Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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10
−3
10
−2
10
−1
10
0
10
1
D
10
−2
10
0
10
2
10
4
10
6
η
10
−3
10
−1
10
1
D
10
−4
10
−2
10
0
10
2
SNR
D B 0
0.1 0.2
0.24 γ = 0.1 Ω = 0.001
a
2
∆U − B
γ a
1
, a
2
∝ exp(−∆U/D)/ cosh(B/D) ≈ a
1
N
D
˙x
i
= f
i
(x) + ξ
i
(t), i = 1, . . . , N,
hξ
i
(t)ξ
j
(0)i = 2D δ
i,j
δ(t)
p(x)
Φ(x)
p(x) ∝ exp
−
Φ(x)
D
.
D
Φ(x)
D 1
ν(D) × exp(∆Φ/D)
T
i
T
i
∝ (a − a
cr
)
γ
,
a a
cr
γ
x
n+1
= (ax
n
− x
3
n
) exp(−x
2
n
/b) + A sin ωn +
√
2Dξ(n).
A 1
δ D
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
x
0.00
0.05
0.10
p(x)
a=2.5
p(x)
x
−2.0 −1.0 0.0 1.0 2.0
x
0.00
0.01
0.02
p(x)
a=2.84
p(x)
x
A = D = 0 b = 10
a 2.5 2.84
a < a
cr
= 2.839 . . .
x
n
= 0 a
cr
a < a
cr
D = D
opt
ω
(D = 0)
a
a > a
cr
a
a
2.85 < a < 2.88
˙x = −σ x + σ y + ξ
1
(t),
˙y = −y + r x − x z + ξ
2
(t),
˙z = −b z + x y + ξ
3
(t),
hξ
i
(t)ξ
j
(0)i = D δ
i,j
δ(t).
σ = 10
r = 210 b = 8/3
0.00 0.01 0.02 0.03 0.04
D
0.0
5.0
10.0
15.0
SNR, (db)
SNR
η
η
D
SNR,
η, dB
2.83 2.84 2.85
a
0.0
10.0
20.0
30.0
40.0
SNR, (db)
SNR
η
η
a
dB
SNR,
η,
a = 2.5 A =
0.05
Ω = 0.1 D = 0 A = 0.005 Ω = 0.1
D 1
c
x,x
(τ) ∝ exp ( −
2r(D))
A sin(Ω t)
δ
SNR ∝
r(D)
D
2
=
1
D
2
exp
−
∆Φ
D
,
∆Φ
0.00 0.10 0.20 0.30 0.40 0.50
D
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
SNR (dB)
D
SNR
A = 1.0 Ω = 0.1 ◦
∆Φ = 0.24
¨u + γ ˙u + u
3
+ (v − 1) u = 0,
˙v = h (βu
2
− α v),
γ = (1 + σ)/
p
σ(r − 1) h = (r − 1)
−1/2
β = (2σ − b)/
√
σ α =
b/
√
σ r 1 v(t)
¨u + γ ˙u +
α + β
α
u
3
− u = 0.
r
R
0
− 5.11 kΩ, R
1,4,6
− 10 kΩ, R
2
−
51.2 kΩ, R
3
− 2.2 kΩ, R
5
− 0.5 kΩ, R
7
− 1.1 kΩ, R
8
− 57 kΩ
C = 150
D1−D4
R
0
− R
8
C D9
A sin(2πft) ξ(t)
R
1
x(t)
U(x)
−1.5 < x < 1.5
x
−
dU(x)
dx
= ax − bx
3
.
a
R
2
b
R
3
, R
5
, R
8
2πf
0
τ
−1
= U
00
(x
min
),
τ
a
b
f 12.5
±5%
|x| ≤ 2 1%
dB
0.0 0.05 0.1 0.15 0.2 0.25
0.000.00.0
10.0
15.0
20.0
0.0
5.0
η,
D
800 Hz
400 Hz
60 Hz
0.20.150.10.050.0
0.0
D
η, dB
f
1
= 0.01492
f
2
= 0.00294
A = 0.02
10.0
15.0
20.0
5.0
A = 20
f
1
f
2