Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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U N N
λ
tr
U
U
U
8
S
1,2
Q
R
S
S
2
Q
1
S
1
1
2
S
2
2
1
2
1
U
Q
1,2
R U
S
1
1,2
S
2
1,2
U
U
U
9
Q
Q
R
.
.
2
1
X
Y
x = y
Q
1,2
x
n+1
= f(x
n
, φ
n
),
φ
n+1
= φ
n
+ Ω + εcos(2πφ
n
+ g(x
n
)), mod 1,
φ Ω
ε
g(x)
δ ·x δ f(x, φ)
f(x, φ) = 1 −a|x|+ ερ sin(2πφ)
2π
θ(t) θ(t) + 2π
R L
10
0.9255 0.9265 0.9275 0.9285
−0.02
0.00
0.02
0.04
0.9345 0.9355 0.9365 0.9375
−0.02
0.00
0.02
0.04
Ω Ω
λ λ
1,2 1,2
Ω
α = β = 0.2 C = 0.02 µ = 3.8 µ = 3.71
R r
θ
L
L R
R
L
L
R
˙ρ
j
= r(1 − ρ
2
j
)ρ
j
+ g(ρ
j−1
cos θ
j−1
+ ρ
j+1
cos θ
j+1
− 2ρ
j
),
˙
θ
j
= δ
j
+ g
ρ
j+2
ρ
j+1
sin θ
j+1
−
ρ
j
ρ
j+1
+
ρ
j+1
ρ
j
sin θ
j
+
ρ
j−1
ρ
j
sin θ
j−1
,
j = 1, 2, . . . , m − 1.
ρ
j
j θ
j
= Φ
j+1
− Φ
j
δ
j
= ω
j+1
− ω
j
δ
j
= ∆ ω
1
= 1
ρ
0
= ρ
1
, Φ
0
= Φ
1
, ρ
m+1
= ρ
m
, Φ
m+1
= Φ
m
,
g 6= 0
˜ω
j
= | <
˙
Φ
j
(t) > | = lim
T →∞
|Φ
j
(t
0
+ T ) − Φ
j
(t
0
)|
T
Ω
j
= ˜ω
j
−ω
1
r = 0.5
m
ω
j
= ω
1
+ (j − 1)∆
(∆, g)
˜ω
j
0 20 40 60 80 100
j
0.00
0.05
0.10
0.15
0.20
Ω
j
0 20 40 60 80 100
j
0.00
0.05
0.10
0.15
0.20
Ω
j
Ω
j
∆ = 0.002
g = 1.45
g = 1.2
g
∆m
2
8g
< 1,
˙ρ
j
= r(1 − ρ
2
j
)ρ
j
+ g(ρ
j−1
cos(Φ
j
− Φ
j−1
) +
D
ρ
j
+
+ ρ
j+1
cos(Φ
j+1
− Φ
j
) − 2ρ
j
) +
√
2Dξ
j
(t),
˙
Φ
j
= ω
j
+ g(
ρ
j+1
ρ
j
sin(Φ
j+1
− Φ
j
) −
−
ρ
j−1
ρ
j
sin(Φ
j
− Φ
j−1
)) +
√
2D
ρ
j
η
j
(t),
j = 1, 2, 3, . . . , m,
ξ
j
(t) η
j
(t)
D
r = 0.5
ε = 0.1
j
g = 0.55 g = 3.8
∆ = 0.002