Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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x
1
(t) x
2
(t)
κ = min
τ
S(τ ),
S(τ )
S
2
(τ) =
h(x
2
(t + τ ) − x
1
(t))
2
i
p
(hx
2
1
(t)ihx
2
2
(t)i)
.
κ = 0
κ
R
x
1
x
2
(τ) =
hx
1
(t)x
2
(t + τ)i − hx
1
(t)ihx
2
(t + τ)i
r
hx
2
1
(t)i − hx
1
(t)i
2
hx
2
2
(t + τ )i − hx
2
(t + τ)i
2
.
η = max
τ
R
x
1
x
2
(τ)
x
1
(t) x
2
(t)
∆Φ(t) = Φ
1
(t) − Φ
2
(t)
D
∆Φ
[−π, π]
h
Φ
)
r
x
1
x
2
(ω) =
|S
x
1
x
2
(ω)|
p
S
x
1
(ω)S
x
2
(ω)
,
S
x
1
, S
x
2
x
1
(t) −hx
1
i x
2
(t) −
hx
2
i S
x
1
x
2
r
x
1
x
2
≡ 0
x
1
(t) x
2
(t) r
x
1
x
2
≡ 1
r
x
1
(t) x
2
(t)
x
1
(t) x
2
(t) T
∆φ(ω) = φ
1
(ω) − φ
2
(ω), ∆φ ∈ [−π, π],
φ
1,2
(ω) = arg
R
T/2
−T/2
x
1,2
(t) exp (−jωt)dt.
∆φ(ω)
p(∆φ, ω)
δ
˙x
1
= −Ω
1
y
1
− z
1
+ γ(x
2
− x
1
),
˙y
1
= Ω
1
x
1
+ αy
1
,
˙z
1
= β + z
1
(x
1
− µ),
˙x
2
= −Ω
2
y
2
− z
2
+ γ(x
1
− x
2
),
˙y
2
= Ω
2
x
2
+ αy
2
,
˙z
2
= β + z
2
(x
2
− µ),
Ω
1,2
= Ω ± ∆
∆ γ α β µ
ω
0
T
0
Φ
0
Φ
0
±2πk, k = 1, 2, . . .
2T
0
ω
0
/2
Φ
0
Φ
0
±2π
2
n
T
0
0.00 0.05 0.10 0.15
0.0
0.5
1.0
0.00 0.05 0.10 0.15
0.0
0.5
1.0
0.00 0.05 0.10 0.15
0.00
0.50
1.00
1.50
γ
γ
r
−
D
0.00
l
l
l
l
1
2
1
2
γ
l l
κ
1
2
eff
0.25
0.50
γ α = 0.165 β = 0.2 m = 10 Ω = 0.97 ∆ = 0.02
κ
D
eff
r
l
1
, l
2
2
n
φ
0
+ 2πm m = 0, 1, 2, . . . , 2
n−1
2
k
x
n+1
= r − x
2
n
+ γ(x
2
n
− y
2
n
),
y
n+1
= r − y
2
n
+ γ(y
2
n
− x
2
n
),
r γ
m m = 0
x = y
m 6= 0
r
r
2
x = y
r
γ = 0.002 m r = 1.3
r = 1.415
m
r
γ
2
n
C
m
2
n
r γ = 0.002
m
2
n
CA
m
2
n
m
m = 0
m 6= 0
2
n
CA
m
2
n
C
m
n m
r
4CA
1
4CA
3
8CA
2
8CA
6
2
n
CA
m
n m
n m
CA
Σ
γ γ
∆ ≤ 0.001 γ ≈ 0.02
∆ 6= 0
∆ ≈ 0.0015
2C
1
, 4C
2
CA
Σ
∆−µ
4CA
0
4CA
1
µ
4CA
1
4CA
0
α =
0.165, β = 0.2, γ = 0.02, Ω = 1
2CA
Σ
∆ = 0
CA
0
U
x
1
= x
2
x
n+1
= f(x
n
) + γg(y
n
− x
n
),
y
n+1
= f(y
n
) + γg(x
n
− y
n
).
U x = y
λ
tn
λ
tr
U
λ
tr
= lim
n→∞
1
n
ln
df(x)
dx
x=x
n
.
λ
tr
U
U U