Midterm 2 Nov 25, Work alone. Maple is allowed but not on a problem or a part of a problem that says no computer.

Transcription

1 Math 416 Name Midterm 2 Nov 25, 2008 Work alone. Maple is allowed but not on a problem or a part of a problem that says no computer. There are 5 problems, do 4 of them including problem 1. Each problem counts 15 points. If necessary, cross out the problem that you do not want graded. Neatness counts, show all work on these sheets going to the back if you need more room. 1. You must do this problem. Do all three parts. No computer. a) The point ( 1, 1) is a fixed point for the the system ẋ 1 = 4x 1 (1 x 2 ), ẋ 2 = x 2 (x 1 + x 2 1). Obtain the linearization of the system at this point and use it to sketch the local phase portrait at ( 1, 1). b) Use the method of isoclines to sketch the global phase portrait of the flow in part a). c) Explain why the linearization of the system at the fixed point (0, 0) cannot be used to determine its local phase portrait at that point.

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3 2. A portion of the velocity field for the flow ẋ 1 = 2x 1 x 2 2 ẋ 2 = x 2 x 2 1 is displayed on the right. a) Obtain the fixed points. b) Linearize the system around each fixed point. If appropriate, use the linearization to obtain the local phase diagram and comment on stability. c) Obtain a strict Liapunov function for the fixed point at the origin. d) Find δ > 0 as large as possible so that the open disk of radius δ and center (0, 0) is contained in the basin of attraction for the fixed point (0, 0).

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5 3. Let ẋ = Ax be a linear system where [ a b A = c d ]. a) Determine conditions on a, b, c, d that guarantee that this system is Hamiltonian. b) Prove that the linearization at a simple fixed point of a planar Hamiltonian system has eigenvalues that are either ±λ or ±i λ where λ is a real number. Sketch the corresponding phase portraits at the fixed point. c) Verify that ẋ 1 = x 2 1 2x 1 x 2, ẋ 2 = x 2 2 2x 1 x 2 is a Hamiltonian system. Obtain the Hamiltonian and use it to sketch the phase diagram for the system.

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7 4. Consider the SIRS model Ṡ = ris + µr I = ris γi Ṙ = γi µr where S, I, and R denote the fraction of susceptible, infected, and recovered individuals in a fixed population. Note that S + I + R = 1. The parameters r, γ, and µ are positive and σ = r/γ > 1. a) Reduce the system to two dimensions using the fact that R = 1 S I. b) Working in the unit square 0 S 1, 0 I 1, obtain the fixed points of the two-dimensional system. c) Do the fixed points lie in the feasible triangle D = { (S, I) S 0, I 0, S + I 1 }? yes no (circle one) Verify your answer and if it is no, then what restrictions on the parameter µ will guarantee that the answer is yes. d) Assuming the fixed points lie in D, is D positively invariant? yes no (circle one) Verify your answer and if it is no, then what restrictions on the parameter µ will guarantee that the answer is yes. Why is it important that D be positively invariant? e) Use a nullcline and fixed point analysis to sketch the phase diagram for the system under the assumption that the fixed points are in the feasible region and it is positively invariant. Partial credit if you choose specific values for the parameters.

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9 5. Consider the coupled and undamped mass-spring system discussed in class on Monday, November 24. Two carts of equal mass m are connected with a spring with constant k and each cart is attached to a wall with springs having the same spring constant k 1. a) Draw a picture and use it to derive the system of two second-order equations governing the motion. Take the positive direction be to the right. Let x 1 be the displacement of the left cart from its equilibrium position and x 2 be the displacement of the right cart from its equilibrium position. Note that x 2 x 1 measures the stretch of the middle spring. b) Divide each second order differential equation by m and rescale the time variable t to τ with the relationship τ = t. This forces the following changes to the derivatives: k m ẋ 1 = dx 1 dt = dx 1 dτ dτ k dt = m x 1 and ẋ 2 = dx 2 = dx 2 dτ k dt dτ dt = m x 2, where x 1 and x 2 denote differentiation with respect to the new time variable τ. A similar calculation will show that ẍ 1 = k m x 1 and ẍ 2 = k m x 2. Take advantage of this to express the second order equations in a more simplified form in terms of τ. c) Replace the two second order equations in part b) with four first order equations in the variables z 1 = x 1, z 2 = x 1, z 3 = x 2, and z 4 = x 2. Simplify this system by letting Note that λ > 1. λ = k + k 1 k d) Identify the matrix A such that the system in part c) is of the form z = Az. e) Make another change of variable, this one in the state vector z, by letting z = P w where P 1 = Find P and obtain the system matrix B for the system: w = P 1 AP w = Bw. [ ] B11 O f) The matrix B should be block-diagonal of the form B = where O represents a 2 2 zero matrix and O B 22 B 11 and B 22 are especially simple 2 2 matrices. Obtain the change-of-basis matrix M so that the substitution w = My converts the system w = Bw into Jordan form y = Jy. Obtain the matrix exponential e Jτ. g) Reverse the substitutions in parts e) and f) to obtain the evolution operator: ψ τ (z) = e Aτ z, for the system z = Az obtained in part c). h) Finally, let ω 0 = k/m, and switch back to time t with the substitution τ = ω 0 t. Note that ω 0 is the natural frequency of the mass m attached to a wall with a spring with constant k. This yields the evolution operator φ t (x) for the first-order system derived from the original second-order system in part a)..