MATEMATIK Datum: Tid: eftermiddag. A.Heintz Telefonvakt: Anders Martinsson Tel.:

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Theodore O’Neal’
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1 MATEMATIK Datum: Tid: eftermiddag GU, Chalmer Hjälpmedel: inga A.Heintz Telefonvakt: Ander Martinon Tel.: Löningar till tenta i ODE och matematik modellering, MMG5, MVE6. Define what i a generalized eigenpace of a matrix. Formulate the theorem on the tability of olution to a linear ytem of ODE d r t = A r t with a contant n n matrix A dt uing generalized egenpace of the matrix A. Decribe main idea of the proof. p Let v be an eigenvector correponding to a poibly multiple eigenvalue λ j, then the generalized eigenvector v i a olution of the equation λ j I A v = v and more generalized eigenvector v k λ j I A v k = v k the equation λ j I A k+ v k = 0. aociated with v are found a nontrivial olution to the equation for k =,..., l if they exit. It i eay to oberve that v k atifie Generalized eigenpace Mλ j, A i a ubpace of all eigenvector and generalized eigenvector correponding to the eigenvalue λ j. Each generalized eigenpace Mλ j, A can in principle include everal linearly independent eigenvector and everal or no one linearly independent generalized eigenvector. Theorem 3.3.5, p. 56 in the book by Hu. Two tatement are formulated there: a if all eigenvalue λ j of the matrix A have trictly negative real part Re λ j < 0, then there are contant 0 < K and 0 < α uch that e At Ke αt for all t 0. b if a weaker etimate i valid Re λ j 0 and thoe egenvalue that have zero real part have the ame algebraic and geometric multyplicity or have imple elementary divior then there i a contant M > 0 uch that e At M. Proof i baed on the repreentation e At = P e Jt P of e At, where matrix P conit of column that are include a maximal poible et of eigenvector v k followed by correponding generalied eigenvector v i k. Thee eigenvector together with generalized eigenvector build a bai in R n. J i a block diagonal matrix with quare Jordan block J k correponding to linearly independent eigenvector of all ditinct eigenvalue of A. Block follow the order in which eigenvector and correponding eigenvalue tand in P. Each eigenvalue λ j with multiplicitet m can have from one to m linearly independent eigenvector. The lat cae mean that the eigenvalue ha the ame algebraic multiplicity and geometric multyplicity. In the lat cae all Jordan block correponding to λ j will conit of matrix and will build up a diagonal block in J. The rool BC B C for the norm of matrix product implie that e At P e Jt P where norm P and P are both bounded. Becaue of the block-diagonal tructure of e Jt following from the block - diagonal tructure of J the following etimate i valid: e Jt max e J kt and the etimating of e At i reduced to the etimating k of all e J k t. e J kt =e λ kt e N kt, where e N kt in cae it ha dimenion larger than, include power of t m, with m < n. N k i the matrix with all element zero exept thoe over the main diagonal that are one. If min Re λ k < 0, and we chooe an α < min Re λ k then product of e λ kt t m < Ke αt

2 for all m n and ome contant K, becaue any any power m, and any ε > 0, t m e εt i bounded for all t 0. It implie the firt tatement. The econd tatement follow from the ame reaoning a above and additional proof for thoe λ k that have Re λ k = 0. We oberve that if λ k ha imple divior the ame algebraic and geometric multyplicity, then N k above will have dimenion and there will be no power of t in the expreion for e λ kn k t. If N k ha dimenion one, then it i jut zero and e Nkt =. Exponential e λkt will be bounded for Re λ k = 0, e λ k t, becaue e λ k t = co Im λ k t + i in Im λ k t. 2. Formulate and prove the theorem on tability and aymptotical tability of equilibrium point to autonomou ODE by Lyapunov function. p See Theorem 5.2., p in the book by Hu. 3. Conider the following ytem of ODE: d r t dt = A r t, with a contant matrix A = Find general olution to the ytem. Find all thoe initial vector r 0 = r 0 that give bounded olution to the ytem. p Eigenvector and eigenvalue are : v = 2 λ = 0, v 2 = λ 2 = 3 Generalized eigenvector v correponding to λ atifie the equation Av = v. there are infinitely many poible olution v =. Jordan form of the matrix i J = A = P JP with P = 0 2 General olution can be found in the general form: xt = e At x 0 =.. n j A λ j I k t k x 0,j e λ jt k! j= k=0 x 0 = j= x0,j where x 0,j Mλ j, A - are projection of x 0 to all generalized eigenpace Mλ j, A of the matrix A correponding to each eigenvector. We chooe x 0 in the form x 0 = C v j +C 2 v +C 3v 2 where C v j +C 2 v = x 0, Mλ, A and v j and v contitute bai for Mλ, A. Mλ 2, A ha dimenion one becaue the eigenvalue λ 2 i imple and x 0,2 = C 3 v 2 in the general formula above. xt = e At x 0 = e λ t C v j + C 2 v + te λt A λ j I e λ t C v + C 2 v + te λt C 2 v + C 3 e λ2t v 2 0 xt = C 2 + C 2 + tc C 3 e 3t 2 C v j + C 2 v = + C 3 e λ 2t v 2 =

3 C + tc 2 + C 3 e 3t 2C + C 2 2tC 2 + C 3 e 3t C C 2 + tc 2 + C 3 e 3t = C + tc 2 + C 3 e 3t 2C + 2t C 2 + C 3 e 3t C + + t C 2 + C 3 e 3t You can get the anwer in infinitely many different form depending on the choice of eigenvector and generalized eigenvector a a bae for initial data in R 3. The olution will be bounded only if initial data are parallel to v, or if C 2 = C 3 = 0 in our repreentation.. Conider the following initial value problem: y = y 2 + t; y = 0. a Reduce the problem to an integral equation. Calculate three firt Picard approximation a in the proof to Picard Lindelöf theorem. b Find ome time interval where the Picard approximation converge. p a The IVP for an equation y = fy, t, yt 0 = y 0 can be rewritten in integral form: yt = yt 0 + t t 0 fy, d. Picard iteration are y n t = yt 0 + t t 0 fy n, d; with y 0 t = yt 0. In our particular cae t 0 =, y = 0, yt = t y 2 + d. Three Picard iteration for the given example are: y 0 t = 0; y t = t d = t2 2 ; y 2 t = t d = t + 2 t2 6 t t b The Picard Lindelöf theorem tate that the IVP for an ODE with the right hand ide fy, t continuou in the domain t t 0 < a, y y 0 < b uch that fy, t < M for thee t and y and atifying Liphitz condition with repect to y with Liphitz contant L ha a unique olution on the interval t t 0 < α for α = mina, b/m, /L. Anwering the quetion reduce to chooing ome a and b, finding an etimate M for the particular right hand ide and finding an etimate for y fy, t that will give an etimate for the Lipchitz contant L max. y fy, t t t 0 <a y y 0 <b Then the time interval i given by the formula t t 0 < α with α = mina, b/m, /L. Chooe a = b = and point out that t 0 =. Then y 2 t + t 3 for t < ; or 0 < t < 2 and y 0 <. We chooe M = 3. y fy, t = 2y, 2y 2 for y 0 <. We chooe L = 2. then the ize of the time interval where iteration converge can be etimated a α = mina, b/m, /L = min, /3, /2 = /3. Anwer. The Picard iteration for the given example converge for the time interval t /3 at leat. 5. For the following ytem of equation find all equilibrium point and invetigate their tability. { x = ln x + y 2 y p = x y [ y ] 2 Jacobian of the right hand ide i Jx, y = x+y 2 x+y 2 There are two tationary point: 0,- and 3,2. 3

4 [ 2 J0, = 0, i table becaue for both eigenvalue Re λ i < 0. [ J3, 2 = becaue one of eigenvalue Re λ 2 > 0. Anwer: 0,- - table 3,2 - untable. ], eigenvalue: λ = i 2, λ 2 = i 2.The tationary point ], eigenvalue: λ = 3, λ 2 =.The tationary point 3, 2 i table 6. Formulate the Poincare-Bendixon theorem and ue it to how that the following ytem of ODE ha a periodic olution xt, yt 0, 0. { x = y y = fx, y y x where f, f x, f y are continuou, f0, 0 < 0 and fx, y > 0 for x 2 + y 2 > b 2. p One of formulation of the Poincare Bendixon theorem i: let ϕt be a olution to an autonomou equation x = fx in plane, bounded for all t > 0. We uppoe that f i Lipchitz to guarantee uniqune of olution. Then the limit et ωϕ of ϕt ha the following property: it either i contain an equilibrium or ii ϕt i periodic itelf or ωϕ i a periodic orbit. The theorem ca an important corollary, that i alo often called Poincare Bendixon theorem: If the equation x = fx in plane ha a compact poitively invariant et a et that no trajectorie can leave that doe not include any equilibrium point, then thi et mut include at leat one periodic orbit. We apply the corollary to the example above and try to find a et in plane atifying requirement i the corollary. We oberve firt that olution exit for initial point everywhere in the plane becaue of the moothne of f. Conider a tet function V x, y = x 2 + y 2 /2 and it evolution along olution to the given ytem. V = xy y 2 fx, y xy = y 2 fx, y. fx, y > 0 for x 2 + y 2 > b 2. It implie that V 0 for x 2 + y 2 = b 2 and that trajectorie of the ytem cannot leave the dic x 2 + y 2 b 2. f0, 0 < 0 and i continuou. x 2 + y 2 δ 2 It implie that there i a mall circle around the origin uch that fx, y < 0 and correpondingly V > 0 for x 2 + y 2 = δ 2. It implie that the ytem cannot leave the ring δ 2 x 2 + y 2 b 2. It i eay to oberve that the only equilibrium point of the ytem i the origin: y = 0 for the firt equation and therefore x = 0 from the econd equation. Alltogether implie that the ring δ 2 x 2 + y 2 b 2 i a poitively invariant et without equilibrium point and mut include at leat one periodic orbit. Max: 2 point; Threholding for mark: for GU: VG: 9 point; G: 2 point. For Chalmer: 5: 2 point; : 7 point; 3: 2 point;

5 One mut pa both the home aignment and the exam to pa the coure. Total point for the coure will be the average of the point for the home aignment 30% and for thi exam 70%. The ame threholding i valid for the exam, for the home aignment, and for the total point. 5

MA 266 FINAL EXAM INSTRUCTIONS May 2, 2005

MA 266 FINAL EXAM INSTRUCTIONS May 2, 2005 MA 66 FINAL EXAM INSTRUCTIONS May, 5 NAME INSTRUCTOR. You mut ue a # pencil on the mark ene heet anwer heet.. If the cover of your quetion booklet i GREEN, write in the TEST/QUIZ NUMBER boxe and blacken