 0.04 0.06 0.08 0.10
D
0.0
0.5
1.0
1.5
2.0
2.5
3.0
<ω>
A=0.01
A=0.05
A=0.1
A=0.15
<ω>
D
A = 0 4 A = 60 2
A = 100 ?
K = 0.2
Ω = 0.5
p(t)
40
60
80
100
120
140
20 30 40 50 60 70 80 90 100
A (mV)
Vn (mV)
Dth
Ath
V ,
A,
D
mV
N
mV
th
A
th
f
0
= 100 ? f
0
= 250 4 f
0
= 500 2
0 0.01 0.02 0.03 0.04 0.05
t, sec
0
200
400
600
P(t)
0 0.01 0.02 0.03 0.04 0.05
t, sec
0
200
400
600
800
P(t)
0 0.01 0.02
t, sec
0
1000
2000
3000
P(t)
Vn=35 mV
Vn=115 mV
Vn=70 mV
p(t)
p(t)
p(t)
t, s
t, t, s s
V V
N N
= 35 mV = 70 mV
= 115 mV
N
V
A = 100 f
0
=
100
V
N
= 35 p(t)
V
N
= 70
t = T
0
/2 V
N
= 115
x(t + ∆t) = sign [ K x(t) − D y(t) − A sin(Ω t)] .
K = 0.2
ξ(t)
0 1000 2000
t
−60
−40
−20
0
20
40
60
φ
D=0.05
D=0.06
D=0.07
t
φ
0.02 0.04 0.06 0.08 0.10
D
10
−8
10
−6
10
−4
10
−2
10
0
10
2
D
A=0.1
A=0.15
A=0.05
eff
D
D
eff
A = 0.1 Ω = 0.5
D
eff
K = 0.2 Ω = 0.5
τ
c
= 10
−2
D
φ
A = 0.1
D = 0.06
D = 0.04 D = 0.08
A ≤ A
th
D
eff
(D)
˙x = α x − x
3
+ γ (y −x) +
√
2D ξ
1
(t),
˙y = (α + ∆) y −y
3
+ γ (x − y) +
√
2D ξ
2
(t).
α
∆
γ
ξ
1
(t) ξ
2
(t)
hξ
i
(t)ξ
j
(t + τ)i = δ
i,j
δ(τ) γ = 0
x(t) y(t)
D
1
= D
2
= D
p(x, y, t)
∂p
∂t
= −
∂
∂x
[αx − x
3
+ γ(y − x)] p
−
∂
∂y
[(α + ∆)y − y
3
+ γ(x − y)] p + D
∂
2
p
∂x
2
+
∂
2
p
∂y
2
.
p(x, y)
p(x, y) = C exp
−
U(x, y)
D
,
U(x, y) = −α
x
2
2
+
x
4
4
− (α + ∆)
y
2
2
+
y
4
4
+
γ
2
(x − y)
2
,
C
U(x, y)
D = 0
x(t) y(t)
x(t)
y(t)
∆ = −0.5
y(t)
γ
Γ
2
(ω) =
|G
x,y
(ω)|
2
G
x,x
(ω) G
y,y
(ω)
,
G
x,y
(ω) x(t)
y(t) G
x,x
(ω) G
y,y
(ω)
γ
x(t) y(t)
γ = 0 r
x
r
y
r
x
=
√
2α
π
exp
−
α
2
4D
, r
y
=
√
2(α + ∆)
π
exp
−
(α + ∆)
2
4D
.
t
x
i
t
y
j
x(t) y(t)
0 1000 2000 3000 4000 5000
−2
−1
0
1
2
x(t)
0 1000 2000 3000 4000 5000
t
−2
−1
0
1
2
y(t)
x(t)
y(t)
t
0 1000 2000 3000 4000 5000
−2
−1
0
1
2
x(t)
0 1000 2000 3000 4000 5000
t
−2
−1
0
1
2
y(t)
x(t)
y(t)
t
x(t) y(t) γ = 0.05
γ = 0.5 α = 1.0 ∆ = −0.5 D = 0.07
r
x,y
= lim
N→∞
1
N
N
X
i=1
2π
t
x,y
i+1
− t
x,y
i
.
γ
t
x
i
t
y
j
Φ
x
(t) = 2π
t − t
x
i
t
x
i+1
− t
x
i
+ 2πi, t
x
i
< t < t
x
i+1
,
Φ
y
(t) = 2π
t − t
y
j
t
y
j+1
−t
y
j
+ 2πj, t
y
j
< t < t
y
j+1
.
10
−2
10
−1
10
0
ω
0.2
0.4
0.6
0.8
1.0
Γ
γ=0.05
γ=0.1
γ=0.3
γ=0.7
ω
Γ
0.0 0.5 1.0 1.5 2.0
γ
0.0
0.2
0.4
0.6
0.8
r
r
r
y
x
r
r
r
y
x
γ
Γ (ω)
r
x,y
α = 1.0 ∆ = −0.5 D = 0.1
φ(t) = Φ
x
(t) − Φ
y
(t) γ > 1
x
n+1
= (ax
n
− x
3
n
) exp(−x
2
n
/b) + A sin(Ωn)