Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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˙m = f(m) + D
m
g(m)g
0
(m),
f(hxi) hδx
2
if
00
(hxi)
U
eff
(m)
U
eff
(m) = U
0
(m) + U
noise
= −
Z
m
dm
0
f(m
0
) −
D
m
g
2
(m)
2
,
U
0
(m) U
noise
D
m
A D
a
T
per
= 2π/Ω
SNR
1
=
πA
2
D
2
a
r
K
,
r
K
r
K
=
p
U
00
eff
(m
min
)|U
00
eff
(m
max
)
2π
exp
−
∆U
eff
D
a
∆U
eff
N G
∆
N
SNR
N
= SNR
1
NG
∆
+ 1.
D
a
G
x,x
(ω) = G
(0)
x,x
(ω) + |χ(ω)|
2
G
s,s
(ω),
G
s,s
(ω)
s(t)
s(t) = A
M
X
k=1
b
k
cos Ω
k
t ,
Ab
k
1 Ω
k
hx(t)i = A
M
X
k=1
b
k
|χ(Ω
k
, D)| cos [(Ω
k
+ ψ
k
) t].
hx(t)i
ψ
k
ψ
k
(Ω
k
, D) = −arctan
Im χ(Ω
k
, D)
Re χ(Ω
k
, D)
.
η (Ω
k
, D) = |χ(Ω
k
, D)|
2
,
SNR (Ω
k
, D) =
π (A b
k
)
2
|χ(Ω
k
, D)|
2
G
(0)
x,x
(Ω
k
, D)
.
E(Ω
k
, Ω
j
, D) =
|χ(Ω
k
, D)|
|χ(Ω
j
, D)|
.
˙s = y, ˙y = −Γ y − Ω
2
s +
√
2εΓ ξ
2
(t),
ξ(t) hξ
2
(t)ξ
2
(t
0
)i = δ(t − t
0
) Γ
ε
G
s,s
(ω)
G
s,s
(ω) =
2ε Γ
ω
2
Γ
2
+ (ω
2
− Ω
2
)
2
.
Ω > Γ/2
ω
p
=
p
Ω
2
−Γ
2
/2
∆ω
in
∆ω
in
=
q
ω
2
p
+ Γ ω
1
−
q
ω
2
p
− Γ ω
1
, ω
1
=
p
Ω
2
− Γ
2
/4.
∆ω
Q
in
=
ω
p
q
ω
2
p
+ Γ ω
1
−
q
ω
2
p
− Γ ω
1
.
˙x = x − x
3
+
√
2D ξ
1
(t) + s(t),
ξ
1
(t) ξ
2
(t)
p(x, s, y, t)
∂ p
∂ t
= −
∂
∂ x
[(x − x
3
+ s) p ] −
∂
∂ s
(y p) +
∂
∂ y
[(Γ s + Ω
2
y) p ]
+ D
∂
2
p
∂ x
2
+ ε Γ
∂
2
p
∂ y
2
.
ε 1
G
x,x
(ω) = G
(0)
x,x
(ω, D) + |χ(ω, D)|
2
G
s,s
(ω).
G
x,x
(ω) =
2hx
2
i
st
λ
m
ω
2
+ λ
2
1 +
hx
2
i
st
λ
m
D
2
ε Γ
ω
2
Γ
2
+ (ω
2
− Ω
2
)
2
.
ω
G
xx
0.05 0.10 0.15 0.20 0.25
ω
0
2
4
6
8
10
12
14
Gxx
D=0.08
D=0.12
D=0.5
•
ε = 10
−3
Ω = 0.2 Γ = 0.008
ω
m
D
ω
m
∆ω
out
D
R
∆ω
out
∆ω
in
R(D) =
∆ω
out
(D)
∆ω
in
.
R
Γ Γ
D
R
0.05 0.15 0.25 0.35 0.45
D
1.00
1.10
1.20
1.30
R
Γ=0.008
Γ=0.004
Γ=0.002
D
0.05 0.15 0.25 0.35 0.45
D
0.0
10.0
20.0
30.0
40.0
50.0
SNR
Γ=0.008
Γ=0.004
Γ=0.002
SNR
R
Γ
ε = 10
−3
Ω = 0.2
Q
out
h
C(D) = Q
out
(D) · h(D),
C(D)
SNR = 1 +
hx
2
i
st
λ
m
D
2
ε Γ
ω
2
m
Γ
2
+ (ω
2
m
− Ω
2
)
2
.
D
s(t)
G
x,x
(ω)
G
s,x
(ω)
G
s,x
(ω) = χ(ω)G
s,s
(ω),
χ(ω)
C =
hs xi
p
hx
2
ihs
2
i
,
hs xi =
R
∞
0
G
s,x
(ω)dω
s(t)
G
s,s
(ω) = γQ/(γ
2
+ ω
2
)
Q → 0
C =
λ
m
Q
D(γ + λ
m
)
,
λ
m
D
σ(t) =
±1
∆U
λ(t) = ±1 B 0 < γ a
0
W
0
(σ, λ) = a
0
exp
−
∆U + σλ B
D
.
d
dt
p(σ λ) = −W
0
(σ, λ) p(σ, λ) + W
0
(−σ, λ) p(−σ, λ)
+ γ [p(σ, −λ) − p(σ, λ)] .
s(t) = A cos(Ωt + ϕ
0
)
Ω < γ a
0
) A ∆U − B
W (σ, λ) = W
0
exp
−
A σ
D
cos(Ωt + ϕ
0
)
≈ W
0
1 −
A σ
D
cos(Ωt + ϕ
0
)
.
∝ (A/D)
2
hσ(t)λ(t
0
)i
hσ(t)
2
i = 1
hσλi
stat
=
a
2
−a
1
a
1
+ a
2
+ 2γ
.
a
0
= 1
a
1,2
= exp [−(∆U ± B)/D] .
ϕ
0
G
σ,σ
(ω) =
G
(0)
σ,σ
(ω) + A
2
πηδ(ω − Ω)
G
(0)
σ,σ
(ω) = 4
a
1
+ a
2
(a
1
+ a
2
)
2
+ ω
2
1 +
(a
2
−a
1
)
2
4γ
2
+ ω
2
−4
(a
2
− a
1
)
2
(a
1
+ a
2
+ 2γ)(4γ
2
+ ω
2
)
η =
1
D
2
a
1
+ a
2
−
(a
2
− a
1
)
2
(a
1
+ a
2
+ 2γ)
2
1
(a
1
+ a
2
)
2
+ Ω
2
.
γ
Ω
B
B = 0