Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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r
r < α(c − 0.5) + 0.5ξ −
p
0.25(ξ − α)
2
− ξ(β − α),
ξ = 1 − 2γ, c =
1 + a, |x| ≤ 1,
1 + b, |x| > 1.
F
1
=
r
1
u
1
u
1
= v
1
= w
1
= 0
X
1
= −X
2
F F
1
X
1
= X
2
F = r(x
1
− x
2
)
α
γ = 0.2
a = −8/7, b = −5/7, β = 22
ε
F (t)
X
1
(t) F (t)
α = 11.4, γ = 0.2
α = 12.7 γ = 0.2
α
α
‘
γ0.3
0.20.10.0
10.6
11.1
12.1
α
l
DS
SA
r
4C
0
2C
0
C
0
11.6
α γ
l
C
0
, 2C
0
, 4C
0
SA
r
DS
X
1
= X
2
X
1
= −X
2
F
1
=
r
1
(x
1
+ x
2
)
α γ
α γ
X
1
= −X
2
F
1
α γ
α = 12.7, γ = 0.2
¨x
1,2
+ α ˙x
1,2
+ f (x
1,2
) − γ(x
2,1
− x
1,2
) = B sin(ωt),
f(x) = (b − 1)x + 0.5(b − a)(|x − 1| − |x + 1|) α
γ B ω
γ = 0
B = 0
P
1
(x
1
= 0, y
1
=
0), P
2,3
(x
2,3
= ±(b − a)/(b − 1), y
2,3
= 0)
0 < a < 1, b >
(1 + α
2
/4), α > 0
P
1
P
2,3
γ > 0
γ
γ
0
γ = γ
0
+ F (t),
F (t) T = 2π/Ω Ω
F (t) = εΩ
2
sgn (sin (Ωt)) ,
ε
˙x
1,2
= y
1,2
˙y
1,2
= −αy
1,2
− f (x
1,2
) +
γ
0
+ εΩ
2
sgn (sin (Ωt))
×(x
2,1
− x
1,2
) + B sin (ωt) .
u = (x
1
−x
2
)/2, u
1
= (x
1
+x
2
)/2, v =
(y
1
− y
2
)/2, v
1
= (y
1
+ y
2
)/2
˙u = v,
˙v = −αv −ω
2
0
(t)u,
˙u
1
= v
1
,
˙v
1
= −αv
1
− f (u
1
) + B sin(ωt),
ω
2
0
(t) = A + 2γ
0
+ 2εΩ
2
0
sgn( sin(Ωt)), f(u
1
) = (f(x
1
) + f(x
2
))/2,
A =
a − 1, |x| ≤ 1,
b − 1, |x| > 1.
u = 0, v = 0
u = 0, v = 0
u = 0, v = 0
T
[0; T/2] [T/2; T ]
0 T/2
T/2 T
u(T/2)
v(T/2)
= [M
1
]
u(0)
v(0)
,
u(T )
v(T )
= [M
2
]
u(T/2)
v(T/2)
.
u = 0, v = 0
[M] = [M
2
] [M
1
]
µ
1,2
= exp(απ/Ω)(0.5S ±
p
0.25S
2
− 1),
S = 2(coshβ
1
coshβ
2
+ Ksinhβ
1
sinhβ
2
),
β
1
= (π/Ω)
p
0.25α
2
− A − 2γ
0
− 2εΩ
2
,
β
2
= (π/Ω)
p
0.25α
2
− A − 2γ
0
+ 2εΩ
2
,
K =
0.25α
2
− A − 2γ
0
p
(0.25α
2
− A − 2γ
0
)
2
− 4ε
2
Ω
4
.
|µ
1,2
| < 1.
A = a − 1, A = b − 1.
ε−Ω
(B = 1.5, α =
0.1, ω = 1, γ = 0.1, a = 0.5, b = 2)
ε = 0
x
1
(0) =
x
2
(0) + ∆x, y
1
(0) = y
2
(0) + ∆y. ∆x = 0.02
∆y = 0.02
ε Ω
(ε = 0)
Ω
ε
Ω
ε
∆x ∆y
ε Ω
Ω
ω =
1
Ω
ε 20%
γ
0
= 0.1 Ω ω = 1
˙x
m
= y
m
,
˙y
m
= −αy
m
− f (x
m
) + (γ/2)(x
m+1
− 2x
m
+ x
m−1
) + B sin(ωt),
x
0
= x
N
, y
0
= y
N
, x
N+1
=
x
1
, y
N+1
= y
1
, m = 1, . . . , N N