Economics Dynamics Problems 215

Economics Dynamics Problems 215 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations 8. Given ˙ x=x ˙ y = 2x + 3y + z ˙ z = 2y + 4z (i) Find the eigenvalues and eigenvectors. (ii) Provide the general solution. (iii) Show that the Wronksian is nonzero. 9. For each of the following systems (a) find the eigenvalues and eigenvectors; (b) solve the system by finding the general solution; (c) obtain the trajectories for the specified initial points; and (d) classify the fixed points. (i) ˙ x = −3x + y ˙ y = x − 3y initial points = (1, 1), (−1, 1), (−1, −1), (1, −1), (2, 0), (3, 1), (1, 3) (ii) ˙ x = 2x − 4y ˙ y = x − 3y initial points = (1, 1), (−1, 1), (4, 1), (−4, −1), (0, 1), (0, −1), (3, 2), (−3, −2) (iii) ˙ x=y ˙ y = −4x initial points = (0, 1), (0, 2), (0, 3) (iv) 10. ˙ x = −x + y ˙ y = −x − y initial points = (1, 0), (2, 0), (3, 0), (−1, 0), (−2, 0), (−3, 0). For the following Holling–Tanner predatory–prey model 6xy x − 6 (8 + 8x) 0.4y ˙ y = 0.2y 1 − x ˙ x=x 1− (i) Find the fixed points. (ii) Do any of the fixed points exhibit a stable limit cycle? 11. Consider the R¨ ssler attractor o ˙ x = −y − z ˙ y = x + 0.2y ˙ z = 0.2 + z(x − 2.5) (i) Show that this system has a period-one limit cycle. (ii) Plot x(t) against t = 200 to 300, and hence show that the system settles down with x having two distinct amplitudes. 199 ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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