Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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˙x = αx − βx
3
+
√
2D ξ(t) + A cos(Ωt + ϕ
0
).
A
Ω ϕ
0
ϕ
0
α, β > 0, ϕ
0
= 0
A
w(t) = x(t)+iy(t)
y(t) x(t)
y(t) = H[x] =
1
π
Z
∞
−∞
x(τ)
t − τ
dτ =
1
π
Z
∞
0
x(t − τ) − x(t + τ)
τ
dτ.
w(t)
w(t) = R(t) exp[iΦ(t)].
w(t)
˙w = α w −
β
4
(3 R
2
w + w
3
) + ψ(t) + A exp(iΩt),
ψ(t) = ξ(t) + i η(t) η(t)
ξ(t)
˙
R = α R −
β
2
R
3
[1 + cos
2
(φ + Ωt)] + A cos φ + ξ
1
(t),
˙
φ = −Ω −
A
R
sin φ −
β
4
R
2
sin[2(φ + Ωt)] +
1
R
ξ
2
(t),
φ(t) = Φ(t) − Ω t
ξ
1,2
(t)
ξ
1
(t) = ξ(t) cos Φ + η(t) sin Φ,
ξ
2
(t) = η(t) cos Φ − ξ(t) sin Φ.
Ω
x
Ω
Φ(t)
hωi = lim
T →∞
1
T
R
T
0
d Φ(t)
dt
dt
A
hωi
A = 0
A ≥ 1 hωi D
A = 2
0 100 200 300 400 500 600
t/T
−100
−80
−60
−40
−20
0
20
40
60
80
100
φ( )
D=0.44
D=0.80
D=1.05
t
0
φ(t)
t/T
0
0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
D
0.00
0.01
0.02
0.03
0.04
0.05
<ω>
1
2
3
4
<ω>
D
φ
A = 3
•
A = 0 A = 1
A = 2 A = 3 α = 5 β = 1
Ω = 0.01
hωi D A = 3
x
D
eff
=
1
2
d
dt
hφ
2
(t)i − hφ(t)i
2
.
D
eff
D
eff
D
eff
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
D
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Deff
A=3.0
A=2.0
A=1.0
D
D
eff
α = 5
β = 1 Ω
0
= 0.01
10
3
φ(t)
Φ
out
(t) = N(t)π Φ
in
(t) = Ωt
φ(t) = Φ
out
(t) − Φ
in
(t).
Φ
out
π
cos(Φ
out
) cos(Φ
in
) =
1
2
[cos(φ) + cos(2Φ
out
− φ)] = cos(φ).
φ
W
out
φ
(t) = r
K
(D) exp
−
A
D
cos[φ(t)]
,
A
2 × 2
mod(2 π)
φ(t) = k(t) π,
k(t)
k → k + 1 k → k − 1
p
k
(t)
kπ t
φ
0
= 0 t
0
= 0
∂p
k
(t)
∂t
= W
in
k+1
p
k+1
− W
in
k
p
k
+ W
out
k−1
p
k−1
− W
out
k
p
k
.
W
out
k
= a
1
= r
K
(D) exp
−
A
D
, W
out
k
= a
2
= r
K
(D) exp
A
D
,
k → k + 1
k
k
W
DPP
k
(t, ϕ
0
) = π
∞
X
n=−∞
δ
t −
nπ + ϕ
0
Ω
.
ϕ
0
hW
DPP
k
i
ϕ
0
=
1
2π
2π
Z
0
W
DPP
±
(t, ϕ
0
) dϕ
0
= Ω .
γ
τ
W
DMP
k
=
2 π
2τ
= πγ.
h
˙
φi
h
d
dt
φi = −hω
in
i+ hω
out
i = −hω
in
i+
π
2
(a
1
+ a
2
) −
π
2
(a
2
−a
1
) hcos(φ)i,
hω
in
i = γπ hhω
in
ii
ϕ
0
= Ω
∆ = π
a
1
+ a
2
2
− hω
in
i,
π(a
2
−a
1
)/2
µ
µ A/D
a
1
= a
2
= r
K
π
hcos φi
lim
t→∞
hcos(φ)i = hσ
stat
i =
a
2
− a
1
2
hω
in
i
π
+ a
1
+ a
2
.
2 × 2
hω
stat
out
i =
π
2
a
1
+ a
2
−
(a
2
−a
1
)
2
2
hω
in
i
π
+ a
1
+ a
2
.
π D
γ = 0.001
A
1,2,3
< ∆U = 0.25
0.00 0.04 0.08
0.0
0.1
0.2
A
0.00 0.02 0.04 0.06 0.08
D
0
0.001
0.002
0.003
<ω
out
*
>/π
π D
γ = 0.001 Ω/π = 0.001
A
1
= 0 A
2
= 0.1 A
3
= 0.2
α
0
= 1 ∆U = 0.25
γ = Ω/π