Ch. 4 연립상미분방정식. 상평면 및 정성법 1 Ch. 4 연립상미분방정식. 상평면 및 정성법 (Systems of ODEs. Phase Plane. Qualitative Methods) 내용 : 행렬과 벡터를 이용한 선형연립방정식의 해법
Nov 10, 2014
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