Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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ω
1
→ ω
1
, ω
2
→ . . . → ω
1
, ω
2
, . . . , ω
k
k
N ≥ 3
α = α
0
C T (α)
α = α
1
2C 2T (α)
2
k
C
α = α
k
, k = 1, 2, 3, . . .
ω
0
= 2π/T
0
α
k
k → ∞
α = α
cr
α > α
cr
˙x = mx + y − xz,
˙y = −x,
˙z = −gz + gΦ(x),
Φ(x) = x
2
x ≥ 0 Φ(x) = 0 x < 0
2 < d < 3
x
n+1
= 1 + y
n
− ax
2
n
,
y
n+1
= bx
n
,
b < 1
a b
−2 −1 0 1 2
−2
−1
0
1
2
0 20 40 60
−2
0
2
−2
−1
0
1
2
0.0 0.5 1.0 1.5 2.0
−70
−50
−30
−10
−4 −2 0 2 4
−2
0
2
4
0 20 40 60
−4
−2
0
2
4
0.0 0.5 1.0 1.5 2.0
−70
−50
−30
−10
−4 −2 0 2 4
−2
0
2
4
0 20 40 60
−4
−2
0
2
4
0.0 0.5 1.0 1.5 2.0
−70
−50
−30
−10
x
y
10 lg (S( )/S )
ω
max
2ω
ω
ω
ω
−
2
ω
−
4
0
0
0
0
0
−3.5 0.5 4.5
−2
0
2
4
0 20 40 60
−4
−2
0
2
4
0.0 0.5 1.0 1.5 2.0
−70
−50
−30
−10
x
t
ω
ω
ω
−
2
0
0
2T
k
0
, k = 1, 2, 3
x
n+1
= f(x
n
) = r − x
2
n
,
r r ≥ 0
x
n+1
= rx
n
(1 − x
n
), x
n+1
= 1 − rx
2
n
.
r x
0
> 0
x
0
= −1/2 +
p
r + 1/4
r ∈ [0; 3/4] µ
1
−2x
0
r = r
1
= 3/4
µ
1
= −1
x
1
, x
2
x
1,2
= 1/2 ±
p
r − 3/4
µ
2
= f
0
x
(x
1
)f
0
x
(x
2
) = 4(1 −r)
r ∈ [3/4; 5/4] r = r
2
= 5/4
µ
2
= −1
r
1
= 3/4, r
2
= 5/4, r
3
≈ 1.368099, r
4
≈ 1.394046, r
5
≈ 1.399637, . . .
r
cr
≈ 1.40115 . . .
k → ∞
δ = lim
k→∞
r
k+1
− r
k
r
k+2
− r
k+1
= 4.669201 . . .
∆
k
a
k
= ∆
k
/∆
k+1
a = lim
k→∞
∆
k
∆
k+1
= −2.5029 . . .
δ a
r = r
cr
x
n+1
= f
0
(x
n
),
f
0
x
n
= 0 f
0
(0) = 1
x
n+1
= f
0
(f
0
(x
n
))
x → x/a
0
a
0
= 1/f
0
(f
0
(0))
0.6 0.8 1.0 1.2 1.4 1.6
r
−0.8
−0.4
0.0
0.4
0.8
1.2
1.6
X
∆
κ
∆
κ+1
x
n+1
= f
1
(x
n
) = a
0
f
0
(f
0
(x
n
/a
0
))
f
i+1
(x) = a
i
f
i
(f
i
(x/a
i
)),
a
i
= 1/f
i
(f
i
(0))
lim
i→∞
f
i
(x) = g(x), lim
i→∞
a
i
= a.
g(x)
ˆ
T g(x) = ag(g(x/a) = g(x),
ˆ
T
a = 1/g(g(0))
g(0) =
1, g
0
x
(0) = 0 g(x)
2
i
i → ∞