527ass8 - you try it. 2. To clarify problem 13 of Section...

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642:527 ASSIGNMENT 8 FALL 2007 Multple-page homework must be STAPLED when handed in. Turn in starred problems Tuesday 10/30/2007. Section 7.4: 7 (a)*, (b), (c) Section 7.5: 4*; 6 Section 9.6: 11; 12 (b), (c), (d)*, 13* Problem 8.A* Two interacting populations x ( t ) , y ( t ) are described by the equations x = (3 - x - y ) x , y = (2 - y ) y . (a) Find all the critical points of this system. You do not need to classify these. (b) Sketch the ±rst quadrant x 0, y 0 of the phase plane, indicating, by arrows or otherwise, regions where x and y are increasing, x is increasing and y decreasing, etc., and where the trajectories are horizontal and vertical. (c) For each initial condition below, ±nd (from your sketch or otherwise) lim t →∞ b x ( t ) y ( t ) B : (i) x (0) = 0 , y (0) = 3; (ii) x (0) = 3 , y (0) = 3; (iii) x (0) = 0 , y (0) = 0. Comments: 1. I have listed problem 6 of Section 7.5 because it is a nice exercise if you want to ±nd out more about the van der Pol oscillator, but it is certainly not required that
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Unformatted text preview: you try it. 2. To clarify problem 13 of Section 9.6: we are considering solutions of n linear equations in n unknowns, that is, the equations have the form A x = b , where A is an n n matrix, x is a column vector of n unknowns, and b is a given vector with n components. Here are the equations written out, but it is much better to work with the matrix/vector notation: a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 . . . . . . . . . . . . . . . a n 1 x 1 + a n 2 x 2 + + a nn x n = b n In (a) we assume that b = ; in part (b) b is arbitrary. To say that the set of solutions is a vector space is to say that a linear combination x (1) + x (2) of two solutions x (1) and x (2) is again a solution. 1...
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