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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 Summer '06
 DeLeenheer
 Derivative, Polar coordinate system, Stability theory, Determine stability, old calc book, variable dilution rate

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