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# hw2(3) - quadrant of R 2(use NW,SW etc arrows Based on the...

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Homework assignment 2 * Due date: Wednesday October 10, 2007 1. Let v be a nonzero vector in R 3 and consider the linear system ˙ x = v × x, where × stand for cross product (see your old calc book in case you forgot). This equation describes the motion of a rigid body rotating around an axis with direction given by the vector v . (a) Write the system in the usual form ˙ x = Ax , and find the general solution. (b) Is the system stable, asymptotically stable or unstable? 2. In class we discussed a bioreactor model with constant input nutrient concentration S 0 (we actually set S 0 = 1) and variable dilution rate D > 0. In this problem, we do things the other way round: We will set D = 1 and let S 0 > 0 be variable. The resulting equations are: ˙ S = S 0 - S - Sx ˙ x = x ( S - 1) (a) Perform phase plane analysis like we did in class, by determining the nullclines, all equilibria and the direction of the vector field in the various regions of the non-negative
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Unformatted text preview: quadrant of R 2 (use NW,SW etc arrows). Based on the resulting sketch, what do you think happens to solutions when t → ∞ ? (b) Linearize the system at the equilibria and discuss the nature of the linearization (stable node, center etc). 3. Consider ˙ x =-y + dx ( x 2 + y 2 ) ˙ y = x + dy ( x 2 + y 2 ) , where d is a real parameter. Determine stability of the equilibrium at (0 , 0) in terms of d . ( Hint: Use polar coordinates. ) 4. Determine stability of the zero solution of the following equations: • ¨ x + ˙ x + sin x = 0 . • ¨ x + ˙ x cos x + sin x = 0 . Note that these are problems 12 . 5 # 13 and # 14 from our text. Ignore the hint given there, and instead try the following function: V ( x, y ) = 2 sin 2 p x 2 P + y 2 2 . * MAP 4305; Instructor: Patrick De Leenheer. 1...
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