Intro Math Modeling (MATH-UA 251) Homework 6 Spring 2022
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Intro Math Modeling (MATH-UA 251)
Spring 2022
Homework 6
Exercise 1. [100 pts] For each of the following affairs, find the eigenvalues and eigenvectors, and sketch (by hand), qualitatively, all possible phase portraits (if applicable, depending on the signs and relative sizes of a and b). Do NOT use computer to make the plots. Specify the stability of the origin [R, J] = [0, 0] (stable node, saddle point, center (for closed orbits), or stable/unstable spirals). Your sketches should show the important qualitative features of each case. Interpret the phase portraits.
(i) (35 pts) R˙ = J, J˙ = -R + J
(ii) (35 pts) R˙ = aJ, J˙ = bR, (a, b 0)
(iii) (30 pts) R˙ = aR + bJ, J˙ = bR + aJ, (a < 0, b > 0, a2 = b2 )
notes: Solutions without details of the work and interpretation of the results will not receive full credits.
2022-04-06