Engineering Mathematics I - Chapter 4. Systems of ODEs 민기복 Ki-Bok Min, PhD 서울대학교 에너지자원공학과 조교수 Assistant Professor,,gy Energy Resources Eng ineerin 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 er ia 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. yt11 y' t yytt ' 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 detAI 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 problem (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 detA 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 itinterva l a<tbt<b yht, , yht – A set of n differentiable functions 11 nn • 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 11n 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' at11y at nnnn yg 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) yyycc12 c 1 y c 2 y c 1 AyAy c 2 (1) (2) Aycc12 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 yycccc11nn y , , arbitrary – Fundamental Matrix : An n x n matrix whose columns are n solutions yy1 , , n 1 n Yy , , y y =Yc – Wronskian of yy1 , , n : The determinant of Y 12 n yy11 y 1 12 n ex) 1 n yy22 y 2 W,yy , (1) (2) 0.5tt 1.5 yy11cc112ee 210.5tt 1.5 12 n y Yc(1) (2) 0.5tt 1.5 c12 e c e yynn y n yy22cc22ee1.5 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 xet yyy'xAeett 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 yxce1 cn x e Constant Coefficient Systems. • Example 1. Improper node (비고유마디점) 31 yyy 3 yAy' y 112 13 yyy213 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 et 1 1 1 2 t 0 1 y2 c2e Constant Coefficient Systems. • Example 4. Center (중심) 01 yy yAy' y 12 40 yy214 1 1 y c e2it c e2it y c e2it c e2it 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 1i t 1 1i 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 y2 xtet uet 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' 22211222dt 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.