Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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n
x
1,i+1
= F
1
(x
1,i
, x
2,i
, . . . , x
n,i
)
. . . . . .
x
n,i+1
= F
n
(x
1,i
, x
2,i
, . . . , x
n,i
) ,
x
j,i
i∆t F
j
F
j
F
j
F
j
F
j
X
k
F
j
(x
1,i
, x
2,i
, . . . , x
n,i
) =
L
X
k=1
C
k,j
X
k
(x
1,i
, x
2,i
, . . . , x
n,i
) .
ν
F
j
(x
i
) =
ν
X
l
1
,l
2
,...,l
n
=0
C
j,l
1
,l
2
,...,l
n
n
Y
k=1
x
l
k
k,i
,
n
X
k=1
l
k
≤ ν ,
C
j,l
1
,l
2
,...,l
n
x
0
F
j
i F
j
N
x
j,i+1
=
ν
X
l
1
,l
2
,...,l
n
=0
C
j,l
1
,l
2
,...,l
n
n
Y
k=1
x
l
k
k,i
, i = 1, . . . , N,
C
j,l
1
,...,l
n
N
ν
n ν K
K =
(n + ν)!/(n!ν!) N K
dx
1
dt
= F
1
(x
1
, x
2
, . . . , x
n
)
. . . . . .
dx
n
dt
= F
n
(x
1
, x
2
, . . . , x
n
) .
x
i
dx
1
dt
= x
2
,
dx
2
dt
= x
3
, . . . ,
dx
n
dt
= f(x
1
, x
2
, . . . , x
n
) .
dx
dt
= y ,
dy
dt
= a (1 − bx
2
) y − x ,
f
f
P/Q
0 10 20
−4.5
4.5
t
x
−4.5 4.5
−4.5
4.5
x
y
0 10 20
−4.5
4.5
t
x
−4.5 4.5
−4.5
4.5
x
y
n = 2
∆t = 0.01 ν = 3
0 50 100
−20
20
x
t
−20 20
−35
20
x
y
0 50 100
−20
20
x
t
−20 20
−35
20
x
y
n = 4
∆t = 0.01 ν = 3
g
j
dx
j
dt
= F
j
(x
1
, x
2
, . . . , x
n
) .
g
j
= g
j
(x) 1, x
j
, x
2
j
x
j
x
k
j, k = 1, 2, . . . , n
n
P T
T P
τ
β
β = ∞
β
x
0.0 1.0 2.0 3.0 4.0
ECG−signal
t,s
P
Q
R
S
T
X1
X2
X3
X1
X2
X3
X1
X2
X3
τ
a(t)
a(t) = O(t) + S(t) + ξ(t) + C ,
O(t)
S(t)
ξ(t) C
a(t)
a(t)
O(t)
a(t)
a
1
(t) =
Z
t
0
a(t
1
)dt
1
=
Z
t
0
O(t
1
)dt
1
,