Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
Подождите немного. Документ загружается.
θ = ω
1
/hωi
θ = 1
ω
1
θ
θ
θ
θ
θ
θ =
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
ω
0.0
0.1
0.2
0.3
0.4
0.5
a
1:1 3:1
5:1
1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
δ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
θ
0.26 0.31 0.36 0.41
0.2
0.3
0.4
0.5
δ
ϕ
n+1
= ϕ
n
+ f (ϕ
n
) , f(ϕ
n
) ≡ f(ϕ
n
+ 2π k) .
ϕ n → ∞
x
k+1
= x
k
+ δ −
K
2π
sin (2π x
k
), mod 1.
δ
K = 0
δ n n =
1, 2, . . .
θ
θ = lim
k→∞
x
k
− x
0
k
,
δ
mod 1
θ =
θ
1
=
θ
2
=
θ =
r+p
s+q
θ(δ)
(K, δ)
0.5 1.0 1.5 2.0 2.5 3.0
p
0.0
0.1
0.2
0.3
0.4
0.5
γ
2:1
3:1
1:1
p = ω
02
/ω
01
ε = 2.0
ω
01
ω
02
¨x
1
− ε(1 − x
2
1
) ˙x
1
+ ω
2
01
x
1
= γ(x
2
− x
1
) ,
¨x
2
− ε(1 − x
2
2
) ˙x
2
+ ω
2
02
x
2
= γ(x
1
− x
2
) ,
γ
θ = m : n
ω
1
/ω
2
= m : n
ϕ(t) = Φ
1
(t) − Φ
2
(t)
˙ϕ = 0, ϕ = const
x(t) ˙x(t)
ξ(t)
A(t) Φ(t) ω(t) =
˙
Φ(t)
p(A, Φ, t |A
1
, Φ
1
, t
1
)
A Φ t > t
1
t
0
A = A
1
Φ = Φ
1
Φ(t)
ϕ(t) = Φ(t) − ω
1
t
U(ϕ)
ϕ(t)
¨x − ε(1 − x
2
) ˙x + ω
2
0
x = a cos(ω
1
t) +
p
2D
0
ξ(t),
ξ(t) δ
hξ(t)i ≡ 0; hξ(t)ξ(t+τ)i = δ(τ ) D
0
A(t) ϕ(t)
˙
A =
ε A
2
1 −
A
2
A
2
0
− µ sin ϕ +
D
A
+
√
2Dξ
1
(t),
˙ϕ = ∆ −
µ
A
cos ϕ +
√
2D
A
ξ
2
(t),
ϕ
µ = a/2ω
1
D = D
0
/2ω
2
1
ξ
1
ξ
2
δ
hξ
1,2
(t)i ≡ 0, hξ
1,2
(t) ξ
1,2
(t+τ)i =
δ(τ) D ξ
1
ξ
2
a ε, D ε
A
0
A(t) A
0
˙ϕ = ∆ − ∆
c
cos ϕ +
√
2D
A
0
ξ
2
(t),
∆
c
= µ/A
0
ϕ
U(ϕ) = −∆ ·
ϕ−∆
c
sinϕ ∆ < ∆
c
ϕ
k
= arccos (∆/∆
c
) + 2πk
U(ϕ)
ϕ(t) ϕ
k
2π
D
D = 0.02
ϕ
D = 0.07
ϕ
∼
=
const
∆ = 0.06 µ = 0.15
A
0
= 1
ϕ
ϕ(t) D = 0.07 D = 0.22
∂p(ϕ, t)
∂t
= −
∂
∂ϕ
(∆ −∆
s
cos ϕ) p(ϕ, t) −Q
∂p(ϕ, t)
∂ϕ
,
Q = D/A
2
0
ϕ
ϕ
P (ϕ, t) [−π, π]
P (ϕ, t) =
∞
X
n=−∞
p (ϕ + 2πn, t) .
P (ϕ, t)
P
st
(ϕ)
P (−π, t) = P (π, t)
R
π
−π
P (ϕ, t)dϕ = 1
P
st
(ϕ) = N exp
∆ · ϕ − ∆
s
sin ϕ
Q
Z
ϕ+2π
ϕ
exp
−
∆ · ψ − ∆
s
sin ψ
Q
dψ ,
−π ≤ ϕ ≤ π,
N ∆ = 0
P
st
(ϕ) =
1
2πI
0
(∆
s
/Q)
exp
∆
s
Q
cos(ϕ + π/2)
, −π ≤ ϕ ≤ π,
I
0
(z)
I
0
(∆
s
/Q) ≈ 1 exp [(∆
s
/Q) cos (ϕ + π/2))] ≈ 1
P
st
(ϕ) = 1/2π
cos(ϕ+π/2) ≈ 1−(ϕ+π/2)
2
/2 I
0
(∆
s
/Q) ≈ exp (∆
s
/Q)/
p
2π∆
s
/Q
P
st
(ϕ) = exp (−∆
s
(ϕ + π/2)
2
/2Q)/
p
2π∆
s
/Q
ϕ
0
= −π/2
Q → 0 δ
lim
Q→0
P
st
(ϕ) = δ(ϕ + π/2)
P
st
(ϕ)
hωi
hωi = h ˙ϕi + ω
1
=
Z
π
−π
(∆ − ∆
s
cos ϕ) P
st
(ϕ)dϕ + ω
1
,
ω
1
ϕ
0
p(ϕ, t = 0) = δ(ϕ −ϕ
0
) hϕ
2
(t = 0)i = 0
hϕ
2
(t)i ∝ D
eff
·t
D
eff
D
eff
=
1
2
d
dt
hϕ
2
(t)i − hϕ(t)i
2
.
D
eff
= 0
U(ϕ)
D
eff
=
p
∆
2
s
− ∆
2
2π
1 + exp
−
2π∆
Q
× exp
−
2
Q
p
∆
2
s
− ∆
2
∆ · arcsin
∆
∆
s
.
2π
Φ(t)
Φ(t)
hωi = lim
T →∞
1
T
Z
t
0
+T
t
0
d Φ(t)
dt
dt = lim
T →∞
1
T
(Φ(t
0
+ T ) −Φ(t
0
)),
D
eff
=
1
2
d
dt
hϕ
2
(t)i − hϕ(t)i
2
,
ϕ(t) = Φ(t) −Ψ(t) Ψ(t)
x(t)
w(t)
w(t) = x(t) + iy(t) = A(t)e
iΦ(t)
.
A(t) =
p
x
2
(t) + y
2
(t) , Φ(t) = arctan
y
x
+ πk, k = 0, ±1, . . .
ω(t) =
d
dt
Φ(t) =
1
A
2
(t)
[x(t) ˙y(t) − y(t) ˙x(t)].
y(t)
x(t) y(t)