Ch. 4 .

Ch. 4 . (Systems of ODEs. Phase Plane.Qualitative Methods)z :

1

Ch. 4 .

4.0

4.0 (Basics of Matrices and Vectors)z ( , Systems of Differential Equations) : Ex.

y1 ' = a11 y1 + a12 y2 , y2 ' = a21 y1 + a22 y2 ,

y2 ' = a21 y1 + a22 y2 + L + a2 n yn , M

y1 ' = a11 y1 + a12 y2 + L + a1n yn ,

yn ' = an1 y1 + an 2 y2 + L + ann yn ,

z : ( ) ( ) . Ex.

y (t ) y (t ) = 1 y2 (t ) y1 ' = a11 y1 + a12 y2 ,

y ' (t ) y ' (t ) = 1 y2 ' (t ) a y y ' a y ' = 1 = Ay = 11 12 1 y2 ' a21 a22 y2

Ex.

y2 ' = a21 y1 + a22 y2 ,

2

Ch. 4 .

4.0

z (Eigenvalue), (Eigenvector)

A = a jk n n , x 0 Ax = x (Eigenvalue) , x (Eigenvector) .

[ ]

x = 0 Ax = x . Ax = x

Ax x = 0

(A I )x = 0

n x1 ,L, xn ( x ) 1 Ax = x x 0 A I

.

det (A I ) = 0

3

Ch. 4 .

4.0

Ex.

a A = 11 a21

a12 a22 a11 a21 a12 a22 = (a11 )(a22 ) a12 a21 = 2 (a11 + a22 ) + a11a22 a12 a21

det (A I ) =

z (Characteristic Equation) : 2 (a11 + a22 ) + a11a22 a12 a21 = 0

A : x A k 0 kx

4

Ch. 4 .

4.1

4.1 (Systems of ODEs as Models)Ex. 1 2 ( 1 ) T1 : 100 T2 : 100 + 150 2 .

(1 gallon = 3.785 liter, 1 lb = 0.45 kg)

y1 : T1 , ? Step 1

y2 : T2

T1 T2

y1 ' = - = y2 ' = - =

2 2 y2 y1 100 100 2 2 y1 y2 100 100

( T1 ) ( T2 )

y1 ' = 0.02 y1 + 0.02 y2 y2 ' = 0.02 y1 0.02 y2

y ' = Ay,

0.02 0.02 A= 0.02 0.02

5

Ch. 4 .

4.1

Step 2 Idea : t

y = xet :

y ' = xet = Axet det (A I ) =

Ax = x

A

0.02 0.02

0.02 2 = ( 0.02 ) 0.022 = ( + 0.04 ) = 0 0.02 (1) 1 (2 ) 1 : x = , x = 1 1

: 1 = 0, 2 = 0.04

1 1 y = c1x(1)e1t + c2 x(2 )e2 t = c1 + c2 e 0.04t 1 1Step 3 : y1 (0 ) = 0,

(c1 c1 )

y2 (0 ) = 150

1 1 c + c 0 1 1 y (0 ) = c1 + c2 = 1 2 = c1 = 75, c2 = 75 y = 75 75 e 0.04t 1 1 c1 c2 150 1 1

Step 4 T1 50 , T1 T2 y1 = 75 75e 0.04t = 50 e 0.04t = 1 3 t = ln 3 0.04 = 27.5

6

Ch. 4 .

4.1

Ex. 2 I1 (t ) I 2 (t ) . t = 0 0 .

Step 1 : Kirchhoff : I1 ' = 4 I1 + 4 I 2 + 12 : 6 I 2 + 4(I 2 I1 ) + 4 I 2 dt = 0 I 2 '0.4 I1 '+0.4 I 2 = 0 I 2 ' = 1.6 I1 + 1.2 I 2 + 4.8 J ' = AJ + g, I 4.0 4.0 12.0 , J = 1 , A = g = 4.8 1.6 1.2 I2

7

Ch. 4 .

4.1

Step 2 J ' = AJ J = xet Ax = x A (1) 2 1 = 2 , x = ; 1 = 0.8 , 1

1 x (2 ) = 0.8

:

2 1 J h = c1 e 2t + c2 e 0.8t 1 0.8

0 a J p ' = J p = 1 , 0 a2

a A 1 + g = 0 a2 a1 = 3, a2 = 0 3 Jp = 0

4.0a1 + 4.0a2 + 12.0 = 0 1.6a1 + 1.2a2 + 4.8 = 0

:

2 1 3 J = c1 e 2t + c2 e 0.8t + 1 0.8 0

I1 = 2c1e 2t + c2e 0.8t + 3 I 2 = c1e 2t + 0.8c2e 0.8t

8

Ch. 4 .

4.1

I1 (0 ) = 2c1 + c2 + 3 = 0 I 2 (0 ) = c1 + 0.8c2 = 0

c1 = 4, c2 = 5

I1 = 8e 2t + 5e 0.8t + 3 I 2 = 4e 2t + 4e 0.8t

79a I1 (t ) I 2 (t ) 79b I1I 2

[I1 (t ), I 2 (t )]

t (parameter)

I1I 2 plane)

(A I )x = 0 (, phase

(Trajectory) .

9

Ch. 4 .

4.1

z n 1 n : y (n ) = F t , y, y' ,L, y (n1) 1 y1 ' = y2 y1 = y, y2 = y ' , y3 = y ' ' ,L, yn = y(n 1)

(

)

yn 1 ' = yn yn ' = F (t , y1, y2 ,L, yn )

y2 ' = y3 M

Ex. 3

y1 = y, y2 = y '

my ' '+ cy '+ ky = 0 det (A I ) = k m

y1 ' = y2 k c y2 ' = y1 y2 m m

y y = 1 y2

0 y ' = Ay = k m

1 y c 1 y m 2

1 c k c = 2 + + = 0 m m m

2.4

10

Ch. 4 .

4.2

4.2 (Basic Theory of Systems of ODEs)

z y2 ' = f 2 (t , y1,L, yn ) yn ' = f n (t , y1,L, yn )z (Solution Vector)

y1 ' = f1 (t , y1,L, yn ) M

y1 f1 y = M , f = M yn fn

y ' = f (t , y )

: z

a < t < b n

y1 = h1 (t ), L, y1 (t0 ) = K1 ,

yn = hn (t ) . yn (t0 ) = K n K1 y (t0 ) = K = M Kn

y2 (t0 ) = K 2 ,

L,

11

Ch. 4 .

4.2

z f1 , L, f n (t0 , K1 , L, K n ) f1 , L, f1 , L, fn R , y1 yn yn . t0 < t < t0 + .

12

Ch. 4 .

4.2

z (Linear System)

y1 ' = a11(t ) y1 + L + a1n (t ) yn + g1 (t ) M yn ' = an1 (t ) y1 + L + ann (t ) yn + gn (t )z : y ' = Ay z : y ' = Ay + g, g 0

a11 L a1n y1 g1 A = M O M , y = M , g = M an1 L ann yn g2

y ' = Ay + g

z

a jk g j t = t0 < t <

t

.

.z

y (1) y (2 ) , y = c1y (1) + c2 y (2 ) .

13

Ch. 4 .

4.2

z (Basis) (Fundamental System)(1) (n ) : J n y , L, y

z (General Solution)(1) (n ) : y = c1y + L + cn y

(c1, L,

cn

)

y ' = Ay a jk (t ) J , y ' = Ay

. y ' = Ay J , . z (Fundamental Matrix) : n y (1) , L, y (n ) n n z Wronskian :

14

Ch. 4 .

4.3 .

4.3 . (Constant-Coefficient Systems. Phase Plane Method)

z : y ' = Ay, A = [a jk ], a jk

Idea

y = xet

y ' = xet = Ay = Axetz

Ax = x ( )

n y (1) = x(1)e t , L, y (n ) = x(n )e t y (1) , L, y (n ) 1 n

,

y = c1x(1)e t + L + cn x (n )e t .1 n

15

Ch. 4 .

4.3 .

z y1 ' = a11 y1 + a12 y2 y1 (t ) y ' = Ay y = y (t ) y2 ' = a21 y1 + a22 y2 2 y1 (t ) y2 (t ) t . t (, ) : y1 y 2 .z

(Trajectory, (Orbit), (Path) : y1 y 2 (Phase Plane) : y1 y 2 (Phase Portrait) : y ' = Ay

16

Ch. 4 .

4.3 .

Ex. 1 () y1 ' = 3 y1 + y2 y2 ' = y1 3 y2

3 1 y ' = Ay = y 1 3

det (A I ) =

3 1

1 = 2 + 6 + 8 = 0 3 1 x (1) = ; 1 1 x (2 ) = 1

1 = 2 , 2 = 4 ,

:

y 1 1 y = 1 = c1y (1) + c2 y (2 ) = c1 e 2t + c2 e 4t 1 1 y2 c1 = 0 c2 = 0

c1 , c2

17

Ch. 4 .

4.3 .

z (, Critical Point) :Ex.dy2 dy2 dt = y2 ' = a21 y1 + a22 y2 = y1 ' a11 y1 + a12 y2 dy1 dy1 dt

dy2

dy1

P = P0 : (0,0 ) P : ( y1 , y2 ) dy2 dy1 . dy2 0 . P0 dy1 0z

: 5 . (Improper Node) (Proper Node) (Saddle Point) (Center) (Spiral Point)

18

Ch. 4 .

4.3 .

Ex. 1 : .

.

3 1 y ' = Ay = y 1 3(1) 1 x = . 1

( t e 4t

e 2t 0 )

(2 ) 1 x = . 1

19

Ch. 4 .

4.3 .

Ex. 2 : , .

1 0 y' = y, 0 1

y1 ' = y1 y2 ' = y 2

:

1 0 y = c1 et + c2 et 0 1

y1 = c1et y2 = c2et

c1 y2 = c2 y1

20

Ch. 4 .

4.3 .

Ex. 3

: , .

1 0 y' = y, 0 1

y1 ' = y1 y2 ' = y2

:

1 0 y = c1 et + c2 e t 0 1

y1 = c1et y2 = c2e t

y1 y2 =

()

21

Ch. 4 .

4.3 .

Ex. 4 : . 0 1 y' = y, 4 0 y1 ' = y2 y2 ' = 4 y1 1 = 2 + 4 = 0 = 2i 4 1

:

det (A I ) =

1 = 2i , x (1) = ; 2i 1 = 2i , 1 x (2 ) = 2i y1 = c1e 2it + c2e 2it y2 = 2ic1e 2it 2ic2e 2it

:

1 1 2it y = c1 e 2it + c2 e 2 2 i i

y1 ' = y2 , y2 ' = 4 y1 4 y1 y1 ' = y2 y2 ' 2 y12 + 1 2 y2 = 2

22

Ch. 4 .

4.3 .

Ex. 5 : t , ( ) .

1 1 y' = y, 1 1 :

y1 ' = y1 + y2 y2 ' = y1 y2 1 1 1 = 2 + 2 + 2 = 0 = 1 i 1 1

det (A I ) =

(1) 1 = 1 + i , x = i ;

1 = 1 i , x (2 ) =

1 i

:

1 1 y = c1 e(1+ i )t + c2 e(1i )t i i

23

Ch. 4 .

4.3 .

y1 ' = y1 + y2 , y2 ' = y1 y2

y1 y1 '+ y2 y2 ' = y12 + y2 2

(

)

r 2 = y12 + y2 2

1 2 r ' = r 2 2

( )

(r ) ' = 2rr '2

rr ' = r 2

~ r = cet ln r = t + c

24

Ch. 4 .

4.3 .

z . (Degenerate Node)

( akj = a jk ) ( akj = a jk , a jj = 0 ) . n n

A (,

det (A I ) = 0 ) ,

( ) y (1) = xet .

y (1) : y (2 ) = xtet + uet

(y ( ) )' = xe2

t

+ xtet + uet = Axtet + Auet

x + u = Au

(A I )u = x

.

25

Ch. 4 .

4.3 .

Ex. 6 4 1 y' = y 1 2 z

:

det (A I ) =

4 1

1 = 2 6 + 9 = 0 = 3, 2 0 u= 1

1 x= 1

1 y (1) = e3t 1

( 3I )u = :

1 1 1 u = 1 1 1

1 0 3t 1 y = c1 e3t + c2 1t + 1 e 1

z (1) c1y

4 2

c1 > 0 c1 < 0

y (2 ) y (2 )

26

Ch. 4 .

4.4 .

4.4 . (Criteria for Critical Points. Stability)

z (Criteria for Types of Critical Points)a a A = 11 12 1 , 2 a21 a22 : det (A I ) =

a11 a21

a12 = 2 (a11 + a22 ) + det A = 0 a22

p = a11 + a22 ( ), q = det A = a11a22 a12 a21 ( ), = p 2 4q( ) : q > 0, 0 : q < 0 : p = 0, q > 0 : p 0, < 0

27

Ch. 4 .

4.4 .

z (Stability) , ( ) t .z

(Stable Critical point) : t = t0 .

z z

(Unstable Critical point) : (Stable and Attractive Critical point)

:

t

28

Ch. 4 .

4.4 .

z : p < 0, q > 0 : p 0, q > 0 : p > 0

q