Skrypt mat

8/17/2019 Skrypt mat

1/98


2/98


3/98


4/98


5/98

n

F

dny

dtn, dn−1ydtn−1

, . . . , y , t

= 0

y ∈ Rm t

dnydtn

= f dn−1y

dtn−1, . . . , y , t

y(t)

I

∀t∈I

dn−1ydtn−1

, . . . , y , t

∈ Dm f

R


6/98

n

Y (t) =

y

dydt

dn−2ydtn−2

dn−1ydtn−1

∈ Rm·n dY (t)dt

=

dydt

d

2

ydt2

dn−1ydtn−1

f dn−1ydtn−1

, . . . , y , t

n y(t)

Y (t)

y = f (t) y(t) =ˆ

f (t) dt

y = sin t y(t) =ˆ

sin t dt = − cos t + C C

Dm f ⊂ Rm·n+1

∆ = (yn−1, . . . , y1, y0, t0) ∈ Dm f.

y

t0 ∈ Dm y dn−1ydtn−1

(t0) = yn−1 . . . dy

dt(t0) = y y(t0) = y0

y = sin t (y0, t0)

y(t) = ´ sin t dt = −cos t + C y0 =

−cos t0 + C

⇒C = y0 + cos t0


7/98

dy

dt =

−ry r > 0 y(0) = m > 0

y0(t) = m

y1(t) = y(0) +

tˆ 0

− ry0(u)

du = m − mrt

n

yn(t) = y(0) +

tˆ 0

− ryn−1(u)

du

yn+1(t) − yn(t) =m − r

tˆ 0

yn(u) du

−

m − r

tˆ 0

yn−1(u) du

= −rtˆ

0

yn(u) − yn−1(u) du

∀n0 yn+1(t) − yn(t) = m (−rt)n+1

(n + 1)!

y1(t) − y0(t) = −rmt yn(t) − yn−1(t) = m (−r)ntnn!

yn+1(t) − yn(t) = −rtˆ

0

yn(u) − yn−1(u)

du = −r

tˆ 0

m(−r)n tn

n! du

= m(−r)n+1tˆ

0

tn

n! du = m(−r)n+1 t

n+1

(n + 1)!


8/98

yn(t) = [yn(t) − yn−1(t)] + [yn−1(t) − yn−2(t)] + . . . + [y1(t) − y0(t)] + y0(t)

= mn

k=1(−rt)k

k!

+ m = mn

k=0(−rt)k

k!

n→∞

−−−→me−rt

d

dt

me−rt

= m(−r)e−rt = −ry(t)

y = f (y, t) y ∈ Rm f

Rm y(t)

(y0, t0) t0 ∈ Dm y (y0, t0) ∈ Dm f

∀t∈Dm y (y(t), t) ∈ Dm f y(t) = y0 +tˆ

t0

f (y(u), u) du

y

y

y(t) = y0 +

tˆ t0

dydt (u) du = y0 +

tˆ t0

f (y(u), u) du

t

dy

dt =

d

dt

tˆ t0

f (y(u), u) du = f (y(t), t)

y(t0) = y0 +

t0

ˆ t0

f (y(u), u) du = y0

K f : K → R

∃m,M ∈K f (m) = inf {f (x) : x ∈ K } f (M ) = sup {f (x) : x ∈ K }


9/98

K f : K → R

∀ε>0 ∃δ>0 ∀x1,x2∈Dm f x1 − x2 < δ ⇒ f (x1) − f (x2) < ε.

δ 0 > 0

δ f : (0, δ 0) →R

+

limx→0+

δ f (x) = 0 0 < x1 − x2 < δ 0 f (x1) − f (x2) δ f

x1 − x2

δ f

δ f (x) = sup {f (x1) − f (x2) : 0 < x1 − x2 < x}.

δ 0 ε = 1

f L

∀x1,x2∈Dm f f (x1) − f (x2) L · x1 − x2

δ f (x) = Lx

f : R ⊃ I → Rm

∀t0


10/98

= f (t) −

mk=1

t́ t0

f k(u) du · f k(t)

t́ t0

f (u) du

= f (t) − t´ t0

f (u) du, f (t)t́ t0

f (u) du

f (t) −

t́ t0

f (u) du

· f (t)t́ t0

f (u) du

= 0

t = t0

y = f (y, t) y(t0) = y0

f B(y0, b) × [t0 − a, t0 + a] a,b > 0 M f (y, t) y(t) [t0 − α, t0 + α] α = min

a, b

M

B(y0, b)

∆t > 0

[u]∆t =

t0 + ∆tu−t0∆t

, u t0

t0 − ∆tt0−u∆t

, u < t0

[·]

[u]∆t

tt0t0 − ∆t

t0 + ∆tt0 − 2∆tt0 + 2∆t

t0 − ∆t

t0 + ∆t

[u]∆t


11/98

∀u∈R |u − [u]∆t| t0 t < t0

y(∆t)(t) = y0 +

t0+k∆tˆ t0

f

y(∆t)([u]∆t), [u]∆t

du +

tˆ

t0+k∆t

f

y(∆t)([u]∆t), [u]∆t

du

= y(∆t)(t0 + k∆t) +

tˆ

t0+k∆t

f

y(∆t)([u]∆t), [u]∆t

du

y(∆t)(t) y(∆t)(t0+k∆t)

u ∈ (t0+k∆t, t0+(k +1)∆t)

u

[u]∆t = t0 + k∆t

y(∆t)(t) = y(∆t)(t0 + k∆t) + (t − t0 − k∆t)f

y(∆t)(t0 + k∆t), t0 + k∆t

y(∆t)(t0 + k∆t) − y0 Mk∆t


12/98

|t − t0| α k∆t α Mk∆t Mα b y(∆t)(t0 + k∆t), t0 + k∆t

f y(∆t)(t)

y(∆t)(t) − y0 y(∆t)(t0 + k∆t) − y0 + |t − t0 − k∆t| · M k∆tM + |t − t0 − k∆t| · M = |t − t0| · M

y(∆t)(t0 + k∆t) − y0 t0 + k∆t < t t0 + (k + 1)∆t

y(∆t) M

y(∆t)(t1) − y(∆t)(t2) =

t1ˆ t2

f

y(∆t)([u]∆t), [u]∆t

du

t1ˆ

t2

M du

= M |t1 − t2|

y : [t0 − α, t0 + α] → B(y0, b)

Φ(y) = sup

y(t) − y0 −

tˆ t0

f (y(u), u) du

: t ∈ [t0 − α, t0 + α].

Φ(y) = 0 ⇔ y(t) = y0 +tˆ

t0

f (y(u), u) du

Φ

yn ⇒ y [t0 − α, t0 + α] ⇒ Φ(yn) → Φ(y).

f B(y0, b) × [t0 − α, t0 + α] δ f

∀ε>0 ∃N 0 ∀t∈[t0−α,t0+α] ∀nN 0 yn(t) − y(t) < η,


13/98

η > 0 δ f (η) < ε2α

ε > 0

yn(t) − y(t) < ε2

|Φ(yn) − Φ(y)|

=

suptyn(t) − y0 −

tˆ t0

f (yn(u), u) du

− supty(t) − y0 −

tˆ t0

f (y(u), u) du

supt

yn(t) − y0 −

tˆ t0

f (yn(u), u) du

−y(t) − y0 −

tˆ t0

f (y(u), u) du

supt

yn(t) − y(t) +tˆ

t0

(f (y(u), u) − f (yn(u), u)) du sup

tyn(t) − y(t) + sup

t

tˆ

t0

(f (y(u), u) − f (yn(u), u)) du<

ε

2 + sup

t

tˆ

t0

f (y(u), u) − f (yn(u), u) du

ε2

+t

tˆ

t0

δ f (η) du

< ε

2 + sup

t

tˆ

t0

ε

2α du

ε

2 +

ε

2α sup

t|t − t0| = ε

2 +

ε

2αα = ε

y(∆t)(t) lim∆t→0+

Φ

y(∆t)

= 0

u ∈ [t0 − α, t0 + α] y(∆t)([u]∆t, [u]∆t − y(∆t)(u), u = y(∆t)([u]∆t) − y(∆t)(u)2 + |[u]∆t − u|2

<√

M 2 + 1∆t

|[u]∆t

−u

|


14/98

tˆ t0

f y(∆t)([u]∆t), [u]∆t − f y(∆t)(u), u du

tˆ

t0

δ f √

M 2 + 1∆t

du

|t − t0| δ f

√ M 2 + 1∆t

δ f f

0 ∆t → 0

Φ

y(∆t)

= sup

y

(∆t)(t) − y0 −tˆ

t0

f

y(∆t)(u), u

du

: t ∈ [t0 − α, t0 + α]

|t − t0| δ f √

M 2 + 1∆t

αδ f √

M 2 + 1∆t ∆t→0−−−→ 0

F

∀ε>0 ∃δ>0 ∀f ∈F ∀x,y x − y < δ ⇒ f (x) − f (y) < ε.

y(∆t) δ f (x) = M x

(∆t)k → 0 y(∆t)k ⇒ y

Φ(y) = limk→∞

Φ y(∆t)k = 0


15/98

y(t)

R t ∈ [t0 − a, t0 + a] t0 ∈ R a > 0

∀t∈[t0−a,t0+a] y(t) K + L t

ˆ t0

y(u) du K, L 0

∀t∈[t0−a,t0+a] y(t) K exp (L |t − t0|)

L = 0

L > 0 K = 0 K = 1n

n ∈ N

∀n∈N y(t) 1n

exp L |t − t0| ⇒ y(t) inf

1

n exp L |t − t0|, n ∈ N

= 0.

K > 0

y(t)

K + L

t́ t0

y(u) du

1 ⇒ y(t)

K L

+

t́ t0

y(u) du

L.

K L

+

tˆ t0

y(u) du = v(t) dvdt =

y(t)

t t0 −y(t) t < t0

tˆ

t0

y(u)

K L

+

ú t0

y(w) dw

du

tˆ t0

L du

sgn(t − t0)t

ˆ t0

y(u)

v(u) dv L |t − t0|ln v(t)v(t0) L |t − t0|

ln

K L

+

t́ t0

y(u) du

K L

L |t − t0|


16/98

K L

+

t́ t0

y(u) du

K L

exp L |t − t0|

y(t) K + L

tˆ t0

y(u) du

K exp L |t − t0|,

y = f (y, t)

f : B(y0, b) × [t0 − a, t0 + a] → Rm y

∃L>0 ∀t∈[t0−a,t0+a] ∀y1,y2∈B(y0,b) f (y1, t) − f (y2, t) L y1 − y2 .

y1 y2 I ⊂ [t0 − a, t0 + a] yk(t0) = y0 k = 1, 2 t ∈ I

y1 y2 I k = 1, 2 t ∈ I

yk(t) = y0 +

tˆ t0

f (yk(u), u) du

y2(t) − y1(t) =

tˆ t0

(f (y2(u), u) − f (y1(u), u)) du

tˆ t0

f (y2(u), u) − f (y1(u), u) du

t

ˆ t0

L y2(u) − y1(u) du = L t

ˆ t0

y2(u) − y1(u) du . K = 0 L = L y(t) = y2(u) − y1(u) y1(t) = y2(t)


17/98

¨

dy

dt = f (y, t) y(t0) = y0

f : Rm+1

→ Rm

B(y0, b) × [t0 − a, t0 + a] = Q

a,b > 0 sup(y,t)∈Q

f (y, t) = M y Q

∃L>0 ∀t∈[t0−a,t0+a] ∀y1,y2∈B(y0,b) f (y1, t) − f (y2, t) L y1 − y2 .

|t − t0| α α = min

a, b

M , 1L

T : X → X (X, d) T

a ∈ X T (a), T (T (a)), . . . T

T c 0

n0 ∞

i=n0

d(ai, ai+1) < ε d(an, am) < ε

m, n n0 (an) (X, d)


18/98

an → p p ∈ X T T (an) → T ( p) an = T (an−1) → T ( p) p = T ( p)

T (q ) = q d( p, q ) = d(T ( p), T (q ))

c·

d( p, q )

d( p, q ) = 0 c


19/98

sup|t−t0|α

F (y1)(t) − F (y2)(t) = sup|t−t0|α

tˆ t0

f (y1(s), s) ds −tˆ

t0

f (y2(s), s) ds

= sup|t−t0|α

tˆ t0

f (y1(s), s) − f (y2(s), s) ds sup|t−t0|α

tˆ t0

L · y1(s) − y2(s) ds

sup|t−t0|α

L ·tˆ

t0

sup|s−t0|α

y1(s) − y2(s) ds sup|t−t0|α

L y1(t) − y2(t) sup|t−t0|α

tˆ t0

ds

Lα sup|t−t0|α

y1(t) − y2(t)

Lα < 1 F

F yn+1(t) = F (yn)(t)

y0(t) = y0 E

W ⊂ Rm × R1 × Rl f : W → Rm y W

Γ f y

∀(y1,t,λ), (y2,t,λ)∈Γ f (y1, t , λ) − f (y2, t , λ) LW y1 − y2 .

LW W

M M = sup {Mh : h = 1} M h Mh =

M · hh · h = h · M · hh h · M f : W

→Rn W

⊂Rm C 1

sup {Df (y) : y ∈ W } = L


20/98

W y(t) = y1 + th h = y2 − y1 y(0) = y1 y(1) = y2

d(f ◦ y)dt

= Df (y(t)) · dydt

= Df (y(t)) · h.

f (y2) − f (y1) = (f ◦ y)(1) − (f ◦ y)(0) =

1ˆ 0

df ◦ ydt

dt

1ˆ

0

df ◦ ydtdt

=

1ˆ 0

Df (y(t)) · h dt 1ˆ

0

L y2 − y1 dt = L · y2 − y1

f : W → Rm C 1 y y t λ

W f V

V ⊂ W V

Dyf W

f : U → Rm (y0, t0) ∈ U I t0 I yk(t0) = y0 k = 1, 2

a b B(y0, b) × [t0 − a, t0 + a] f y

y = f (y, t) f : U → Rm U ⊂ Rm×R U (y0, t0) ∈ U y1 y2 yk(t0) = y0 k = 1, 2 y1 ≡ y2


21/98

I

α = inf {t ∈ I : t t0 ∧ y1(t) = y2(t)}

α∈

I

y1(α) = y2(α) y1 y2

[t0, α)

(y1(α), α) α y1 ≡ y2 α y1 ≡ y2 α

t0

y(t)

a

(y(a), a) ∈ U α > 0 y : [a − α, a + α] → Rm y(a) = y(a)

y = y [a

−α, a] ỹ = y

∪y

y

(a, a+α) ỹ t1

y y

f y

U (y0, t0) ∈ U y(t) y(t0) = y0

(yα) α ∈ A yα(t0) = y0

y =α∈A

yα


22/98

yβ

I β I β β ∈

A

I β y

yγ γ ∈ A y ≡ yγ Dm y ⊃ Dm yγ yγ y ≡ yγ

K ⊂ U ⊂ Rm K U ε > 0

L ⊂ U ∀x∈K B(x, ε) ⊂ L

f : U → Rm U ⊂ Rm × R f y y(t)

(a, b)

Y (t) : (a, b) → U, Y (t) = (y(t), t).

K

⊂U Y −1(K )

K

0t Y −1(K ) Y −1(K ) (a, b)

Y a b

Y −1(K ) = (a, b)

Y −1(K ) {a} {b}

K

∃η>0 ∃α>0 ∀(y0,t0)∈K B(y0, η) × (t0 − α, t0 + α) ⊂ L ⊂ U

L f L M

y(t) y(t0) = y0

[t0−α, t0+α] α = min {α, ηM } t0 ∈ Y −1(K ) |a − t0| |b − t0|


23/98

α y ∪ y y

dy

dt = A(t)y + B(t)

I A(t) B(t)

y

y(t) = y0 +

tˆ t0

A(τ )y(τ ) dτ +

tˆ t0

B(τ ) dτ

y(t) y0 +

tˆ t0

A(τ )y(τ ) dτ +

tˆ

t0

B(τ ) dτ .

t

∈ I τ t t0

A(τ )

B(τ )

β = sup B(τ ) α = sup A(τ ) τ t t0

y(t) y0 + β |t − t0| + αtˆ

t0

y(τ ) dτ.

y(t)

y(t)

(

y0

+ β

|t−

t0|)eα|t−t0|

I

I


24/98

0y

(x, y)

x0 = x1 = 1 xn+2 = xn+1 + xn n 0

n > 1 xn > (1, 6)n−1

1 y = x2

x(t) 0y

g

v2/2 gy

x dxdt

x x(0) = 2

dxdt (t̂)

t̂ x(t̂) = −1

α > 0

α

x(t) r

c

dx

dt = rx − cxt.

t = 0 x(0) = 1

t̂ x(t̂)


25/98

h1 = 1500

h2 = 500 50 /

0xy y = c1 exp(c2x)

dx

dt − x ctg t = 0 x

π2

= 1

(0, π)

dydt = ry 0 < r 0

y = t(1 − cos

|y|)

y = P (y, t) P

R

y = y2

y+t y(0) = 2

0 y(10) 2e10

x

= y

y = tg x + 3√

t

x(0) = 0 y(0) = 1 [0, +∞) y(t) > 12 t 0

(x0, y0) ∈ R2 x

= x2 + y2 x(0) = x0

y = 2xy2 y(0) = y0

(0, x0, y0)


26/98

y = f (x, y)

{(x, y) : y > 0} f C 1 x

f : R → R [f (a) − f (b)](a − b) 0 a, b ∈ R y1 y2

y = f (y), y(t0) = y0,

t t0 y1(t) = y2(t)


27/98

dy

dt = f (y,t,λ)

f : G →Rm

G = U × B(λ0, c) ⊂Rm+1

×Rl

.

(y0, t0) ∈ U y(t, λ) y(t0, λ) = y0 λ

[b, a]

t0 ∈ [b, a] y(t, λ0) [b, a]

dy

dt = f (y,t,λ)

f : T D × B(λ0, c) →Rm

T D × B(λ0, c) ⊂Rm+1

×R

c,D > 0

T D = {(y, t) : t ∈ [b, a], y − y(t, λ0) D}

y(t, λ0) λ = λ0

y(t0, λ0) = y0


28/98

∀λ∈B(λ0,c) f (y,t,λ) (y, t) y

∃L>0 ∀(y1,t),(y2,t)∈T f (y1, t , λ0) − f (y2, t , λ0) L y1 − y2

∀ε>0

∃δ>0

∀(y,t)

∈T

λ

−λ0

< δ

⇒ f (y,t,λ)

−f (y,t,λ0)

< ε

c1 > 0 λ λ − λ0 < c1 y(t, λ) (y0, t0) [b, a]

λn → λ0 y(t, λn)⇒ y(t, λ0) [b, a]

α(λ) λ ∈ B(λ0, c)

α(λ) = inf {t ∈ [t0, a] : y(t, λ) − y(t, λ0) D}

α(λ) = a

y(t, λ) [t0, α(λ)]

y(t, λ) β ∈ (t0, α(λ)) (y(t, λ), t) ∈ T t β α(λ) < β < α(λ)

λ ∈ B(λ0, c)

∀(y,t)∈T f (y,t,λ0) − f (y,t,λ) < ε

∀t∈[t0,α(λ)] y(t, λ) − y(t, λ0) ε · (a − t0)eL(a−t0).

y(t, λ) − y(t, λ0) =

tˆ t0

[f (y(u, λ),u ,λ) − f (y(u, λ0),u ,λ0)] du

tˆ

t0

f (y(u, λ),u ,λ) − f (y(u, λ),u ,λ0) du

+

tˆ t0

f (y(u, λ),u ,λ0) − f (y(u, λ0), u , λ0) du

ε(t − t0) + Ltˆ

t0

y(u, λ) − y(u, λ0) du ε(a − t0) + Ltˆ

t0

y(u, λ) − y(u, λ0) du.


29/98

ε ε[(a − t0)eL(a−t0)] D2 c1 > 0 λ − λ0 < c1 ∀(y,t)∈T f (y,t,λ) − f (y,t,λ0) < ε

∀λ∈B(λ0,c) ∀t∈[t0,α(λ)] y(t, λ) − y(t, λ0) D

2 ,

α(λ) α(λ) = a y(t, λ)

t ∈ [t0, a] λn → λ0 εn → 0 n ∈ N

∀(y,t)∈T f (y,t,λ0) − f (y,t,λn) < εn.

supt∈[t0,a]

y(t, λ0) − y(t, λn) εn[(a − t0)eL(a−t0)] → 0.

b = t0

b < t0 t0 = a

f C 1 y

f : U → Rm dydt

= f (y, t) f C 1 y

(y0, t0) ∈ U (ỹ, t̃) ∈ B((y0, t0), c) ⊂ U y(t, ỹ, t̃) y(t, y0, t0) [b, a]

(ỹ, t̃) (y0, t0) [b, a]

y(t, ỹ, t̃)⇒ y(t, y0, t0) (ỹ, t̃)

→(y0, t0)

λ = (ỹ, t̃) F (Y , T , λ) = f (Y + ỹ, T + t̃)

F [U − (y0, t0)] × B((y0, t0), c) c = dist {(y0, t0), ∂U } U −(y0, t0) U (y0, t0) F C 1 Y

dY (T, λ)

dT = F (Y , T , λ), Y (0, λ) = 0


30/98

Y (t, λ)

y(t, ỹ, t̃) = Y (t − t̃, λ) + ỹ

y(t̃, ỹ, t̃) = Y (t̃ − t̃, λ) + ỹ = Y (0, λ) + ỹ = ỹ.

y(t, ỹ, t̃)

dy(t, ỹ, t̃)

dt =

dY (t − t̃, λ)dt

.

T = t − t̃ dY (T, λ)

dt =

dY (T, λ)

dT ·

dT

dt = F (Y , T , λ) = f (Y + ỹ, T + t̃)

= f (Y (T, λ) + ỹ, T + t̃) = f (y(t, ỹ, t̃), t).

λ

dy

dt = λ2 + y2, y(0) = 1, λ0 = 0.

(−∞, 1) y(t, 0) = 11−t λ > 0 ∃α λ = α2

yˆ 1

1

α2 + u2 du = t.

y = α · tgtα + arctg 1α .

limα→0 α · tg

tα + arctg

1

α

=

1

1 − t


31/98

t 0

[0, ∞) ∩ Dm y(t, λ) =

t 0 : tα + arctg 1α

< π

2

.

t∗ t∗α + arctg 1α

= π2

tg(t∗α) · tg

arctg 1

α

= 1

tg(t∗α) = α

t∗ = arctg α

α K > −∞ [−K, t∗]

λ0 ∈ (λ0 − c, λ0 + c) c > 0 dy

dt = f (y,t,λ) f : U × (λ0 − c, λ0 + c) → Rm, U ⊂ Rm+1,

f C 1 y λ

y(t, λ0) (y0, t0) ∈ U I

λ λ0 y(t, λ)

I

t ∈ I

z (t, λ0) = ∂y(t, λ)

∂λ

λ=λ0

t λ0

∂z

∂t(t, λ0) = Dyf (y,t,λ0)

y=y(t,λ0)

· z (t, λ0) + ∂ f ∂λ

(y(t, λ0), t , λ)λ=λ0

z (t0, λ0) = 0


32/98

ε0 > 0

T ε0 = {(y, t) : t ∈ I, y − y(t, λ0) ε0}

U η > 0

|λ

−λ0

|< η

y(t, λ) t ∈ I

y(t, λ), t ∈ T ε0

t ∈ I

w(t, λ) = y(t, λ) − y(t, λ0)

λ − λ0 0 < |λ − λ0| < η

w(t, λ) =

t

´ t0 f

y(τ, λ), τ , λ

− f

y(τ, λ0), τ , λ0

dτ

λ − λ0

=

tˆ t0

f

y(τ, λ), τ , λ

− f

y(τ, λ), τ , λ0

λ − λ0 dτ

+

tˆ t0

f

y(τ, λ), τ , λ0

− f

y(τ, λ0), τ , λ0

λ − λ0 dτ

g1(w,τ ,λ) = f y(τ, λ0) + (λ − λ0)w,τ ,λ− f y(τ, λ0) + (λ − λ0)w,τ ,λ0

λ − λ0

g2(w,τ ,λ) =f

y(τ, λ0) + (λ − λ0)w,τ ,λ

− f

y(τ, λ0), τ , λ0

λ − λ0− Dyf (y,τ ,λ0)

y=y(τ,λ0)

· w.

w < ε0λ−λ0

w(t, λ) =

tˆ t0

g1(w(τ, λ), τ , λ) dτ +

tˆ t0

g2(w(τ, λ), τ , λ) dτ

+

tˆ t0

Dyf (y,τ ,λ0)y=y(τ,λ0)

w(τ, λ) dτ


33/98

g1 λ = λ0

g1(w,τ ,λ0) = ∂f

∂λ(y(τ, λ0), τ , λ)

λ=λ0

.

w

∀R,ε>0 ∃δ1(ε,R)>0 ∀τ ∈I ∀0


34/98

|λ − λ0| < δ0(ε)√ R2+1 j

∂f

∂λ(y(τ, λ0) + (λ − λ0)w,τ, λ̄)|λ=λj

[ j] −

∂f

∂λ(y(τ, λ0) + (λ − λ0)w,τ, λ̄)|λ=λ0

[ j]

<

ε

√ m,

g1(w,τ ,λ) − g1(w,τ ,λ0) < ε.

δ 1(ε, R) = minε0R

, δ0(ε)√ R2+1

g2(w,τ ,λ)

∀R,ε>0 ∃δ2(ε,R)>0 ∀τ ∈I ∀0


35/98

g2(w,τ ,λ) = g2(w,τ ,λ) − g2(0, τ , λ) sup {Dwg2(w,τ ,λ) : w R} · w < ε.

δ 2(ε, R) = minε0R

, δ 0(ε, R)

η1 = min {δ 1(1, R), δ 2(1, R)}

∃M 0


36/98

∀t∈I λ w(t, λ) (M + 2) |I | exp{L|I |}.

I λ

= I I λ

w(t, λ) I λ = I M 0 = R η1 = δ min

w(t) := w(t, λ)

dw

dt = g1(w,t,λ) + g2(w,t,λ) + Dyf (y,t,λ0)

y=y(t,λ0)

· w

w(t0) = 0 g1 λ = λ0

g2 g(w,t,λ0)≡

0

λ = λ0 g1 g2 f w Dyf (y,t,λ0)

y=y(t,λ0)

C 1 f

λ = λ0 g1 g2 w g1 t

C 1 f

w λ = λ0 g1

g2 w L

λ → λ0 λ g1 g2

δ 1(ε, R) δ 2(ε, R)

R = M 0 |λ − λ0| < η1

δ = min

η1, δ 1

ε

2, M 0

, δ 2

ε

2, M 0

.

w(t, λ) I

λ → λ0 dw

dt = g1(w,t,λ0) + Dy(y,t,λ0)

y=y(t,λ0)

· w


37/98

w(t0) = 0 I

I

limλ→λ0 w(t, λ) = ∂y(t, λ)∂λ ,

z (t, λ0)

z (t, λ0) λ0

λ0

y(t, λ0) λ0

λ→

λ0 z (t, λ)⇒ z (t, λ0)

t z (t, λ0) (t, λ0)

λ λ = (λ(1), . . . , λ())

∂y(t,λ)

∂λ(j) |λ=λ0 j = 1, . . . ,

Dλy(t, λ)|λ=λ0

∂

∂tDλy(t, λ)|λ=λ0 = Dyf (y,t,λ0)|y=y(t,λ0)Dλy(t, λ)|λ=λ0 + Dλf (y(t, λ0), t , λ)|λ=λ0.

λ0 = (ỹ0, t̃0)

λ = (y0, t0) T = t − t0 Y = y − y0 dY

dT = F (Y , T , λ) = f (Y + y0, T + t0), Y (0) = 0

y(t, λ) = Y (t − t0, λ) + y0 y(t0, (y0, t0)) = y0

y0 t

∂y(t, (y0, t0))

∂y0|λ=λ0 =

∂

∂y0[Y (t − t0, (y0, t0)) + y0]|λ=λ0 =

∂

∂y0Y (t − t0, (y0, t0))|λ=λ0 + I.


38/98

z (t, λ0) z

∂z (t, λ0)

∂t = DY F (Y , T , λ)|Y =Y (t−t̃0,λ0) · z (t, λ0) +

∂

∂λF (Y , T , λ)|λ=λ0

= DY f (Y + y0, T + t0)|Y =y(t,λ0)−ỹ0 · z (t, λ0) + ∂ ∂λ f (Y + y0, T + t0)|λ=λ0= Dyf (y, T + t̃0)|y=y(t,λ0) · z (t, λ0) +

∂

∂λf (y, t)|λ=λ0.

z (t, λ0) = ∂y(t, λ)

∂y0|λ=λ0 − I

∂ ∂t∂y(t, λ)∂y0 |λ=λ0 − I = ∂ ∂t ∂y(t, λ)∂y0 |λ=λ0

= Dyf (y, t)|y=y(t,λ0)

∂y(t, λ)

∂y0|λ=λ0 − I

+ Dyf (y, t)|y=y(t,λ0)

= Dyf (y, t)|y=y(t,λ0) · ∂ y(t, λ)

∂y0|λ=λ0

∂y(t,λ)∂y0

|λ=λ0 = I z (t0, λ0) = 0

∂

∂t

∂y(t, (y0, t0))

∂t0 |λ=λ0 = Dyf (y, t)

|y=y(t,λ0)

∂y(t̃0,(y0,t0))∂t0

|λ=λ0 = −f (ỹ0, t̃0)

∂x∂µ

µ=0

x = −x2t + 2µ x(1) = 1 x = x + t sin(xt) x(1) = µ x = x + sin(xt) x(−1) = µ x = 2t + µx2 x(0) = µ − 1

x − x = (x + 1)2 − (µ + 1)x2 x(0) = 12

x(0) = −1

y = −1t − 1

t2 + ty2 ∂y

∂t0

t0=1

y(t0) = 1


39/98

y = x3 + 2

xy − y

2

x

(1, µ) µ = −1

dxdt = x + µy

2

dydt = x + y

x(0) = 1 y(0) = 0 µ = 0


40/98


41/98

f C 1 G dydt

= f (y)

ϕt(x) t ∈ R G

ϕt(x) = y(t, (x, 0)).

y(t, (x, 0))

y(0) = x t ∈ R y(t)

ϕt ◦ ϕs = ϕt+s.

dϕt

dt =

dy(t, (x, 0))

dt = f (y(t, (x, 0))) = f (ϕt(x))

dϕt ◦ ϕsdt

= f (ϕt ◦ ϕs)dϕt+s

dt =

d(t + s)

dt

dϕt+s

d(t + s) = 1 · f (ϕt+s)


42/98

ϕt ◦ ϕs|t=0 = ϕ0 ◦ ϕs = ϕs

ϕt+s|t=0 = ϕs.

ϕt

◦ϕs ϕt+s dy/dt = f (y)

t ϕt(x)

C 1 x Dxϕt(x)

d

dtDxϕ

t(x) = Dyf (y)|y=ϕt(x) · Dxϕt(x)

Dxϕ0(x) = I

y = y2 + 1

y(0) = x

arctg y(t)

−arctg x = t

y(t) = tg(t + arctg x) = ϕt(x).

ϕt ◦ ϕs = tg(s + arctg ϕtx) = tg(s + arctg (tg(t + arctg x))) = tg(s + t + arctg x)

ϕt+s


43/98

y = 1 + y + y2

y = ay + by

y

y

= φt

y0

y0

= exp

t 0 1

b a

y0

y0

R2

y =

0 1

−1 0

y +

0

−1

y = f (y) Rm divf = 0

m φt

E Vol(φt(E )) = Vol(E )

φt(x) = tg(t + x)?

y = (1 + y) sin(sin y)

ddx

φt(x)|x=0


44/98


45/98

dy

dt = A(t) · y + B(t)

Rm

A(t) m × m I B(t) m

I y(t) m

an(t)y(n) + an−1y(n−1) + . . . + a1y + a0y = b(t)

n R ai(t) an(t) ≡ 0

dy

dt = A(t)y + B(t)

I A(t) B(t)

V y = Ay V B

y = Ay + B


46/98

V C (I )

V B V C (I ) V + v v ∈ C (I ) y1 ∈ V B y2 ∈ V B y1 − y2 ∈ V

y1, y2 ∈ V y = y1 + y2 dy

dt =

d(y1 + y2)

dt =

dy1dt

+ dy2

dt = Ay1 + Ay2 = A(y1 + y2) = Ay

day

dt = a

dy

dt = aAy = A(ay)

V

C (I )

y1

∈ V B

dydt

= Ay1 + B

y2 − y1 ∈ V dy2dt

= d(y2 − y1)

dt +

dy1dt

= A(y2 − y1) + Ay1 + B = Ay2 + B

y2 ∈ V B y2 ∈ V B

d(y2 − y1)dt

= dy2

dt − dy1

dt = Ay2 + B − Ay1 − B = A(y2 − y1)

y2 − y1 ∈ V

{y1, . . . , yk}

{y1, . . . , yk} V t ∈ I {y1(t), . . . , yk(t)} Rm t ∈ I {y1(t), . . . , yk(t)} Rm

⇒ {y1, . . . , yk}

k j=1

α jy j ≡ 0 ⇒ α1, . . . , αk = 0.


47/98

t0 ∈ I (α1, . . . , αk) = 0 k

j=1

α jy j(t0) = 0.

y(t) =k

j=1α jy j

I

y(t0) = 0 y(t) = 0

y(t) ≡ 0 ⇔ k j=1

α jy j ≡ 0 (α1, . . . , αk) = 0

⇒ ⇒ (α1, . . . , αk) = 0

k j=1

α jy j ≡ 0.

∀t∈I k

j=1α jy j(t) = 0

V

{y1, . . . , yk

} ⊂V

{y1, . . . , yk} k = m

⇒ k m Rm k < m t0 ∈ I {y1(t0), . . . , yk(t0)} x ∈ Rm {y1(t0), . . . , yk(t0), x(t0)}

yk+1

yk+1(t0) = x(t0)

{y1, . . . , yk+1} {y1, . . . , yk}

⇐ {y1, . . . , ym} ym+1 {y1, . . . , ym+1}


48/98

t ∈ I {y1(t), . . . , ym+1(t)} Rm

dim V = m

I

M (t)

M (t) ∈ M m×m

M (t) I C 1

t0

∈I M (t0)

dM dt

= AM

⇒ M

⇐ dM dt

= AM M

m

M

P m

M (t)P

d(M (t)P )

dt =

dM (t)

dt · P = AM (t)P = A(M (t)P ).

U m×m M (t)U U M (t)

dU M (t)

dt = U

dM (t)

dt = U AM (t) = A(U M (t))

M (t, t0) = M (t)M (t0)−1

M (t0, t0) = I


49/98

M (t) y0 Rm

M (t, t0)y0

y(t0) = y0

y(t) = M (t, t0)y0

y(t) = M (t)C C ∈ Rm

P = y0

M (t0, t0) = I

C = M (t0)−1y0

y1, . . . , ym ∈ Rm

W (y1, . . . , ym)(t) = det(y1(t), . . . , ym(t)).

y1, . . . , ym

C m−1

W (y1, . . . , ym)(t) = det

y1(t) . . . ym(t)

y1(t) . . . ym(t)

y(m−1)1 (t) . . . y

(m−1)m (t)

.

W (t)

y1, . . . , ym

∃t0∈I W (y1, . . . , ym)(t0) = 0 ∀t∈I W (y1, . . . , ym)(t) = 0


50/98

y1(t), . . . , ym(t)

dydt

= Ay

dW (t)

dt = trA(t) · W (t).

M = [xij]

det M =m

j=1

(−1)i0+ jxi0 jM i0 j =mi=1

(−1)i+ j0xij0M ij0

i0, j0 = 1, . . . , m M ij

i j M

∂ det M

∂xij= (−1)i+ jM ij.

M (t) y1(t), . . . , ym(t)

xij i y j A = [aij]

dW

dt =

d

dt det M (t) =

mi=1

m j=1

∂ det M (t)

∂xij

dxijdt

=m

i,j=1(−1)i+ jM ij(t)

·

m

k=1 aik(t)xkj =m

i,k=1m

j=1(−1)i+ jaik(t)M ij(t)xkj(t)

=m

i,k=1

aik(t)m

j=1

(−1)i+ jM ij(t)xkj(t) = . . . .

j det M (t) i

k

. . . =

m

i,k=1 aik ·

W (t) k = i

0 k = i =m

i=1 aii(t)W (t) = trA(t) · W (t).


51/98

m

y(m) + am−1(t)y(m−1) + . . . + a0(t)y = 0,

dW

dt = −am−1(t)W (t).

Y (t)

Y (t) =

y

y

y(m−2)

y(m−1)

dY (t)dt =

y

y

y(m−1)

−am−1(t)y(m−1) − . . . − a0(t)y

.

A(t)

0 1 0 0 . . . 0

0 0 1 0 . . . 0

0 0 0 . . . 0 1

−a0 −a1 −a2 . . . −am−2 −am−1

−am−1

W (t0)

W (t) = W (t0) · expt

ˆ t0

trA du

W (t) = W (t0)exp

tˆ t0

−am−1(u) du.


52/98

y(t) = A(t)y(t) + B(t)

M (t, u) M (u, u) = I

y(t) =

tˆ t0

M (t, u)B(u) du

dy

dt =

d

t

´ t0

M (t, u)B(u) du

dt = M (t, t)B(t) +

t

ˆ t0

d

dtM (t, u)B(u) du

= B(t) +

tˆ t0

A(t)M (t, u)B(u) du

= B(t) + A(t)

tˆ t0

M (t, u)B(u) du = B(t) + A(t)y

A

σ(A) Sp(A)

0 (f )

f (z ) =∞

k=0akz

k.

f (z ) B(0, (f )) (f ) = ∞


53/98

A

λ1, . . . , λm q 1, . . . , q m

M k, 1 k m 0 q k − 1 f (z )

f (A) =mk=1

qk−1=0

M k, · df

dz (λk)

M k,

f (z ) = z r r = 0, . . . , m − 1

M k,l

Ar =mk=1

qk−1=1

M k, · r · . . . · (r − + 1)λr−k

cr

∀k,m−1r=0 r · . . . · (r − + 1)λ

r

−

k cr = 0

w(z ) =m−1r=0

crz r

∀k, w()(λk) = 0 λk w q k

w mk=1

q k = m w

m − 1

m

χA(A) = 0

χA(z ) A

χA(z ) = z m − wA(z ) deg wA < m


54/98

r 0 z m+r z rwA(z )

z = A

z = λk 1

k

n

z = λk < q k

z rχA(z )

d

dz (z rχA(z )) =

s=0

s

(z r)(−s)(χA(z ))(s)

χ(s)A (λk) = 0 s λk q k − 1 s < q k

r 0 deg f < m + r r = 0 f = z m+r

f (A) = ArwA(A) deg wA m − 1

z rwA(z ) m + r

f (A) = ArwA(A) =mk=1

qk−1=0

M k,dl

dz (z rwA(z ))|z=λk =

mk=1

qk−1=0

M k,f ()(λk)

f

m+r

f m + r

f


55/98

f (z ) f (z ) =∞n=0

cn(z − z 0)n

f (A) = limN →∞

N n=0

cn(A − z 0E )n

z 0 = 0

A λ1, . . . , λm q 1, . . . , q m

M k, 1 k m 0 q k − 1

f A

f (A) =mk=1

qk−1=0

M k, · df

dz (λk).

f (A) = limN →∞

N n=0

cnAn = lim

N →∞

N n=0

mk=1

qk−1=0

M k, · cn · n · . . . · (n − + 1)λn−k

=mk=1

qk−1=0

M k, · limN →∞

N n=0

cn · n · . . . · (n − + 1)λn−k =mk=1

qk−1=0

M k,f ()(λk)

λk A

E k := {v ∈ Rn : ∃q0 (A − λkE )qv = 0}

= 1


56/98

Rm

Rm =

nk=1

E k

n

M k,

M k,

∀k Im M k, ⊂ E k k M k,0 E k

v =n

k=1

, vk ∈ E k ⇒ M k,0v = vk,

k = k 0 q k − 1 E k ⊂ ker M k,

dydt

= Ay A exp(At)

exp(0) = I

ddt

exp(At) = A exp(At)

f (z ) = exp(zt)

exp(At) =

nk=1

qk−1=0

M k,dezt

dz |z=λk =n

k=1

qk−1=0

M k, · t · etλk .

d

dt exp(At) =

nk=1

qk−1=0

M k,d

dt(t exp(tλk)) =

nk=1

qk−1=0

M k,(t + λkt

)exp(tλk).

g(z ) = z exp(tz )

A exp(At) = g(A) =n

k=1

qk−1=0

M k,dg(z )

dz |z=λk

dg

dz = zt exp(tz ) + t−1 exp(tz )

nk=1

qk−1=0

M k,dg(z )

dz |z=λk =

nk=1

qk−1=0

M k,(λkt + t−1)exp(tλk) =

d

dt exp(At).


57/98

exp{At}

det

eA

= etr(A)

det

eA

= det (exp A · 1) = exp

det1 ·1ˆ

0

tr(A) du

= exp(tr(A)).

ψi = z i

i = 0, . . . ,n

j=1q j − 1

n

A p =n

k=1

qk−1=0

M k,dz p

dz (λk) =

nk=1

qk−1=0

M k, · p · ( p − 1) · . . . · ( p − l + 1)λ p−k .

c p k = 0 m−1 p=0

c pλ pk = 0

λk w(z ) =m−1 p=0

c pz p

0 =

m−1

p=0 c p p( p − 1) . . . ( p − + 1)λ

p−k =

m−1

p=0 c p p( p − 1) . . . ( p − + 1)w

()

(λk)

w q k

m − 1

eAt

λ̄(A) = max{Re λk : λk ∈ Sp(A)}.

̄(A) = max{ : 0 q k − 1, Re λk = λ̄(A), M k, = 0}.


58/98

exp{At}

A = sup{Ax : x 1} A = [a jk ] max{|a jk| } A m max{|a jk|}

A λ̄(A) = λ̄ ̄(A) = ̄

0 0.


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a(t)y + b(t)y + c(t)y = 0

a,b,c

x(t) =∞n=0

xn(t − t0)n

x xn

xn = x(n)(t0)

n!

t0 a(t0) = 0

y + p(t)y + q (t)y = 0

p(t) = b(t)a(t)

q (t) = c(t)a(t)

t0

(t − t0)2y + (t − t0) p(t)y + q (t)y = 0.

t0

sin ty + cos ty + tg ty = 0.

t0 = 0

tsin t

t y + cos ty + tg ty = 0


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t2

sin t

t2y + t cos t t

sin ty + t tg t

t

sin ty = 0.

t0 = 0

t−t0 = z t0 = 0 y(t) y

y =∞k=0

cktk.

y

t

y(t) =∞k=1

kcktk−1 =

∞k=0

(k + 1)ck+1tk,

y(t) =∞k=2

k(k − 1)cktk−2 =∞k=0

(k + 1)(k + 2)ck+2tk.

∞

k=0 [ck+2(k + 2)(k + 1) + p(t)ck+1(k + 1) + q (t)ck] = 0. p(t) =

∞m=0

pmtm

q (t) =∞

m=0q mt

m

∞k=0

p(t)ck+1(k + 1)tk =

∞k=0

∞m=0

pmtmck+1(k + 1)t

k = . . .

= m + k m = − k

=∞

=0t

k=0ck+1 p−k(k + 1).

∞k=0

q (t)cktk =

∞=0

k=0

tckq −k.

∞=0

c+2( + 2)( + 1) +

k=0

ck+1(k + 1) p−k +

k=0

ckq −k

t ≡ 0


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c+2( + 2)( + 1) +

k=0

ck+1(k + 1) p−k +

k=0

ckq −k = 0.

ck

+ 2

c+2

+ 1

c+2 = −1

( + 1)( + 2)

k=0

ck+1(k + 1) p−k +

k=0

ckq −k

.

c0 = y(0) c1 = y(0)

y1 : c10 = 1, c

11 = 0,

y2 : c20 = 0, c

21 = 1,

y c0, c1

y

y + y = 0 p(t) = 0 q (t) = 1

c+2 = −1

( + 1)( + 2)c.

c0 = 1, c1 = 0 ⇒ c2k+1 = 0 c2 = −11·2 c4 = 11·2·3·4 cos t

c0 = 0, c1 = 0 ⇒ c2k = 0 c1 = 1 c3 = �

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