Lösningsförslag till tenta i ODE och matematisk modellering, MMG511, MVE162 (MVE161)

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1 MATEMATIK Datum: Tid: GU, Chalmers Hjälpmedel: - Inga A.Heintz Teleonvakt: Jonatan Kallus Tel.: 55 Lösningsörslag till tenta i ODE och matematisk modellering, MMG5, MVE6 (MVE6). Formulate and prove the theorem about properties o solutions to linear systems o ODEs with periodic coe cients: = A(t); A(t + p) = A(t), when t!. (p) Theorem on boundedness and zero limits o solutions to periodic linear systems. ) Every solution to a periodic linear system is bounded on R + i and only i the absolute value o each Floquet multiplier is not greater than and any Floquet multiplier with absolute value is semisimple. ) Every solution to a periodic linear system tends to zero at t! i and only i the absolute value o each Floquet multiplier is strictly less than. Floquet multipliers are eigenvalues to the monodromy matri ( p; ) o the system. By Floquet theorem any solution (t) to system satisying initial conditions () =, is represented as (t) = A(t)(t); A(t + p) = A(t), 8t R () (t) = ( t; ) = (t) ep(tf )( ; ) = (t) ep(tf ) where F = plog(( p; )); ( ; ) =. (t) is a p - periodic continuous or piecewise continuous matri valued unction. (t) is invertible or all t as a product o invertible matrices. We de ne y(t) = ep(tf ) as a solution to the equation y (t) = F y; y() = () y(t) = (t)(t), and (t) = (t)y(t). The mapping (t) determines a one to one correspondence between solutions to the periodic system () and the autonomous system ().The periodicity and continuity properties o (t) and (t) imply that there is a constant M > such that k(t)k M and (t) M or all t R. It implies that k(t)k M ky(t)k and ky(t)k M k(t)k. Thereore )k(t)k is bounded on R + i and only i corresponding ky(t)k = kep(tf )k is bounded on R + : ) k(t)k! when t! i and only i corresponding ky(t)k! when t! : Since Log (( p; )) = G = pf, it ollows that (( p; )) = ep(p) : (F )g (F ) = Log() : (( p; )) p and that algebraic and geometric multiplicities o each (F ) coincide with those o ep(p) (( p; )) :We use now that Log(z) = ln(jzj) + i arg(z) ep(z) = ep(re z)(cos(im z) + i sin(im z) Here we choose the argument arg(z) [:) to make Log an invertible unction that is called the principle logarithm. The ollowing connections between properties o Floquet multipliers and properties o corresponding eigenvalues to the matri F = plog(( p; )) are a direct consequence: a) The Floquet multiplier (( p; ));hasjj < i and only i Re Log() < that is i the corresponding eigenvalue = plog() to F has Re Log() < : b) The Floquet multiplier (( p; ));has jj i and only i Re Log() that is i the corresponding eigenvalue = plog() to F has Re Log() : c) The Floquet multiplier (( p; )); with jj = is semisimple i and only i the corresponding eigenvalue = plog() to F has Re Log() = is semisimple. Known relations between properties o solutions to an autonomous system and the spectrum o corresponding matri applied to the system y (t) = F y and to the spectrum (F ) o the matri F together with statements ),), a),b),c) in the present proo imply the statement o the theorem.

2 . Formulate and give a proo to Lyapunov s theorem on stability o equilibrium points. (p) Lyapunov s theorem on stability Let be an equilibrium point or the system = () with : G! R n ; Lipschitz, () =. Suppose there is a positive de nite continuously di erentiable, C (U) unction V : U! R ; such that U G; open; U and V (z) = rv (z) 8z U. Then is a stable equilibrium point. Remark. A unction V with these properties is usually called the Lyapunov unction o the system. Proo. Take an arbitrary " > such that B("; ) U. Let = min zs(";) V (z) be a minimum o the continuous unction V on the boundary o B("; ); that is the sphere S("; ) = z : kzk = "g and is a compact set (closed and bounded). Then > because V (z) > outside the equilibrium point. By continuity o the unction V and the act that V () = one can nd a < < " such that 8z B(; ) we have V (z) < =. On the other hand or any part o the trajectory (t) = '(t; ), inside U the unction V ('(t; )) is nonincreasing because d dtv ('(t; )) = (rv ) ((t)). Thereore all trajectories '(t; ) with initial points B(; ) satisy V () < =. Thereore V ('(t; )) < = and '(t; ) cannot reach the sphere S("; ) where V (z) = min zs(";) V (z). Thereore any such trajectory stays within the ball B("; ) and by the de nition, the origin is stable. It implies also that R + I, where I is the maimal interval or initial point, because the trajectory stays inside a compact set.. Consider the ollowing system o ODE: d! r (t) = A! r (t), with a constant matri A having eigenvalues dt and : A = 5. Find a general solution to the system and a canonical Jordan orm o the matri A. Find all initial conditions such that solutions to I.V.P. will be bounded. (p) Solution. We nd eigenvectors corresponding eigenvalues. =. an eigenvector can be taken as v = corresponding to the eigenvalue =. = A I = 5 :There are no other linearly independent eigenvectors an eigenvector can be taken as v = 5 We check i =, has an generalized eigenvector, namely i Av () eigenvector can be chosen as v () = The canonical Jordan orm or the matri A is J = 5 5, = v has a solution. 5, reduced row echelon orm is: 5 : Vectors v, v (), and v build a basis in R. General solution to the system with initial data = C v + C v () + C v is 5 : (t) = C v + tc v + C v () + e t C v 5 :Generalized Solutions are bounded i and only i = C v. Find maimal eistence time interval or solution o the I.V.P. = t with arbitrary initial data () =. (p) d = t, is the equation with separable variables.

3 R d = R tdt ; = t + C ; general solution is: (t) = C t : To satisy the initial condition we put t = and nd () = = C, thereore C = = gives a constant solution (t) = or all t R. I ma () = R and (t) = t. For < we observe that the solution (t) = t is de ned or all t R. Thereore I ma () = R: For > we observe that the solution (t) = is de ned only or t ( t up in the points p. Thereore in this case I ma () = ( p ; p ): p ; p ). The solution blows 5. Consider the ollowing system o ODEs representing a one-dimensional mechanical system with velocity y and coordinate. Find all equilibrium points, investigate their stability properties, and nd their possible domains o attraction. = y y = y (p) Equilibrium points are (; ) and ( ; ): Jacobi matri is A(; y) = Jacobi matri at the origin is, characteristic polynomial is p() = + +, eigenvalues are ip ; ip. Real parts o eigenvalues are negative and thereore the origin isstable ocus, asymptotically stable equilibrium. Jacobi matri at the point ( ; ) is, characteristic polynomial is p() = p + ;eigenvalues are 5 ; p 5. One is negative, another is positive, the equilibrium point is a saddle point and is unstable. We try to nd the domain o attraction or the asymptotically stable point in the origin. R The system represents the one dimensional Newton equation with potential orce with potential G() = z + z dz = + : A natural Lyapunov unction to such a system is the sum o kinetic and potential energy: V (; y) = y + + Calculating derivative o V along trajectories we arrive to V (; y) = rv = + (y) + y y = y We can apply LaSalle s invariance principle to nd domain o attraction o the equilibrium in the origin. The set V () denotes the set where V (; y) = : I a trajectory '(t; ) has the closure o the positive semi orbit O + () compact and contained in the open set U where V (; y), then '(t; ) approaches the union o invariant sets (orbits) in V () as t!. I this union o invariant sets consists o one point, then the trajectory '(t; ) tends to this point as t!. This condition on the trajectory '(t; ) is easy to check by nding a compact positively invariant set K or the system where the condition V (; y) is satis ed.then all trajectories starting in K will have the closure o the positive semi- orbit O + () compact and contained in K. I this positively invariant set includes only one equilibrium as an invariant set in V (), then this equilibrium is an attractor or all trajectories starting in K that is it s domain o attraction. Positively invariant sets o our system can be ound as sets bounded by level sets o the Lyapunov unction V (; y) = y + + (with not very large values to have them as closed curves). We need to nd one such positively invariant set that does not contain the equilibrium point ( ; ): The largest such set would be one with boundary going eactly through the point ( ; ). Taking the level set o V going through this point we get corresponding value or V : V ( ; ) = = 6. The level set V (; y) = 6 can be epressed by the implicit equation y + + = 6 and is evidently symmetric with respect to ais. We need to investigate it s shape. Considering the derivative d d + 6 = o the unction g() = + 6 we observe that g() is rising between and, g( ) = ; g() = 6, and g() is decreasing or < : In the point = :5 it is equal to zero. Thereore or [ ; :5] the unction g() is non-negative. It implies that or [ ; :5] the level set V (; y) = 6 consists o two curves symmetric with respect to q ais: y = + : It gives the boundary o the bounded domain U symmetric with respect to the ais. The part o the level set V (; y) = 6 in the hal plane < does not bound a compact set and is o no interest or consideration o the equilibrium in the origin.

4 y This domain is positively invariant because V (; y) or all (; y). The same is valid also or any smaller set bounded by level sets V (; y) = C < =6: Points (; y) U where V (; y) = are on the - ais. Ecept the origin, they all have velocities (; y) pointed across the - ais and cannot belong to an invariant set in V ().The second equilibrium point ( ; ) o the system was ecluded beore by the choice o the level set o V (; y) = =6. Thereore the origin is the only invariant set in V () \ U. It implies that all trajectories starting in U tend to the origin as t!, and thereore U is the domain o attraction or the asymptotically equilibrium point in the origin. 6. Show that the ollowing system o ODEs has a periodic solution. = y y = y ln + y (p) We will consider a simple test unction V (; y) = ( + y ). V (; y) = y y y ln + y = y ln + y. The ellipse + y = (red in the picture) is the boundary between the domains where V (; y) has negative and positive values. For + y we get V (; y) For + y > we get V (; y) < Thereore a circle with radius r so small that + y or + y < r will have the property that no one trajectory enters it: Thereore a circle with radius R so large that + y > or + y R will have the property that no one trajectory escapes it: The largest circle inside the ellipse + y = is + y =, or + y = The smallest circle outside the ellipse + y = is + y =. + y = y Thereore choosing r = and R = we observe that these two circles bound an annulus shaped set K = (; y) : r + y R that is positively invariant and compact. The system has no equilibrium in this set. Thereore according to the Poincare-Bendison theorem K must contain at least one periodic orbit. Ma. points; Threshold or marks: or GU: VG: 9 points; G: points. For Chalmers: 5: points; : 7 points; : points; One must pass both the home assignments and the eam to pass the course. Total points or the course are calculated as: T otal = :6 Assignment + :6 Assignment + :68 Eam - that is the average o the points or the home assignments (%) and or this eam (68%). The same threshold is valid or the eam, or the home assignments, and or the total points or the course.

5 Points that you have got or the assignments and or the eam are valid and are kept up to the moment when you will collect all necessary points.

(4p) See Theorem on pages in the course book. 3. Consider the following system of ODE: z 2. n j 1

(4p) See Theorem on pages in the course book. 3. Consider the following system of ODE: z 2. n j 1 MATEMATIK Datum -8- Tid eftermiddag GU, Chalmers Hjälpmedel inga A.Heintz Telefonvakt Aleei Heintz Tel. 76-786. Tenta i ODE och matematisk modellering, MMG, MVE6 Answer rst those questions that look simpler,