Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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a
i,j
ˆ
A
λ
i
= Reρ
i
N = 2
ˆ
A
ρ
1
ρ
2
ρ
1
ρ
2
ρ
1
ρ
2
ρ
1
ρ
2
ρ
1
ρ
2
Reρ
1,2
< 0 ρ
1
ρ
2
Reρ
1,2
> 0
ρ
1
ρ
2
ρ
1,2
= ±iω
ˆ
A
Det
ˆ
A = ρ
1
ρ
2
Sp
ˆ
A = ρ
1
+ ρ
2
N ≥ 3
R
3
ρ
1
ρ
2,3
Reρ
2,3
> 0 ρ
1
ρ
2,3
Reρ
2,3
< 0
λ
i
≥ 0 λ
j
≤ 0
x
0
(t)
x
0
(t) = x
0
(t + T ) ,
T
ˆ
A(t)
x
0
(t)
ˆ
A(t) =
ˆ
A(t + T ) .
y(t
0
) T
y(t
0
+ T ) =
ˆ
M
T
y(t
0
) ,
ˆ
M
T
Det[
ˆ
M
T
−µ
ˆ
E] = 0,
x
0
(t)
T
ˆ
A(t
0
)
µ
i
x
0
(t)
|µ
i
| ≤ 1, i = 1, 2, . . . , N
+1
N
X
i=1
µ
i
= Sp
ˆ
M
T
,
N
Y
i=1
µ
i
= Det
ˆ
M
T
.
λ
i
=
1
T
ln |µ
i
|.
λ
i
≤ 0 λ
i
> 0
x
0
(t)
k ω
j
j = 1, 2, . . . , k
x
0
(t) = x
0
(ϕ
1
(t), ϕ
2
(t), . . . , ϕ
k
(t))
= x
0
(ϕ
1
(t) + 2πm, ϕ
2
(t) + 2πm, . . . , ϕ
k
(t) + 2πm),
m ϕ
j
(t) = ω
j
t, j = 1, 2, . . . , k
ˆ
A(t)
t → ∞
ϕ
j
k
k
k
k
1
k
k ≥ 3
x(n + 1) = P (x(n), α) ,
x ∈ R
N
n
P (x, α) P
j
, j = 1, 2, . . . , N
α ∈ R
m
x
0
(n)
y(n) = x(n)−x
0
(n)
x
0
(n)
y(n + 1) =
ˆ
M(n)y(n) ,
ˆ
M(n)
m
j,k
=
∂P
j
(x, α)
∂x
k
x∈x
0
(n)
.
y(n + 1) =
ˆ
M(n)
ˆ
M(n − 1) . . .
ˆ
M(1)y(1) .
x
0
(n)
λ
i
= lim
n→∞
1
n
ln
y
i
(n)
y
i
(1)
.
ˆ
M(n)y
i
(n) = µ
i
(n)y
i
(n) ,
µ
i
ˆ
M(n)
i
λ
i
= lim
n→∞
1
n
n
X
k=1
ln |µ
i
(k)|.
x
0
1
, x
0
2
, . . . , x
0
l
l
l
x
0
1
= P
l
(x
0
1
) .
l = 1
x
0
= P (x
0
) ,
x
0
1
ˆ
M
x
0
(n)
ˆ
M(n + l) =
ˆ
M(n)
y
i
(1) l
y
i
(l + 1) =
ˆ
M(l)
ˆ
M(l − 1) . . .
ˆ
M(1)y
i
(1) =
ˆ
M
l
y
i
(1) .
ˆ
M
l
µ
l
i
ˆ
M
l
l
ˆ
M
l µ
l
i
λ
i
=
1
l
ln |µ
l
i
|.
l
|µ
l
i
| < 1 i = 1, 2, . . . , N
N −1
N
µ
l
i
, i =
1, 2, . . . , (N − 1)
ˆ
M
l
l
µ
l
N
= 1
α
1
, α
2
, . . . , α
M
α
α
α < α
∗
Q S α = α
∗
α > α
∗
˙x = α − x
2
.
x
0
1,2
= ±
√
α
ρ
1,2
=∓2
√
α
x
0
1
x
0
2
α = 0,
S
Q
x
0
1
= x
0
2
= 0
ρ(α) = 0
Q
1
Q
2
Q
0
Q
0
˙x = α
1
+ α
2
x + x
3
.
α
2
> 0
α
1
> 0
Q
0
ρ
Q
0
< 0
Q
Q
Q
0
2
1
Q
0