Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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x
a
g(x) = 1 − 1.5276330x
2
+ 0.1048152x
4
+ 0.0267057x
6
− 0.0035274x
8
+
+ 0.0000816x
10
+ 0.0000254x
12
− 0.0000027x
14
.
a
−2.502907876 . . .
f(x
n
)
ˆ
T
g(x)
ˆ
T
g(x) r = r
cr
ˆ
L
g
h(x) ρ
ˆ
L
g
h(x) = a [g
0
(g(x/a))h(x/a ) + h(g(x/a ))] = ρh(x).
ρ
1
= δ =
4.6692016091 . . .
a δ
ω
0
/2
k
ω
0
/2
k+1
k → ∞
r = r
cr
λ
r > r
cr
λ ∼ ε
γ
, γ =
ln 2
ln δ
≈ 0.4498,
ε = r − r
cr
γ
1.32 1.35 1.38 1.41 1.44
−0.075
−0.050
−0.025
0.000
0.025
0.050
1.30 1.35 1.40 1.45 1.50
−0.40
−0.30
−0.20
−0.10
0.00
0.10
0.20
0.30
r m
λλ
1,2
λ
g = 0.2
r
k
k = 1, 2, . . .
r
2
i
2
i
−4 −2 0 2 4 6
−3
−1
1
3
5
0.0 0.5 1.0 1.5 2.0
−70
−50
−30
−10
−4 −2 0 2 4 6
−3
−1
1
3
5
0.0 0.5 1.0 1.5 2.0
−70
−50
−30
−10
−4 −2 0 2 4 6
−4
−2
0
2
4
6
0.0 0.5 1.0 1.5 2.0
−70
−50
−30
−10
10 lg (S(
y
ω
)/S
max
)
x
ω
ω
ω
−
2
ω
0
−
4
ω
ω
−
2
ω
0
0
0
0
0
ω
0
T
0
2
i
a r
k
δ
M
N M
N
M → N
3 → 5 → 7 → 3 ∗ 2 → 5 ∗ 2 → 7 ∗ 2 → . . . → 3 ∗ 2
2
→
→ 5 ∗ 2
2
→ 7 ∗ 2
2
→ . . . → 2
3
→ 2
2
→ 2 → 1.
6, 5, 3, 6, 5, 6, 4, 6, 5, 6, . . .
α < α
cr
C
α = α
cr
C α > α
cr
C
α < α
cr
C
α = α
cr
α > α
cr
C
20 30 40 50 60
−60
−40
−20
0
20
40
60
t
x
˙x = σ(y − x), ˙y = −xy + rx − y,
˙z = xy − bz
r = 166.1, σ = 10, b = 8/3
C
α = α
cr
α > α
cr
x
n+1
= f(x
n
) = ε + x
n
+ β|x
n
|
p
+ “ ”.
C
C
ε α−α
cr
ε = 0 p
x
AB
ε = 0
ε & 0
AB
x
n
x
n
f( )
x
n
f( )
x
n
A
B
0
0
p
p = 2
x
n
∈ [0; 1]
x
n+1
= f
0
(x
n
)
f
0
(0) = 0, f
0
0
(0) = 1
g(0) = 0; g
0
(0) = 1
T
l
∼ ε
−ν
, ν =
p − 1
p
.
p = 2 T ∼ 1/
√
ε
r
n+1
= (1 + ε)r
n
+ βr
3
n
+ “ ”,
φ
n+1
= φ
n
+ Ω, mod 1.
ε = 0
N ≥ 4
T
l
∼
1
ε
,
ε = α − α
cr
x
n+1
= −(1 + ε)x
n
− βx
2
n
+ “ ”,
ε = 0
T
2
2+d, d ∈ [0; 1]
T
2
→ CA
θ
θ
µ
1,2
= exp jφ µ
1,2
θ = φ/2π