Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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l
l
A
1
2
Q
2
Q
0
Q
1
Q
0
A
x
α
2
α
1
α
1
α
2
α
2
< 0
α
1
Q
0
, Q
1
Q
2
Q
0
ρ
Q
0
> 0 Q
1
Q
2
ρ
Q
1,2
≤ 0
l
1
l
2
Q
1,2
Q
0
l
1
l
2
A α
1
= α
2
= 0
ρ
Q
1
(α
1
, α
2
) = 0
ρ
Q
2
(α
1
, α
2
) = 0
N ≥ 2
Reρ
1,2
= 0 Imρ
1,2
6= 0
N = 2
F
F
F
C
0
α<α∗ α=α∗ α>α∗
F
F
C
0
F
α<α∗ α>α∗α=α∗
N
α < α
∗
F α = α
∗
ρ
1,2
= ±jω
0
α > α
∗
F
Reρ
1,2
> 0
C
0
α = α
∗
N > 2
C
0
F
α < α
∗
˙a = (α + jω
0
)a + L
1
a|a|
2
, ω
0
6= 0, L
1
6= 0 ,
a L
1
L
1
< 0 L
1
> 0
˙
A = αA + L
1
A
3
,
˙
Φ = ω
0
,
A = |a| Φ = Arg(a)
αA + L
1
A
3
= 0
A
F
= 0 A
0
=
p
−α/L
1
α/L
1
< 0 ω
0
T = 2π/ω
0
A = A
F
A = A
0
ρ
F,0
= α + 3L
1
A
2
F,0
L
1
< 0 α > 0
α < 0 α > 0 L
1
> 0
α < 0
α > 0
µ
1,2
|µ
1
| = 1
µ
1
(α
∗
) = +1 µ
1
(α
∗
) = −1 µ
1,2
(α
∗
) = exp (±jϕ)
α
∗
2
L
1
= 0
a
S
Q
1
Q
2
1
2
C
C
S
C
α<α∗
α=α∗
α
α = α
∗
µ
1
+1
N = 3 α < α
∗
C
1
C
2
Q
1
Q
2
µ
1
= 1
α = α
∗
C
1
C
2
C
α > α
∗
α
C
1
C
2
α =
α
∗
µ
1
(α
∗
) −1
dµ/dα|
α
∗
6= 0
α < α
∗
C
0
T
0
α > α
∗
C
0
C T T
0
T ≈ 2T
0
C
0
C
C
0
C
α < α
∗
C
0
α = α
∗
µ
1,2
(α
∗
) = exp (± jϕ)
ϕ ∈ [0, 2π] ϕ(α
∗
) 6= 0, π/2, π/3
C
0
2D
T
2
C
0
T
2
C
0
S
C
C
0
x
t
T
0
T
1 2
C
0
C
α = α
∗
y C
0
C
0
|µ
1,2
(α
∗
)| = 1
L
θ(α) = ϕ/2π
T
2
θ(α
∗
)
C
θ(α
∗
) = p/q p q
p/q
C
0
S
L
L
C
0
Γ C
0
µ
1
(α
∗
) = ±1 µ
1,2
(α
∗
) =
exp (±jϕ)
U
˙
x = F (x, α) + γg(y, x),
˙
y = F (y, α) + γg(x, y),
x, y ∈ R
N
α
γ g
g(x, x) = 0
x = y
U
U
U
µ
1
(α
∗
) = −1
µ
1,2
(α
∗
) = exp (±jϕ)
µ
1
(α
∗
) = +1
C
0
∈ U
U
x
s
1
−y
s
1
y
1
x
1
1
0
2
C
C
C
x
1
y
1
−
s
s
α
Q
W
S
Q
W
u
Q
α
α = α
∗
Γ
0
α 6= α
∗
σ
Q
σ
Q
(α) = ρ
1
(α) + ρ
2
(α) ρ
1,2
Q σ
Q
(α
∗
) < 0
A
C
B
Q
A
B
Q
C
Γ
0
α=α∗
α>α∗
σ
Q
(α
∗
) > 0 Γ
0
Γ
0
C
Q Γ
0
N ≥ 3
N = 3 Q
σ
1
(α) = Reρ
1,2
(α) + ρ
3
(α) ρ
1,2
= Reρ
1,2
± jImρ
1,2
ρ
3
Q
α = α
∗
Γ
0
σ
1
(α
∗
) 6= 0
• σ
1
(α
∗
) < 0
A
Γ B
• σ
1
(α
∗
) > 0 Γ
0
A B
A
B
Q
W
Q
W
Q
S
0
ç
U
t −t
N = 2 α < α
∗
Q
1
Q
2
α = α
∗
Γ
0
α > α
∗
C
C
α → α
∗