Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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x
n+1
= x
n
+ y
n
+ δ cos 2πy
n
, mod 1,
y
n+1
= x
n
+ 2y
n
, mod 1.
δ < 1/2π
(x
n
, y
n
)
J
J =
1 1 − 2πδ sin 2πy
n
1 2
6= 0, δ <
1
2π
.
|J| < 1
n → ∞
1 < D
I
< 2 δ = 0.05
D
I
' 1.96 δ = 0.10 D
I
' 1.84
D
C
= 2
1 ≤ D ≤ 2
x
n+1
= 2λ tanh(x
n
) cos 2πϕ
n
,
ϕ
n+1
= ω + ϕ
n
, mod 1,
δ = 0.15
ω
ω = 0.5(
√
5 −1)
λ > 1
0.0 0.2 0.4 0.6 0.8
−4.0
−2.0
0.0
2.0
x
φ
λ =
1.5
x(t)
˙x
ξ
= f (x
ξ
, ξ(t)).
ξ(t)
x
ξ
(t) → x
ξ
(t + dt) ξ(t)
x
ξ
(t)
ξ(t)
x
ξ
(t)
N
ξ
j
(t) j = 1, 2, . . . , N N → ∞
x
ξ
(t)
x
1
, x
2
, . . . , x
n
t
1
, t
2
, . . . , t
n
p(x
1
, t
1
; x
2
, t
2
; . . . ; x
n
, t
n
)
P (x
ξ
(t) < X)
p(x, t)
∂
∂t
p = L p ,
L
L
x
ξ
t
x
P
x
(x
ξ
(t) < x) = lim
N→∞
1
N
N
X
i=1
θ(x − x
ξ
i
(t)) = hθ(x − x
ξ
(t) )i.
h · i N → ∞
ξ
i
(t) θ
x 0 1
p(x, t) x
ξ
(t)
t
[x, x + dx) p
x
(x, t) dx = P
x
(x ≤ x
ξ
(t) < x + dx)
x
p
x
(x, t) =
d
dx
P
x
(x
ξ
(t) < x) = hδ(x − x
ξ
(t) )i,
δ(·)
R
X
p
x
(x, t) dx = 1
X
x y = f(x)
y =
R
f(x
0
)δ(x − x
0
) dx
0
f x hf(x)i = hy(t)i =
R
f(x
0
)p
x
(x
0
, t) dx
0
y
p
y
(y, t) = hδ( y − y
ξ
(t)) i = hδ( y − f (x
ξ
(t)) ) i
=
Z
X
δ( y − f(x))p
x
(x, t) dx.
y x =
f
−1
(y) = g(y) δ
p
y
(y, t) = p
x
(x, t)
1
df(x)
dx
,
x = g(y)
x
ξ
(t)
t
i
i = 1, 2, . . . , n t
0
< t
1
< t
2
< . . . < t
n
x
ξ
(t)
n
p
n
(x
1
, t
1
; . . . ; x
n
, t
n
) = < δ(x
1
− x
ξ
(t
1
)) . . . δ(x
n
− x
ξ
(t
n
)) >,
n = 2, 3, 4, . . .
p
n
(x
n
, t
n
|x
1
, t
1
; . . . ; x
n−1
, t
n−1
) = hδ(x
n
−x
ξ
(t
n
))i
x
ξ
(t
1
)=x
1
;...;x
ξ
(t
n−1
)=x
n−1
,
t
n
x
ξ
x
ξ
t
i
i = 1, . . . , n −1
p
n
(x
1
, t
1
; . . . ; x
n
, t
n
)
= p
n
(x
n
, t
n
|x
1
, t
1
; . . . ; x
n−1
, t
n−1
) p
n−1
(x
1
, t
1
; . . . ; x
n−1
, t
n−1
).
n
p
n
(x
n
, t
n
|x
1
, t
1
; . . . ; x
n−1
, t
n−1
) = p
2
(x
n
, t
n
|x
n−1
, t
n−1
).
p
n
(x
1
, t
1
; . . . ; x
n
, t
n
) = p
2
(x
n
, t
n
|x
n−1
, t
n−1
) . . . p
2
(x
2
, t
2
|x
1
, t
1
)p(x
1
, t
1
),
p(x
1
, t
1
)
t
1
< t
2
< t
3
p
3
t
2
p
2
(x
1
, t
1
; x
3
, t
3
) =
Z
X
dx
2
p
3
(x
1
, t
1
; x
2
, t
2
; x
3
, t
3
).
p
2
p
3
p
2
(x
3
, t
3
|x
1
, t
1
) =
Z
X
dx
2
p
2
(x
3
, t
3
|x
2
, t
2
) p
2
(x
2
, t
2
|x
1
, t
1
),
x ∆x
t
W (x → x + ∆x, t) = lim
dt→0
1
dt
p
2
(x + ∆x, t + dt|x, t).
dt
p
2
(x, t + dt|x + ∆x, t) =
1 − dt
Z
dx
0
W (x + ∆x → x
0
, t)
δ(∆x)
+ W (x + ∆x → x, t) dt + O(dt
2
),
x ∆x = 0
dt → 0
∂
∂t
p
2
(x, t|x
1
, t
1
)
=
Z
dx
0
( W (x
0
→ x, t) p
2
(x
0
, t|x
1
, t
1
) − W (x → x
0
, t) p
2
(x, t|x
1
, t
1
) ).