Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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N = 2
x
m
= ¯x, y
m
= ¯y.
¯x
¯y
˙η
(1)
m
= ˙η
(2)
m
,
˙η
(2)
m
= −αη
(2)
m
− [A + 2Γ (t)] η
(1)
m
+ Γ (t)
h
η
(1)
m+1
+ η
(1)
m−1
i
,
η
(1)
m
= x
m
−¯x, η
(2)
m
= y
m
−¯y, Γ (t) = (1/2)[γ
0
+εΩ
2
sgn( sin(ωt))]
A = f
0
(¯x) =
a −1, |x| ≤ 1,
b −1, |x| > 1.
ξ
(1)
j
=
1
N
N
X
m=1
e
−km
η
(1)
m
, ξ
(2)
j
=
1
N
N
X
m=1
e
−km
η
(2)
m
,
k = i(2πj/N), i
2
= −1, j = 1, . . . , N
˙
ξ
(1)
j
= ξ
(2)
j
,
˙
ξ
(2)
j
= −αξ
(2)
j
− ω
2
0
(t, j)ξ
(1)
j
,
ω
2
0
(t, j) = A + [γ
0
+ εΩ
2
sgn( sin(Ωt))][1 − cos(2πj/N)]
x
m
= ¯x, y
m
= ¯y
j = N
j = 1, . . . , N −1
N = 2, j = 1
µ
1,2
j
T = 2π/Ω
µ
1,2
(j) = e
−απ/Ω
h
0.5S(j) ±
p
0.25S
2
(j) − 1
i
,
S(j) = 2cosh[β
1
(j)]cosh[β
2
(j)] + K(j)sinh[β
1
(j)]sinh[β
2
(j)],
β
1,2
(j) = (π/Ω)
p
C
1
(j) ∓C
2
(j),
K(j) =
2C
1
(j)
p
C
2
1
(j) − C
2
2
(j)
,
C
1
(j) =
α
2
4
− A − γ
0
[1 − cos(2πj/N)],
C
2
(j) = εΩ
2
[1 − cos(2πj/N)].
j = 1, . . . , N −
1 |µ
1,2
(j)| < 1 A = a − 1
A = b − 1
ε
N Ω = 25
B = 1.5 ω = 1 α = 0.1
γ
0
= 0.1 a = 0.5 b = 2
ε = 0
N
j
N
ε
ε
ε N > 10
A = a −1 A = b − 1
Ω = 15; 25; 50; 100 8; 10; 15; 21
x
n+1
(i) = α − x
2
n
(i) + γ[2x
2
n
(i) − x
2
n
(i − 1) − x
2
n
(i + 1)],
x
n+1
(i) = α − x
2
n
(i) + γ[2x
n
(i) − x
n
(i − 1) − x
n
(i + 1)],
α γ x
n
(i)
i n N
i = 1, 2, . . . , N n = 0, 1, 2, . . .
x
n
(1) = x
n
(N +1)
T = s
λ = 2
T = 2 λ = 3 T = 2 λ = 5 T = 4
T = s
i
¯x
k
s
x
n+k−1
(i) = ¯x
k
+ ˜x
n+k−1
(i), x
n+s
(i) =
¯x
1
+ ˜x
n+s
(i) k = 1, . . . , s, ˜x
n+k−1
(i)
¯x
k
α
0
α = α
n+k−1
(i) = α
0
+ ˜α
n+k−1
(i), k = 1, . . . , s.
¯x
k+1
+ ˜x
n+k
(i) = α
0
+ ˜α
n+k−1
(i) − [¯x
k
+ ˜x
n+k−1
(i)]
2
+ γ[2(¯x
k
+
+ ˜x
n+k−1
(i))
2
− 2¯x
2
k
] − γ[(x
2
n+k−1
(i + 1) − ¯x
2
k
) +
+ (x
2
n+k−1
(i − 1) − ¯x
2
k
)],
k = 1, . . . , s
(i − 1)
(i+ 1)
¯x
k
˜x
n+k−1
(i)
i
¯x
k+1
= α
0
− ¯x
2
k
, ¯x
s+1
= ¯x
1
,
˜x
n+k
(i) = ˜α
n+k−1
(i) − 2¯x
k
(1 −2γ)˜x
n+k−1
(i) −
− γ[(x
2
n+k−1
(i + 1) − ¯x
2
k
) + (x
2
n+k−1
(i − 1) − ¯x
2
k
)],
k = 1, . . . , s
i
˜α
n+k−1
(i) = a
1
˜x
n+k−1
(i)
+a
2
[(x
2
n+k−1
(i + 1) − ¯x
2
k
) + (x
2
n+k−1
(i − 1) − ¯x
2
k
)].
a
1
a
2
i
T = s
˜α
n+k−1
(i) = 2(1 − 2γ)¯x
k
(x
n+k−1
(i) − ¯x
k
)
+γ[(x
2
n+k−1
(i + 1) − ¯x
2
k
) + (x
2
n+k−1
(i − 1) − ¯x
2
k
)],
k = 1, . . . , s
T = 1 α = 1.6 γ = 0.01
¯x
1
ε
¯x
1
γ[(x
2
n
(i + 1) − ¯x
2
1
) +
(x
2
n
(i − 1) − ¯x
2
1
)] ε
¯x
1
¯x
2
T = 2 T = 4
λ = 2 T = 2 λ = 3 T = 2
λ = 5 T = 4
λ
pλ (p = 1, 2, . . .)
x
n
(i) = x
n
(i + pλ)
x
n+1
(λm + 1) = α − x
2
n
(λm + 1) + γ[2X
2
n
(λm + 1) − x
2
n
(λm + 2) −
− x
2
n
(λm)],
x
n+1
(λm + 2) = α − x
2
n
(λm + 2) + γ[2X
2
n
(λm + 2) − x
2
n
(λm + 3) −
− x
2
n
(λm + 1)],
x
n+1
(λm + λ) = α − x
2
n
(λm + λ) + γ[2x
2
n
(λm + λ) − x
2
n
(λm + λ + 1) −
− x
2
n
(λm + λ − 1)],
(m = 0, 1, . . . , (N/λ) − 1).
λ = 2
T = 2
˜α
n
(i) = 2p
1
(i)(1 − 2γ)[x
n
(i) − p
1
(i)] +
+ γ[(x
2
n
(i + 1) − p
2
2
(i)) + (x
2
n
(i − 1) − p
2
2
(i))],
˜α
n+1
(i) = 2p
11
(i)(1 − 2γ)[x
n+1
(i) − p
11
(i)] +
+ γ[(x
2
n+1
(i + 1) − p
2
21
(i)) + (x
2
n+1
(i − 1) − p
2
21
(i))],
p
1
(i) =
¯x
1
,
¯x
2
,
p
2
(i) =
¯x
2
, i = 2m + 1,
¯x
1
, i = 2m + 2,
p
11
(i) =
¯x
11
,
¯x
21
,
p
21
(i) =
¯x
21
, i = 2m + 1,
¯x
11
, i = 2m + 2,
¯x
1
, ¯x
2
, ¯x
11
, ¯x
21
m = 0, 1, . . . , N/2−
1.
λ = 3 T = 2 λ = 5
T = 4
λ = 2, T = 2
λ = 3, T = 2 λ = 5, T = 4
i
n
α
α γ
1.6 < α < 1.9 0 < γ < 0.4
N = 20
γ > 0.04