Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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0.220 0.225 0.230 0.235 0.240
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0.50
0.60
0.70
F
x
F
x
ε = 0.1, θ = θ
g
α =
3.271 α = 3.272
T
3
α
α = α
cr1
φ
0
n → ∞
α
cr1
≤ α ≤ α
cr2
α ∈ [α
cr1
, α
cr2
]
θ
H
K
x(t)
x(t) = R
0
[1 + ρ(t)] cos [ω
0
t + φ(t)].
R
0
ω
0
ρ(t) φ(t)
ρ(t)
cos ω
0
t
φ(t)
˙
φ(t) =
√
2Bξ(t),
ξ(t) hξ(t)i ≡ 0
hξ(t + τ)ξ(t)i = δ(τ ) B
ψ(τ) =
R
2
0
2
[1 + K
ρ
(τ)] exp(−B|τ|) cos ω
0
τ,
K
ρ
(τ)
ρ(t)
x(t) = ±a
t
k
t
k
θ
p(θ) = η exp(−ηθ), θ ≥ 0,
η
ψ(τ) = a
2
exp(−2η|τ|).
x(t) = ±a
t
n
= nT
0
+ η, n = 1, 2, 3, . . . T
0
= const η
[0, T
0
] p
q
ψ(τ) = a
2
n −
|τ|
T
0
(p − q)
n−1
+ a
2
|τ|
T
0
−(n − 1)
(p − q)
n
,
(n −1)T
0
≤ |τ| < T
0
.
p = q = 1/2
3
R
2
0
2
[1 + K
ρ
(τ)] K
A
(τ)
A(t) = R
0
[1 + ρ(t)]
ψ(τ) = a
2
1 −
|τ|
T
0
|τ| < T
0
; ψ(τ) = 0 |τ| ≥ T
0
,
ψ(τ)
˙x = −y − z +
√
2Dξ(t), ˙y = x + αy, ˙z = β + z(x − µ),
(α = β = 0.2, µ = 6.5),
ξ(t) δ
D
A(t)
Φ(t)
x(t) = A(t) cos Φ(t), y(t) = A(t) sin Φ(t).
4
XY
x(t) y(t)
˙
A =
1
2
αA −
1
2
αA cos 2Φ − z cos Φ +
√
2Dξ cos Φ,
˙
Φ = 1 +
1
2
α sin 2Φ +
1
A
z sin Φ −
√
2D
A
ξ(t) sin Φ,
˙z = β + z(A cos Φ − µ).
D = 0 D 6= 0
φ(t) = Φ(t) −ω
0
t
ω
0
= h
˙
Φ(t)i
σ
2
φ
(t)
B
eff
=
1
2
dσ
2
φ
(t)
dt
.
Ψ
x
(τ) = ψ
x
(τ)/ψ
x
(0)
K
A
(τ)
B
eff
Ψ
x
(τ) K
A
(τ)
B
eff
Ψ
x
(τ)
τ < 20
Ψ
x
(τ)
K
A
(τ) B
eff
Γ (τ)
Γ (τ) =
K
A
(τ)
K
A
(0)
exp(−B
eff
|τ|).
x(t) α = β =
0.2, µ = 6.5
D = 0 B
eff
≈ 0.0002 D = 10
−3
B
eff
≈ 0.0003
D = 0 D = 0.001 D = 0.01
Γ (τ)
Ψ
x
(τ) K
A
(τ)/K
A
(0)
τ ≥ 20 0 < τ < 20
x(t)
α = β = 0.2, µ = 6.5
D = 0 D = 10
−3
φ(t)
τ > τ
c
τ
c
exp (−B
eff
|τ|)
ω
0
B
eff
S(ω) = C
B
eff
B
2
eff
+ (ω − ω
0
)
2
, C = const.