Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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u
n+1
= f
0
(v
n
)u
n
,
v
n+1
= (1 − 2γ)f(v
n
).
γ
Λ
k
= lim
N→∞
1
N
N
X
n=1
ln |(1 − 2γ)f
0
(v
n
)|.
v
u = 0
Λ
⊥
= lim
N→∞
1
N
N
X
n=1
ln |f
0
(v
n
)|.
Λ
⊥
= Λ
k
− ln |1 − 2γ|.
0 ≤ γ ≤ 0.5 ln |1 − 2γ| ≤ 0
Λ
⊥
≥ Λ
k
.
r (f(x
n
) + f (y
n
))
x
n+1
= f(x
n
) + γ (f(y
n
) − f(x
n
)) + r (f (x
n
) + f (y
n
)) ,
y
n+1
= f(y
n
) + γ (f(x
n
) − f(y
n
)) .
u, v
u
n+1
= (1 + r)f
0
(v
n
)u
n
,
v
n+1
= (1 −2γ)f(v
n
) + rf
0
(v
n
)u
n
.
u
n
= 0
Λ
a
⊥contr
= lim
N→∞
1
N
N
X
n=1
ln |(1 + r)f
0
(v
n
)|
Λ
a
⊥contr
= Λ
⊥
+ ln |1 + r|.
r −1
−1.6 −1.4 −1.2
−1
−0.8 −0.6 −0.4
r
−12.0
−10.0
−8.0
−6.0
−4.0
−2.0
0.0
Λ
transversal
tangent
−1.6 −1.4 −1.2
−1
−0.8 −0.6 −0.4
r
−0.20
−0.15
−0.10
−0.05
0.00
0.05
0.10
0.15
Λ
noise=0
noise=0.0001
noise=0.001
r a = 3.8 γ = 0.04
r
r = 0
a = 3.8 γ = 0.04
ρ = |x + y|
δ ρ > δ
ρ ≤ δ
r Λ
⊥contr
< 0
δ = 0.01
r = 0
x = −y
r
r
r
1
0.5
0
−0.5
−1
1
0.5
0
−0.5
−1
−1 −0.5
0
0.5 1
−1 −0.5
0
0.5 1
y
y
1
0.5
0
−0.5
−1
−1 −0.5
0
0.5 1
0
200
400
600
800
1000
y
x
x
x
n
n
n
0
200
400
600
800
1000
2
1
0
−1
−2
x+y
2
1
0
−1
−2
x+y
2
1
0
−1
−2
0
200
400
600
800
1000
x+y
r = 0 r = −0.48
r = −0.5
r
−1 < r < −0.525
r (r > −0.525)
r
r = −0.406
r r < −1.472
(u = 0)
(−0.005 ≤ u ≤ 0.005)
a =
3.8, γ = 0.04
r = −1.485
˙x
1,2
= α [y
1,2
− x
1,2
−h(x
1,2
)] ,
˙y
1,2
= x
1,2
− y
1,2
+ z
1,2
+ γ [(x
2,1
− x
1,2
) − (y
2,1
− y
1,2
) + (z
2,1
− z
1,2
)] ,
˙z
1,2
= −βy
1,2
,
x
1,2
, y
1,2
z
1,2
h(x
1,2
) = bx
1,2
+0.5(a−b)(|x
1,2
+1|−|x
1,2
−1|) α, β, a b
γ
X
1
= (x
1
, y
1
, z
1
) ←→ X
2
= (x
2
, y
2
, z
2
) X
1
=
(x
1
, y
1
, z
1
) ←→ −X
2
= (−x
2
, −y
2
, −z
2
)
α γ a = −8/7
b = −5/7 β = 22
α
X
1
= X
2
X
1
= X
2
X
1
= X
2
X
1
= −X
2
u = (x
1
−x
2
)/2, v = (y
1
−y
2
)/2, w = (z
1
−z
2
)/2, u
1
= (x
1
+x
2
)/2, v
1
=
(y
1
+ y
2
)/2, w
1
= (z
1
+ z
2
)/2
˙u = α [v − u − c(u)] ,
˙v = u − v + w −2γ [u − v + w] ,
˙w = −βv,
˙u
1
= α [v
1
− u
1
− c(u
1
)] ,
˙v
1
= u
1
− v
1
+ w
1
,
˙w
1
= −βv
1
,
c(u) = [h(x
1
) − h(x
2
)] /2, c(u
1
) = [h(x
1
) + h(x
2
)] /2
X
1
= X
2
X
1
= −X
2
X
1
=
X
2
F (u) =
ru
r u = v = w = 0
X
1
= X
2