Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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N ≥ 4
T
3
T
3
T
2
T
3
T
2
T
3
T
3
T
2
T
3
T
3
x
n+1
= Φ(x
n
, y
n
), mod 1,
y
n+1
= Ψ(y
n
, x
n
), mod 1.
T
3
T
3
T
3
θ
x
= lim
n→∞
Φ
n
(x
n
, y
n
)
n
,
θ
y
= lim
n→∞
Ψ
n
(x
n
, y
n
)
n
.
T
3
T
3
T
3
T
3
x
n+1
= x
n
+ Ω
x
−
K
2π
sin 2πy
n
, mod 1,
y
n+1
= y
n
+ Ω
y
−
K
2π
sin 2πx
n
, mod 1.
Ω
x
Ω
y
θ
x
θ
y
K
K < 1
K > 1
T
3
K
K < 1
K < 1
T
3
0.0 0.5 1.0
0.0
0.5
1.0
0.0 0.5 1.0
0.0
0.5
1.0
0.0 0.5 1.0
0.0
0.5
1.0
x x
x
y
y y
x
n+1
= x
n
+ y
n
, mod 1,
y
n+1
= x
n
+ 2y
n
, mod 1.
δ sin 2πx
n
x
n+1
= x
n
+ y
n
+ δ sin 2πx
n
, mod 1,
y
n+1
= x
n
+ 2y
n
, mod 1,
δ < 1/2π
δ
D
C
= 2
δ
T
3
T
3
T
3
T
3
T
2
x
n+1
= F(x
n
, φ
n
, α),
φ
n+1
= φ
n
+ θ, mod 1,
x ∈ R
N
F ∈ R
N
φ α
φ θ
θ
θ = 0.5(
√
5−1)
N = 1
θ
T
2
α
α = α
0
T
2
α = α
1
T
2
α = α
cr1
α = α
cr2
> α
cr1
α
cr1
< α < α
cr2
N = 1
x
n+1
= f(x
n
, φ
n
, α),
φ
n+1
= φ
n
+ θ, mod 1.
θ
k
=
p
k
/q
k
, lim
k→∞
θ
k
= θ
φ
0
k
φ
0
θ = θ
k
x(φ
0
), φ
0
∈
[0; 1/q
k
]
φ
0
θ = θ
k
∂x
n
/∂φ
0
∂x
n
∂φ
0
=
n
X
k=1
f
φ
µ
n−k
(x
k
, φ
k
) + µ
n
(x
0
, φ
0
)
∂x
0
∂φ
0
,
µ
m
(x
k
, φ
k
) =
m−1
Y
i=0
f
x
(x
k+i
, φ
k+i
), µ
0
= 1,
λ = lim
n→∞
1
n
ln |µ
n
|.
n
∂x
n
∂φ
0
=
n
X
k=1
f
φ
µ
n−k
(x
k
, φ
k
).
Γ
n
Γ
n
= min
x
0
,φ
0
max
0≤i≤n
|
∂x
i
∂φ
0
|,
Γ
n
n → ∞ ∂x
n
/∂φ
0
Γ
n
n
x
n+1
= 2σ(tanh x
n
) cos (2πφ
n
) + α cos (2π(φ
n
+ β)),
φ
n+1
= φ
n
+ θ, mod 1
θ = θ
g
= 0.5(
√
5 − 1) α
α = 0
Γ
n
Γ
n
∼ n
η
.
η
η = 1
Λ
n
(x, φ) =
1
n
ln |µ
n
(x, φ)|,
µ
n
(x, φ)
(x, φ)
Γ
n
n θ =
θ
g
, σ = 1.5, β = 1/8
µ
n
Λ
n
n
lim
n→∞
Λ
n
= λ < 0
α = α
cr1
α = α
cr1
2
k
x
n+1
= α(1 + ε cos 2πφ
n
x(1 − x)),
φ
n+1
= φ
n
+ θ, mod 1.
α < α
cr1
α = α
cr1
α > α
cr1
T
3
2