Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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x
2
x
1
x
2
x
1
x
1
x
2
t → ∞
¨x +
dU(x)
dx
= 0 ,
U(x)
H(x, ˙x)
U(x)
Q
t → ±∞
Q
W
s
Q
W
u
Q
t −∞ +∞
t
Ψ = ω
ex
t
˙
Ψ = ω
ex
U(x)
x
O
2
1
x =x
1
.
x =x
2
u
W
W
Q
s
Q
Q
O
U(x)
Ψ
Ψ ∈ [0, 2π]
x
¨x − (α − x
2
) ˙x + ω
2
0
x = 0 .
α > 0 t → ∞
t → ∞
α
x(t)
α
α > 2ω
0
x
0
x
0
Ψ
Ψ
π
˙x = f(x) + B sin ω
ex
t
Ψ = ω
ex
t (−∞, +∞) Ψ ∈ [0, 2π]
−1.0 −0.5 0.0 0.5 1.0
x
1
−1.0
−0.5
0.0
0.5
1.0
x
2
0 20 40 60 80
t
−1.0
−0.5
0.0
0.5
1.0
x
1
−10 −5 0 5 10
x
1
−60
−40
−20
0
20
40
60
x
2
0 20 40 60 80
t
−10
−5
0
5
10
x
1
ω
0
= 1 α = 0.1 α = 10 x
1
= x, x
2
= ˙x
x
x
x
2
1
3
S
N − 1
x
∗
(t)
ε > 0 δ(ε) > 0
x(t) kx(t
0
) − x
∗
(t
0
)k < δ
t > t
0
kx(t) − x
∗
(t)k < ε k...k
R
N
δ
kx(t)−x
∗
(t)k → 0 t → ∞
t Γ
∗
x
∗
(t)
t → ∞
x
0
t
0
x(t) = T
∆t
x
0
t T
∆t
∆t = t − t
0
V L ∈ V V
x
0
x(t) ∈ L
t → +∞ t → −∞ L
x
0
∈ V L t → +∞
L
V
V L t → −∞
L
V W
s
W
u
W
s
L
W
u
L
L
W
s
W
u
t → −t
N ≥ 3
V = L
˙
x = F (x, α) ,
x ∈ R
N
α ∈ R
m
x
0
(t)
y = x(t) − x
0
(t)
˙
y = F (x
0
+ y) − F (x
0
) .
F (x
0
+ y) x
0
y
˙
y =
ˆ
A(t)y ,
ˆ
A
a
j,k
=
∂f
j
∂x
k
x(t)=x
0
(t)
, j, k = 1, 2, . . . , N ,
x
0
(t) f
j
F
ˆ
A
e
i
ρ
i
ˆ
Ae
i
= ρ
i
e
i
, i = 1, 2, . . . , N.
ρ
i
Det
h
ˆ
A − ρ
ˆ
E
i
= 0 ,
ˆ
E
t
∗
i
y
i
(t) = y
i
(t
∗
) exp [(t − t
∗
)ρ
i
] .
ky
i
(t)k
ρ
i
ˆ
A
t
∗
ρ
i
y(t) =
P
N
i=1
y
i
(t)
x
0
(t)
y
i
(t)
i
ˆ
A
e
i
(t)
λ
i
λ
i
= lim
t→∞
1
t − t
0
ln
y
i
(t)
y
i
(t
0
)
,
x
0
(t)
N
N × N N
N N
λ
1
≥ λ
2
≥ . . . ≥ λ
N
x
0
(t)
ρ
i
(t)
t
∗
= t
0
∆t =
t − t
0
x(t
0
+ ∆t)
y
i
(t
0
+ ∆t) = y
i
(t
0
) exp [ρ
i
(t
0
)∆t] .
∆t
e
i
y
i
(t
0
+ ∆t) i
y
i
(t
0
)
x
0
(t) ∆t
i
y
i
(t) ≈ y
i
(t
0
) exp
"
X
k
ρ
i
(t
k
)∆t
#
.
ky
i
(t
0
)k → 0 ∆t → 0
y
i
(t) = y
i
(t
0
) exp
Z
t
t
0
ρ
i
(t
0
)dT
0
.
λ
i
= lim
t→∞
1
t − t
0
Z
t
t
0
Re ρ
i
(t
0
)dt
0
.
i λ
i
ρ
i
ˆ
A(t)
N
X
i=1
λ
i
=
lim
t→∞
1
t − t
0
Z
t
t
0
divF (t
0
)dt
0
.
x
0
(t)
y(t
0
)
x
0
(t)
F (x(t))
N
X
i=1
λ
i
< 0 .
x
0
(t)
F (x
0
, α) = 0
ˆ
A
Det A
^
Sp A
^
Sp A
^
( )
2
1
__
4
Saddle
Center
Saddle
Stable node
Unstable focusStable focus
Unstable node