常微分方程定性理论幻灯片1

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• Ωp(AP ) Thm 1.1 Ωp(AP ) L+ p Weierstrass Ωp Ωp
− L+ ( L p p)
mathbbRn mathbbR2 ... Poincare ´- . . .
Ωp )
(Bolzano16 49
1.Ωp
{qn} ⊆ Ωp, lim qn =
n→∞
mathbbRn mathbbR2 ... Poincare ´- . . .
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49
•
1. R2 x d dt = X (x, y ), dy = Y (x, y ). dt X (x, y ), Y (x, y ) ∈ C (D, R2), D ⊆ R2 , x,y Lipschitz (2.1) D
25 49 mathbbRn mathbbR2 ... Poincare ´- . . .
mathbbRn mathbbR2
... Poincare ´- . . .
(1) t N1 N2 (2) t AB, CD ABCD
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49
N1 N2 N1 N2
•
1. N1 N 2
P0 P0 N1 N 2
(2.1)
mathbbRn mathbbR2
N 1 N2 P0 S (P0, ε) S (P0, ε) P 0 (x 0 , y0 ) N1 N 2 X (x0, y0)(x − x0) + Y (x0, y0)(y − y0) = 0. S (P0, ε) λ(x, y ) N 1 N2
(2.1)
mathbbRn mathbbR2
... Poincare ´- . . .
Defn 2.1 (1) N1N2 (2) (2.1) N 1 N2
N1 N2 ⊂ D N 1 N2 (2.1)
(2.1) ,
,
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Defn 2.2
N1 N2 (2.1) BC, AD (2.1) N1, N2 AB, CD AB DC N1N2 ABCD
n→∞
mathbbR2
= f ( lim f (p, tn), t) = f (q, t),
n→∞
f (q, t) ∈ Ωp,
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∴ f (Ωp, t) ⊆ Ωp, ∀t ∈ R. ∀t ∈ R Ωp = f (f (Ωp, t), −t) ⊆ f (Ωp, −t) = f (Ωp, t).
3.Ωp Ωp 2 Ω1 , Ω p p 2 1 2 Ω1 Ω = Ω , Ω Ω P p p p p = ∅. 2 d(Ω1 , Ω p p) = d > 0 d S (Ωi , ω p 3 ), i = 1, 2. i ∃{ti } , i = 1 , 2 , n → ∞ t n n → +∞, f (p, ti n) ∈
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mathbbRn mathbbR2
... Poincare ´- . . .
1 q ∈ Ωp ⇒ Lq ∪ Ωq ∪ Aq ⊂ Ωp. 2 Ωp ⊆ L+ p.
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•
(1.1) 1 Ωp(Ap) = ∅
mathbbRn mathbbR2 ... Poincare ´- . . .
Ωp(Ap) = 2 ∅, Ωp ∩ f (p, R+) = ∅(Ap ∩ f (p, R−) = ∅) Poisson ( P +(P −) ) Ωp ∩ f (p, R+) = ∅(Ap ∩ f (p, R−) = ∅). 3
( P31 Thm4.2)
: M1 ∈ L+ p Ωp = M1 M1 N1 N 2 L+ M1 p N1 N 2 M2 M1, M2 M1 ∈ Ωp M1 N 1 N2
M1 ∈ Ωp L+ p = M1 M1 ∈ Ωp
mathbbRn mathbbR2
... Poincare ´- . . .
32
mathbbRn mathbbR2
... Poincare ´- . . .
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2. N 1 N2 − L+ ( L p p ) N 1 N2 M1, M2, M3 M2 N1 N 2 L+ p M1, M2 ,
A.L+ p B.L+ p N1 N2 R N1 N2 M2 M2 R
N1 N2 (2.1) M1, M3
2 ¯n}, t ¯n ∈ (t1 {t , t n n) 2 d S (Ωp, ).
n→
mathbbRn mathbbR2 ... Poincare ´- . . .
d 1 ¯ ¯ S (Ωp, ) f (p, tn)∈ 3 L+ p
3 ¯n)} {f (p, t ¯ni ) → q, i → ∞. f (p, t ¯ Ωp q ∈ Ωp q∈
∈
mathbbRn mathbbR2
... Poincare ´- . . .
t = 0 p t f (p, t)
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f (·, t) : D → D.
(1.1) (i) f (p, 0) = p (ii) f (p, t) p, t ( ( ), ), ).
mathbbRn mathbbR2 ... Poincare ´- . . .
22 49
mathbbRn mathbbR2
... Poincare ´- . . .
2
R2
´-Bendixson Poincare
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49
mathbbRn
... Poincare ´- . . .
L
R2 R2
Jordan
R2 L P-B
mathbbR2
´ I.Bendixson H. Poincare
q q ∈ Ωp ε . qm ∈ Ωp ∀ε, ∃m > 0, d(qm, q ) < 2 ω ∀T > 0, ∃tm > ε T d(f (p, tm), qm) < 2 ∴ d(f (p, tm), q ) d(f (p, tm), qm)+d(qm, q ) < ε, ∴ q ∈ Ωp.
17
... Poincare ´- . .ቤተ መጻሕፍቲ ባይዱ.
Rn
mathbbR2
X (1.1)
(1.1) ¯ ∈D X
= Rn
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F (X ) ∈ C (D, Rn) Lipschitz X d dt = F (X ),
X Cauchy
mathbbRn mathbbR2
... Poincare ´- . . .
: ”⇒” ” ⇐ ” ∵ ∀p ∈ A, ∀t ∈ R ∴ f (A, t) ⊆ A,
f (p, t) ∈ A,
∴ f (A, −t) ⊇ f (f (A, −t), t) = f (A, 0) = A. ( f (p, t) t ) ,f (A, t) f (A, −t) A f (A, t) = A.
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4 t X (t2, 0, X (t1, 0, X0)) = X (t1 + t2, 0, X0)). t = 0 X0 t1 X1 t2 X2 X0 t1 + t2 X2
(1.1) F (X ) D C (D, Rn) X Lipschitz f (p, t) (1.1) D→D
R
F (X )
... Poincare ´- . . .
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49
X (x0, y0)(x−x0)+Y (x0, y0)(y −y0) = c.
λ
(2.1)
t
(2.1)
dλ |(2.1) = X (x0, y0)X (x, y )+Y (x0, y0))Y (x, y ). dt dλ 2 2 | = X ( x , y ) + Y (x0, y0)) > 0, 0 0 dt (x0,y0) , ∃δ ∈ (0, ε] dλ |(2.1) > 0, (x, y ) ∈ S (P0, δ ). dt (2.1) S (P 0 , δ ) t c λ(x, y ) = c N1 N2 ⊂ S ( P 0 , δ ) N1 N2 P0 ABCD N1N2 N1 N2
(iii)f (f (p, t1), t2) = f (p, t1+t2) ( t
{f (·, t)| t ∈ R } D → D (1.1) f D×R → D f D
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p
f (p, t)
p
p
f (p, R) = {f (p, t)| t ∈ R } Lp
mathbbRn mathbbR2
... Poincare ´- . . .
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2.Ωp ∀t ∈ R, f (Ωp, t) = Ωp. q ∈ Ωp ∃{tn}, n → ∞ t → +∞ lim f (p, tn) = q f (p, tn + t)
mathbbRn ... Poincare ´- . . .
n→∞
n→∞
lim f (p, tn + t) = lim f (f (p, tn), t)
X∈ (1.1) X X (t, t0, X0) Rn+1 Rn
dX = F (X ) (1.1) dt D ⊆ Rn. F (X ) ∈ C (D, Rn). Rn t (1.1) D F (X ) Rn t X =
4
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D X (t, t0, X0) ¯) = 0 F (X
(1.1) t
Rn
mathbbRn
mathbbRn mathbbR2 ... Poincare ´- . . .
:
M1 M2 N1 N 2
L+ p
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49
N1 N2 N1 N 2
M2 R
mathbbRn mathbbR2
... Poincare ´- . . .
2 N1 N 2 f (p, tn)
L+ p ω
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49
•
´-Bendixson Poincare − 2.1: L+ ( L p p) ω (α ) Lp
f (p, t + T ) = f (p, t),
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Defn 1.1 +∞(−∞),
∃{tn}, n
→
∞ ,tn
→
mathbbRn mathbbR2 ... Poincare ´- . . .
n→∞
lim f (p, tn) = q ∈ D, ω ω (α ) (α )
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q
f (p, t) ω (α )
p
+ + L+ = f ( p, R ) = { f ( p, t ) | t ∈ R }; p
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− − L− = f ( p, R ) = { f ( p, t ) | t ∈ R }. p
T >0 f (p, t) T Rn
∀t
mathbbRn mathbbR2
... Poincare ´- . . .
(1.2) (1.1)
6 49
X (t ) = X , t ∈ R, X ∈ D. 0 0 0 0
1 X (t)
(1.1) (1.1)
t
X X = X (t + τ ) τ
=
D
2 (1.1) 3 ¯ X ¯ ∈D X ¯ X
X0 ∈
mathbbRn mathbbR2
... Poincare ´- . . .
f (p, t) f (p, t)
Ωp(Ap)
1 Defn1.1 ⇔ ∀ε > 0, T d(f (p, t), q ) < ε 2 Lp f (p, t)
f (p, t) ω > 0, ∃t > T, Ωp = Ap =
q
mathbbRn mathbbR2
... Poincare ´- . . .
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Defn 1.2 A
f (A, t) = A f (·, t) f (·, t)
∀t ∈ R f (·, t)
mathbbRn mathbbR2
... Poincare ´- . . .
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: A f (p, t) ∈ A
⇔ ∀p ∈ A, ∀t ∈ R
mathbbRn mathbbR2 ... Poincare ´- . . .
i d S (Ωp, ), i
Ωp
mathbbRn mathbbR2
... Poincare ´- . . .
3
= 1, 2.
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{ti n}
2 1 2 1 2 0 < t1 < t < t < t < · · · · · · < t < t 1 1 2 2 n n < ··· .
∞
¯n → +∞ t
mathbbRn mathbbR2
... Poincare ´- . . .
1
49
mathbbRn mathbbR2
... Poincare ´- . . .
2
49
mathbbRn mathbbR2
... Poincare ´- . . .
1
Rn
3
49
(
)
mathbbRn mathbbR2 ... Poincare ´- . . .
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M1, M2 A,B A B
N1 N2 N1 N2
mathbbRn mathbbR2
... Poincare ´- . . .
P+
P−
P
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− 2.2: L+ ( L p p) − L+ ( L p p)