Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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τ
D
c
τ
τ i i + 1
τ
τ
τ
N ∆t
N∆t
D
c
D
max
=
2 ln N
ln(1/r)
, r =
ε
ε
max
.
D
max
N r = 0.1 N = 1000 D
max
6 6
N = 100000 D
max
6 10
x(t) =
a(t),
da(t)
dt
, . . . ,
d
n−1
a(t)
dt
n−1
= (x
1
, x
2
, . . . , x
n
) .
a
i
i∆t x
j
x
∆t
x(t) =
Z
t
0
a(t
1
) exp
t − t
1
β
dt
1
.
x
i
(k) =
1
2k + 1
k
X
j=−k
a
i+j
,
k
∆t
i∆t
a(t)
R
a(t)
T
0
T
i
Z
T
i+1
T
i
a(t)dt = θ , I
i
= T
i+1
− T
i
,
θ I
i
θ
θ
a = θ a(t)
a(t)
0.0 40.0
20.0
60.0
t
S
0.0 40.0
−20.0
20.0
Θ
t
S
a(t)
a(t)
x, y
ϕ
m
(t) = 2π
t − T
i
T
i+1
− T
i
+ 2πi , T
i
6 t < T
i+1
,
T
i
ϕ
p
= arctan
y
x
.
ϕ
m
ϕ
p
a(t)
a
H
(t)
z(t) = a(t) + i a
H
(t) = A(t) exp (i ϕ
H
(t))
a
H
(t) =
1
π
Z
∞
−∞
a(τ)
t − τ
dτ ,
A(t) = (aa + a
H
a
H
)
1/2
, ϕ
H
(t) = arctan
a
H
(t)
a(t)
.
lim
A→−∞,B→∞
R
B
A
f(x) dx
lim
A→∞
R
A
−A
f(x) dx
A(t)
z(t) ϕ
H
(t)
−π/2 a
H
(t)
a(t)
a, a
H
(x, y)
a(t)
a
H
(t)
ω
H
(t) =
dϕ
H
(t)
dt
=
˙a
H
a − ˙aa
H
a
2
+ (a
H
)
2
a(t) a
H
(t)
(a(t) = 0 a
H
(t) = 0)
a, a
H
ω
H
, A
ω
H
(t) A(t)
a(t)
θ
a(t) = θ
ϕ
i
ω
i
T
i
ω
i
= hω
H
(T
i
)i =
2π
I
i
,
ω
i
I
i
ω
H
(t)
ω
i
ω
H
(t)
ω
i
T
i
ω
i
(T
i
)
dx
dt
= −(y + z) ,
dy
dt
= x + ay ,
dz
dt
= b + z(x − c) ,
a = 0.15, b = 0.2, c = 10.0
x(t)
ω
H
(t)
hω
H
i(t) =
1
I
i
Z
T
i+1
T
i
ω
H
(t) dt,
ω
i
(t) hω
H
i(t) ω
i
(T
i
)
I
i
x(t)
(a(t), a
H
(t)) (A(t), ω
H
(t)) (ω
i
(t), ω
i
(t + τ ))
0 50
0.8
1.7
t
ω
< >
H
H
ω
10 75
0.98
1.13
t
<
ω
H
>,
ω
i
ω
H
I
i
hω
H
i
hω
H
i
ω
i
(t)
−12 15
−10
10
x
x
H
4.0 12.0
0.8
1.8
A
ω
H
1.0 1.1
1.0
1.1
ω
τ
i
(t)
i
(t+ )
ω
x, x
H
x
x
H
x
A, ω
H
A ω
H
`
ω
i
(t), ω
i
(t + τ)
´
ω
i
(t)
x
ω
i
(t)
log M log ε M ε
θ
λ
1
x = θ
θ λ
1
I
i
= T
i+1
− T
i
ω(T
i
) = 2π/I
i
−3.0 −0.5
−5.0
−1.0
−lg (M)
lg (e)
0.0 18.0
0.0
0.1
1
λ
Θ
x
I
i
T
i
ω(T
i
)
ω
int
(t)
λ
1
ω
int
(t)
I =
1
N
P
N
i=1
I
i
|θ| 6 13 λ
1
12%
λ
1
(θ)
λ
1
I
0.0 7.0
0.0
7.0
y
n+1
y
n
5.5 6.4
5.5
6.5
I
n+1
I
n
5.0 20.0
5.0
20.0
I
n+1
I
n
Θ = 0
I