Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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∆ω
Ω
P
s
= 2[G
x,x
(Ω) − hG
(0)
x,x
i]∆ω
hG
(0)
x,x
i
SNR =
1
G
(0)
x,x
(Ω)
2
Z
Ω+∆ω
Ω−∆ω
G
x,x
(ω)dω ,
G
(0)
x,x
ω =
Ω
∆ω → 0
p(τ)
p(τ)
τ = nT
s
/2, n =
1, 3, 5, . . . T
s
p(τ)
n
n = 1
hτi
˙x = x − x
3
+ A cos(Ωt + φ) +
√
2D ξ(t)
U
0
(x) = −x
2
/2 + x
4
/4
ξ(t) D
f(t) = A cos(Ωt + φ)
p(x, t; φ)
∂ p
∂ t
= −
∂
∂ x
[(x −x
3
+ A cos(Ωt + φ))p] + D
∂
2
p
∂ x
2
.
∂ p
∂ t
= [L
0
+ L
ext
(t)] p,
L
0
= −
∂
∂ x
(x − x
3
) + D
∂
2
∂ x
2
A = 0 L
ext
(t) = −A cos(Ωt + φ)
∂
∂ x
t
0
→ −∞
p
as
(x, t; φ) =
∞
X
n=−∞
a
n
(x) exp [ i (nΩt + φ)].
hp
as
i
φ
=
a
0
(x)
θ = Ωt + φ x, θ
θ
hx(t)i
hx(t)i
asy
=
∞
X
n=−∞
M
n
exp[i (nΩt + φ)],
M
n
Ω A
|M
1
|
2
Ω
η
η =
2|M
1
|
A
2
.
σ(t) = ±x
m
p(x, t|x
0
, t
0
) = p(σ, t|σ
0
, t
0
) δ(σ − x
m
) + p(−σ, t|σ
0
, t
0
) δ(σ + x
m
).
p(σ, t|σ
0
, t
0
)
σ t
W
σ
(t)
dp(σ, t|σ
0
, t)
dt
= −W
σ
(t) p(σ, t|σ
0
, t) + W
−σ
(t) p(−σ, t|σ
0
, t).
W
σ
(t) = r
K
exp
−
Aσ
D
cos Ωt
.
A = 0
r
K
r
K
=
1
√
2 π
exp
−
1
4D
.
∆U
eff
= ∆U
0
± Ax
m
cos Ωt
∆U
0
−A D
A x
m
D
D
p(σ, t|σ
0
, t
0
) =
1
2
exp[−2r
K
(t − t
0
)]
2 δ
σ
0
,σ
− 1 −
2r
K
Ax
m
cos(Ωt
0
+ ψ)
D
q
4r
2
K
+ Ω
2
+ 1 +
2r
K
Ax
m
cos(Ωt + ψ)
D
q
4r
2
K
+ Ω
2
,
ψ = −arctan(Ω/2r
K
)
hx(t)|x
0
, t
0
i =
R
xp(x, t|x
0
, t
0
) dx
t
0
→ −∞
hx(t)i
as
= A
1
(D) cos [ Ωt + ψ(D)].
A
1
(D) ψ(D)
A
1
(D) =
Ax
2
m
D
2 r
K
(D)
q
4r
2
K
(D) + Ω
2
,
ψ(D) = −arctan
Ω
2 r
K
(D)
.
η =
4r
2
K
x
4
m
D
2
(4r
2
K
+ Ω
2
)
.
D
hx(t + τ )x(t)|x
0
, t
0
i =
RR
xyp(x, t + τ|y, t)p(y, t|x
0
, t
0
)dx dy
t
0
→ −∞
τ t
G
σ,σ
(ω)
G
σ,σ
(ω) = G
(0)
σ,σ
(ω) +
π
2
A
2
1
(D) [ δ(ω − Ω) + δ(ω + Ω)].
G
(0)
σ,σ
(ω)
G
(0)
σ,σ
(ω) =
4r
K
x
2
m
4r
2
K
+ ω
2
1 −
A
2
1
(D)
2x
2
m
.
A = 0
A
2
SNR = π
Ax
m
D
2
r
K
D = D
SNR
max
= ∆U/2 = 1/4
η
Ω→0
=
4x
4
m
D
2
, η
Ω→∞
=
4r
2
K
x
4
m
D
2
Ω
2
.
η(D)
2r
2
K
= Ω
D
SPA
max
hx(t)i
f(t)
hx(t)i
asy
= hxi
st
+
Z
∞
−∞
κ(t −τ, D) f(τ) dτ.
hxi
st
f(t) = 0 f (t)
hxi
st
= 0
κ(t)
κ(s) = 0 s < 0
χ(ω)
hx(t)i = A |χ(ω)| cos (Ωt + ψ),
ψ
ψ = −arctan
Im χ(Ω)
Re χ(Ω)
.
η = |χ(ω)|
2
.
G
(0)
xx
(ω)
δ
G
x,x
(ω) = G
(0)
x,x
(ω) +
π
2
A
2
|χ(Ω)|
2
[ δ(ω −Ω) + δ(ω + Ω)].
ω = Ω > 0
δ ±Ω
SNR =
πA
2
|χ(Ω)|
2
G
(0)
x,x
(Ω)
,
χ(ω)
χ(ω)
κ(t)
c
(0)
x,x
(t)
κ(t) = −
Θ(t)
D
d
dt
c
(0)
x,x
(t),
Θ(t)
c
(0)
x,x
(τ, D)
c
(0)
x,x
(τ, D)
g
j
exp(−λ
j
τ) λ
j
g
j
λ
m
λ
m
= 2 r
K
=
√
2
π
exp
−
1
4D
.
c
(0)
x,x
(τ, D) ≈ hx
2
i
st
exp(−λ
m
τ), G
(0)
x,x
(ω) =
2λ
m
hx
2
i
st
λ
2
m
+ ω
2
,
hx
2
i
st
λ
m
c
(0)
x,x
(τ, D) = g
1
exp(−λ
m
τ) + g
2
exp(−ατ),
α
α = 2
g
1,2
τ = 0
g
1
= hx
2
i
st
−g
2
, g
2
=
λ
m
hx
2
i
st
λ
m
− α
+
hx
2
i
st
− hx
4
i
st
λ
m
−α
.
χ(ω, D) =
1
D
λ
m
hx
2
i
st
λ
2
m
+ ω
2
(λ
m
− i ω),
χ(ω, D) =
1
D
g
1
λ
2
m
λ
2
m
+ ω
2
+
g
2
α
2
α
2
+ ω
2
− i ω
g
1
λ
m
λ
2
m
+ ω
2
+
g
2
α
α
2
+ ω
2
.
η (Ω, D) =
1
D
2
(hx
2
i
st
λ
m
)
2
λ
2
m
+ Ω
2
, SNR =
πA
2
2D
2
hx
2
i
st
λ
m
.
η(Ω, D) =
(g
1
λ
m
)
2
(α
2
+ Ω
2
) + (g
2
α)
2
(λ
2
m
+ Ω
2
) + 2g
1
g
2
αλ
m
(αλ
m
+ Ω
2
)
D
2
(λ
2
m
+ Ω
2
)(α
2
+ Ω
2
)
,
SNR =
π A
2
2D
2
(g
1
λ
m
)
2
(α
2
+ Ω
2
) + (g
2
α)
2
(λ
2
m
+ Ω
2
) + 2g
1
g
2
αλ
m
(αλ
m
+ Ω
2
)
g
2
α(λ
2
m
+ Ω
2
) + g
1
λ
m
(α
2
+ Ω
2
)
.