Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
Подождите немного. Документ загружается.
h(r) = ε
−δ
2
r
2
/2 + r
4
/4 − 2δ
4
, |r| < 2δ,
0 δ
ε
εδ
4
2εδ
4
1 h(r)
δ 2δ
u
0
h(r)
−7.0 −2.0 3.0
R
−0.10
0.10
0.30
U(R)
h(r)
U(R) δ = 4 r
0
= 0.5 r
0
= 1.0 r
0
= 1.5
τ
∂u
∂t
= 2u(1 − u
2
) + r
2
0
∂
2
u
∂r
2
+ 2h(r)u + A cos (Ωt + ϕ
0
) +
√
2Dξ(t).
R(t)
u(R(t), t) = u
2
∂
∂t
u(R, t) +
∂
∂R
u(R, t)
˙
R = 0.
r = R(t)
˙
R
φ = r − R(t)
R(t)
u
0
r
(φ)
u(r, t) → u
0
(φ)
R(t)
(u
r
(φ))
2
r = R φ = 0
Z
∞
−∞
dr
∂u
0
∂t
∂u
0
∂r
=
Z
∞
−∞
dr
∂u
0
(r,t)
∂t
∂u
0
(r,t)
∂r
∂u
0
∂r
2
≈ −
˙
R
Z
∞
−∞
dr ( u
0
r
)
2
.
τ
˙
R =
2
R
+∞
−∞
dru
0
r
(φ)u
0
h(r)
R
+∞
−∞
dr(u
0
r
)
2
+
˜
A cos(Ωt + ϕ
0
) +
p
2
˜
Dη(t),
˜
D = 9r
2
0
D/4
˜
A =
3r
0
A/2
c = 0
τ
˙
R = −
dU
r
0
(R)
dR
+
˜
A cos(Ωt + ϕ
0
) +
p
2
˜
Dη(t).
R(t) φ(t)
−
dU
r
0
(R)
dR
=
3
2
Z
+∞
−∞
h(r)
3
(φ/r
0
) sinh(ξ/r
0
)dr.
u
0
r
(r −R) −u
0
R
(r −R)
U
r
0
(R) =
3r
2
0
2
Z
+∞
−∞
h(r)
1
2r
0
2
[(r −R)/r
0
]dr,
r
0
→ 0
2
[(r − R)/r
0
]/2r
0
→ δ(r − R)
r
0
U
0
(R) = 3r
2
0
h(R)/2
r
0
x = (r −R)/r
0
R < 2δ
U
r
0
(R) =
3
2
εr
2
0
−
1
2
δ
2
−
3r
2
0
π
12
R
2
+
R
4
4
.
h(r)
U
r
0
(R)
r
0
2δ/
√
π R > 2δ
r
0
r
K
=
r
2
0
2
√
2τπ
ε
4δ
2
− πr
2
0
exp
−
1
6D
τε
δ
2
−
π
4
r
2
0
2
.
SNR =
A
2
τ
2
6D
2
√
2
ε
4δ
2
− πr
2
0
exp
−
τε
6D
h
δ
2
−
π
4
r
2
0
i
2
.
D
max
=
1
12
τε
δ
2
−
π
4
r
2
0
2
,
D
r
0
r
0
r
0
0.05 0.15 0.25 0.35 0.45
τ
5.0
10.0
15.0
SNR, dB
2
D
0.0 0.2 0.4 0.6 0.8
τ
0.0
5.0
10.0
15.0
SNR, dB
2
D
r
0
r
0
= 1.5
1 0.5
D r
2
0
= 1.5 N r
2
0
= 1 ◦
r
2
0
= 0.5
r
0
= 1 D = 0.2
δ = 4 ε = 0.0005 A = 0.01 Ω = 0.1 τ = 0.05
U
r
0
(R)
δ
˙x
i
= f(x
i
) +
p
2D
m
g(x
i
)ξ
i
(t) +
γ
2d
X
j
(x
j
−x
i
) +
p
2D
a
ζ
i
(t).
x
i
(t) i
i = 1, . . . , L
d
N = L
d
L ×d
γ
ξ
i
(t) ζ
i
(t)
hξ
i
(t)ξ
j
(t
0
)i = δ
i,j
δ(t − t
0
), hζ
i
(t)ζ
j
(t
0
)i = δ
i,j
δ(t − t
0
).
f(x) g(x)
f(x) = −x(1 + x
2
)
2
, g(x) = 1 + x
2
.
x
j
m
m
m = const
P
st
P
st
(x, m) =
C(m)
p
D
m
g
2
(x) + D
a
exp
x
Z
0
f(y) − γ(y − m)
D
m
g
2
(y) + D
a
dy
,
C(m)
m
m =
Z
∞
−∞
xP
st
(x, m) dx,
0 2 4 6 8 10
2D
m
5
10
15
20
γ
1
2
3
(D
m
; γ)
D
a
= 0 0.5 2.5
γ = 20
D
m
= 1.5
m
m 6= 0
m = 0
0 2 4 6 8 10
2D
m
0.0
0.2
0.4
0.6
0.8
1.0
m
1
2
3
|m|
γ = 20 D
a
= 0 0.5 2.5
D
m
D
a
= 0
m
1
= −m
2
6= 0
˙x
i
= f(x
i
) +
p
2D
m
g(x
i
)ξ
i
(t) +
γ
2d
X
j
(x
j
− x
i
) +
p
2D
a
ζ
i
(t)
+ A cos(Ωt + ϕ
0
),
A Ω ϕ
(D
m
; γ)
L
2
= 18 × 18
D
a
m(t) =
1
L
2
N
X
i=1
x
i
(t)
D
a
m
1
m
2
D
a
0 500 1000
t
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
m(t),10F(t)
−1.0
−0.5
0.0
0.5
1.0
F (t)
D
a
= 0.005 0.5 2.5
A = 0.1 Ω = 0.1 γ = 20 D
m
= 1.5
S(ω)
m(t) m(t) = +1 m(t) = −1
D
a
0 1 2 3 4 5
2D
a
0
50
100
SNR, SNR
N
•
◦ SNR
N
A D
a
h˙xi = hf (x)i + D
m
hg(x)g
0
(x)i.
δ m(t)