Engineering Mathematics I - Chapter 4. Systems of ODEs

민기복 Ki-Bok Min, PhD 서울대학교 에너지자원공학과 조교수 Assistant Professor,,gy Energy Resources En gineerin g Ch.4 Systems of ODEs. Phase Plane. Qua lita tive Me tho ds

• Basics of Matrices and Vectors • Systems of ODEs as Models • Basic Theory of Systems of ODEs • Constant-Coefficient Systems. Phase Plane Method • Crit eri a for Criti cal Poi nt s. Stability bilit • QQyualitative Methods for Nonlinear Systems • Nonhomogeneous Linear Systems of ODEs Basics of Matrices and Vectors

•Systems of Differential Equations – more than one dependent variable and more than one equation

y ',ay ay ay yayay', 1111122 1nn 1111122 yayay'   ay, yayay', … 2211222 2nn 2211222 

ynn',ay11 ay n 2 2  ay nnn • Differentiation – The derivative of a matrix with variable entries is obtained by differentiating each entry.

 yt  y' t  11 yytt  '   yt22 y' t Basics of Matrices and Vectors Eivenvalues (고유치) and Eivenvectors (고유벡터)

 • Let A   a j k be an n x n matrix. Consider the equation Ax   x where  is a scalar and x is a vector to be determined. – A scalar  such that the equation Ax   x holds for some vector x ≠ 0 is called an eigenvalue of A, – And this vector is called an eigenvector of A corresponding to this eigenvalue .

Axx  Basics of Matrices and Vectors Eivenvalues (고유치) and Eivenvectors (고유벡터)

Ax  x Ax  Ix  0 AIx    0

 – n linear algebraic equations in the n unknowns xx 1 ,, n (the components of x). – The determinant of the coefficient matrix A  I must be zero in order to have a solution x ≠ 0. • Characteristic Equation detAI   0

– Determine λ1 and λ2

– Determination of eigenvector corresponding to λ1 and λ2 Basics of Matrices and Vectors Eivenvalues (고유치) and Eivenvectors (고유벡터)

• Example 1. Find the eigenvalues and eigenvectors

 4.0 4.0 A    1.6 1.2

• If x is an eigg,envector, so is kx Systems of ODEs as Models

10 gal/min 10 gal/min VS. 10 gal/min 10 gal/min

Single ODE Systems of ODEs Systems of ODEs as Models Exampl e 3. MiMiixing probl em (Sec 13)1.3)

– Initial Condition: 1000 gal of water, 100 lb salt, initially brine runs in 10 gal/min, 5 lb/gal, stirring all the time, brine runs out at 10 gal/min – Amount of salt at t?

10 gal/min

10 gal/min ? Systems of ODEs as Models

– Tank T1 and T2 contain initially 100 gal of water each.

–In T1 the water is pure, whereas 150 lb of fertilizer are dissolved in T2.

– By circulating liquid at a rate of 2 gal/min and stirring the amounts of fertilizer yt 1  

in T1 and y 2  t  in T2 change with time t. 2 gal/min – Model Set Up 2 gal/min

22 yyyTyyy' Inflow / min - Outflow / min  Tank ' 0.02 0.02 1211112100 100 22 yyyTyyy' Inflow / min - Outflow / min  Tank ' 0.02 0.02 2122212100 100

0020.02 0020.02  yAyA ' ,    0.02 0.02 Systems of ODEs as Models

y1-y2 plane - phase plane (상평면 ) Trajectory (궤적): - 상평면에서의 곡선 - 전체 해집합의 일반적인 양상을 잘 표현함. Phase portrait (상투영): trajectories in phase plane Systems of ODEs as Models

• Example 2. Electrical Network

y1-y2 plane - phase plane (상평면 ) Trajectory (궤적): - 상평면에서의 곡선 - 전체 해집합의 일반적인 양상을 잘 표현함. Phase portrait (상투영): trajectories in phase plane Systems of ODEs as Models Conv ersion of an nth-order ODE to a ssystem stem

•2nd order to a system of ODE – Example 3. Mass on a Spring my''cy'ky  0

 y1  y  y, y  y' y    1 2 y2  y1' y2  0 1  y  y' Ay   k c  1 k c     y '  y  y  y2  2 m 1 m 2  m m

   1  c k detA  I  k c  2     0  Calculation same     m m m m as before!

Ex)''2'0750)''2'0yy.75 y  0 Systems of ODEs as Models Conv ersion of an nth-order ODE to a ssystem stem

– solve singgyle ODE by methods for s ystems – includes higher order ODEs into …first order ODE Basic Theory of Systems of ODEs CtConcepts ((lanalogy to siilngle ODE)

yf  • First-Order Systems 11   yf , yf111',,, t yy n   yfnn  yf',,, t yy  221 n y ', f t y 

yfnn',,, t yy1  n

• SltiSolution on some iitnterva l a

• Initial Condition: yt10  K 1, yt 20  K 2 ,  , ytnn 0 K Basic Theory of Systems of ODEs EitExistence and UUiniqueness

• Theorem 1. Existence and Uniqueness Theorem

– Let f 1 ,  , f n be continuous functions having continuous partial

fff 11n derivatives , , , , in some domain R of tyy 12  y n - yy1 nn y

tK, , , K space containing the point  01 n . Then the first-order system

has a solution on some interval ttt 00    satisfying the

initial condition , and this solution is unique. Basic Theory of Systems of ODEs Homogeneous vs. NhNonhomogeneous

aa11 1n y 1 g 1 • Linear Systems     Ayg , , aannnn12  y g yaty1111'    atygt 1nn  1   yAyg'  

ynn' at11y  at nnnnyg t 

– Homogeneous: yAy'  – Nonhomogeneous: yAygg0',  Basic Theory of Systems of ODEs CtConcepts ((lanalogy to siilngle ODE)

• Theorem 2 (special case of Theorem 1)

• Theorem 3

 (1) (2) (1) (2) (1) (2) yyycc12  c 1 y c 2 y c 1 AyAy  c 2 (1) (2) Aycc12 y Ay Basic Theory of Systems of ODEs GlGeneral soltilution

미분 아님!! • Basis, General solution, Wronskian – Basis: A linearly independent set of n solutions yy 1  ,,,  , n  of the homogeneous system on that interval – General Solution: A corresponding linear combination

1 n yycccc11nn y  , , arbitrary – Fundamental Matrix : An n x n matrix whose columns are n solutions yy1, , n  1  n    Yy  , , y y =Yc – Wronskian of yy1, , n : The determinant of Y

12  n   yy11 y 1 12   n ex) 1 n yy y W,yy ,  22 2  (1) (2) 0.5tt 1.5 yy11cc112ee  210.5tt   1.5 12  n   y Yc(1) (2)  0.5tt 1.5   c12 e  c   e yynn y n yy22cc22ee1.5  11.5   Constant Coefficient Systems.

y '  Ay • Homogeneous linear system with constant coefficients

  – Where n x n matrix A   a jk  has entries not depending on t –Try y  xet  yyy 'xAeett y Ax  Ax  x • Theorem 1. General Solution – The constant matrix A in the homogeneous linear system has a linearly independent set of n eigenvectors, then the corresponding solutions yy  1 ,  ,  n  form a basis of solutions, and the 1 n corresponding general solution is   1t   nt yxce1  cn x e Constant Coefficient Systems.

• Example 1. Improper node (비고유마디점)

31 yyy  3 yAy'  y 112 13  yyy213 2

 y1  1 2 1 2t  1  4t y     c1y  c2y  c1 e  c2  e y2  1 1 Constant Coefficient Systems.

• Example 2. Proper node (고유마디점) 10 yy  yAy'    y 11 01  yy22

t 1 t 0 t y1  c1e y  c1  e  c2  e  t 0 1 y2  c2e Constant Coefficient Systems.

• Example 3. Saddle point (안장점) 10 yy  yAy'    y 11 01  yy22

1 0 y  c et y c et c et 1 1  1    2    t 0 1 y2  c2e Constant Coefficient Systems.

• Example 4. Center (중심) 01 yy  yAy'    y 12 40  yy214

 1   1  y  c e2it  c e2it y c e2it c e2it 1 1 2  1   2    2it 2it 2i  2i y2  2ic1e  2ic2e Constant Coefficient Systems.

• Example 5. Spiral Point (나선점) yyy   11 112 yAy'   y   11 yyy212 

1 1i t  1  1i t  y  c1 e  c2  e i  i Constant Coefficient Systems.

• Example 6. Degenerate Node (퇴화마디점): – 고유벡터가 기저를 형성하지 않는 경우: 행렬이 대칭이거나 반대칭(skew-symmetric)인 경우 퇴화마디점은 생길 수가 없음.

1 t  4 1 y  xe y'  y  1 2 y2  xte t  uet

 1    1  0 3t   3t y  c1 e  c2  t   e 1  1 1 Constant Coefficient Systems. Summary Δ is discriminant (판별식) of determinant of (A-λI)

• Improper node (비고유마디점): 0,12  0 – Real and distinct eigenvalues of the same sign

* • Proper node (고유마디점): 0, 12  – Real and equal eigenvalues • Saddle point (안장점): 0,12  0 – Real eigenvalues of opposite sign

• Center (중심):   0, pure imaginary – Pure imaginary eigenvalues •Spppiral point (나선점): 0, complex – Complex conjugates eigenvalues with nonzero real part

* : when two linearly independent eigenvectors exist. Otherwise, degenerate node Name Δ Eigenvalue Trajectories

Improper node (비고유마디점)   0 12  0

Proper node   0 (고유마디점) 12

Saddle Point   0 (안장점) 12  0

Center Pure imaginary (중심) e.g., i   0 Spiral Point Complex number (나선점) e.g., 1 i Criteria for Critical Points. Stability

• Critical Points of the system dy dyyyyyy2 ' a y  a y yAy'   22211222dt dy dy111111221 y' a y a y dt

dy2 –dy1 A unique tangent direction of the trajectory passing

PPyy:,  through 12 , except for the point PP 0 :000,0

dy2 – Critical Points: The point at which d y 1 becomes undetermined, 0/0 Criteria for Critical Points. Stability

• Five Types of Critical Points

– Depending on the geometric shape of the trajectories near them

 Improper Node

 Proper Node

 Saddle Point

 Center

 Spiral Point Criteria for Critical Points. Stability

• Stable Critical point (안정적 임계점):

– 어떤 순간t  t0 에서 임계점에 아주 가깝게 접근한 모든 궤적이 이후의 시간에서도 임계점에 아주 가까이 접근한 상태로 남아 있는 경우. • Unstable Critical point (불안정적 임계점): – 안정적이 아닌 임계점 • Stable and Attractive Critical point (안정적 흡인 임계점): – 안정적 임계점이고, 임계점 근처 원판 내부의 한 점을 지나는 모든 궤적이t   를 취할 때 임계점에 가까이 접근하는 경우 Criteria for Critical Points. Stability

Stabl e & UtblUnstable Unstable attractive

Stable Stable & Attractive Qualitative Methods for Nonlinear StSystems

• Qualitative Method – Method of obtaining qualitative information on solutions without actually solving a system. – particularly valuable for systems whose solution by analytic methods is difficult or impossible.

NliNonlinear systems yfyy',  yyy'  f y, thus 1112 yfyy2212'  f  , 

linearization yayayhyy'  ,  y' Ay h y, thus 1111122112 y2211222212',a y a y h yy Qualitative Methods for Nonlinear StSystems

• Theorem 1. Linearization

– If f1 and f2 are continuous and have continuous partial derivatives in a neighborhood of the critical point (0,0), and if det A  0 , then the kind and stability of the critical point of nonlinear systems are the same as those of the linear ize d system

yyyy'  a y  a y y'A y, thus 1111122 yayay2211222'. – Exceptions occur if A has equal or pure imaginary eigenvalues; then the nonlinear system may have the same kind of critical points as linearized system or a spiral point. Qualitative Methods for Nonlinear StSystems

• Example 1. Free undamped pendulum – Determine the locations and types of critical points – Step 1: setting up the mathematical model

 q : the angular displacement, measured counterclockwise from the equilibrium position

 The weight of the bob : mg

 A restoring force tangent to the curve of motion of the bob : mg sin q

 By Newton’s second law, at each instant this force is balanced by the force of acceleration mLq``

g  mLθ '' mg sin 0 θ'' k sin 0  k L Qualitative Methods for Nonlinear StSystems

– Step 2: Critical Points  0,0,    2,0,     4,0,    Linearization set yy  ,   ' yy'  θ'' k sin 0 12 12 y21'i'  k sin y

yy210, sin 0 infinitely many critical points :  nπ ,n 0 ,  0, 1, 2,  1 01 yy'  sin yy y3  y yAy'  y, thus 12 116 1 1  k 0 yky21'   – Step 3: Critical Points    ,0,   3 ,0,    5 ,0,   Linearization.

Consider the critical point (p , 0) set yy   ,   ' ' 01 12 yAy'   y θ'' k sin  0 1 3  sin sinyyyyy sin  k 0 111116   critical points are all saddle points. Qualitative Methods for Nonlinear StSystems Nonhomogeneous Linear Systems of ODEs

• Nonhomogeneous of Linear Systems: y ', Aygg 0 – Assume g(t) and the entries of the n x n matrix A(t) to be continuous on some interval J of the t-axis.

– General solution : yyhp y  h  y : A general solution of the homogeneous system y '  A yg  on J p  y : A particular solution(containing no arbitrary constants) of yAy '  on J • Methods for obtaining particular solutions – Method of Undetermined Coefficients – Method of the Variation of Parameter Nonhomogeneous Linear Systems of ODEs

• Method of undetermined coefficients: – Components of g : – (1) constants – (2) positive itinteger powers of t – (3) exponential functions – (4) cosines and sines. Nonhomogeneous Linear Systems of ODEs

• Example 1. Method of undetermined coefficients. Modification rule 31 62t yAyg' ye  13 2 11 yh  cec24tt e – A general solution of the homogeneous system : 12   11 – Apply the Modification Rule by setting yup te22tt v e 2t 1 – Equating the t e -terms on both sides: 2uAu  u aa  with any  0 1 – Equating the other terms: 6k uv 22h02 Avv  a 2,    choose k 0 24 k   – General solution :

11122422tttt      y cec12  e 2 te e 1112 Nonhomogeneous Linear Systems of ODEs

• Example 1. solution by the method of variation of parameters

31 62t yAyg' ye  13 2 – General solution of the homogeneous system : hn 1   yycct1  n yYcYAY    '  p – Particular solution : yYu   tt  

p y ' Y ' u Yu '  Y ' u  Yu ' AYu  g

t  Yu ' g u ' Y11 g u Yttdt g  t0 Nonhomogeneous Linear Systems of ODEs

• Example 1. solution by the method of variation of parameters (cont.)

11ee44tt ee 22 tt Y1  6t 22tt 44 tt 22e ee ee 11ee22tt6 e 2 t 42 uYg' 1   44tt 2 t 22tt 22ee 2 e 84ee t 22t  u dt  22tt  0 422ee  ee24tt2222tt222te224ttt e e  yYup  ee24tt  24tt2t 224ttt   ee 22et222te e e    22  2