(a) Consider the two-dimensional system dr dt = r( r)( r2), d dt = 1....

(a) Consider the two-dimensional system dr dt = r(μ − r)(μ − r2), dθ

dt = −1.

Show how the phase portrait changes as the parameter μ varies and draw a bifurcation diagram.

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(b) Prove that none of the following systems has a limit cycle:

18.1. Dynamical Systems with Applications 423

(i) dx dt = y − x3, dy dt = x − y − x4y;

(ii) dx dt = y2 − 2xy + y4, dy dt = x2 + y2 + x3y3;

(iii) dx dt = x + xy2, dy dt = x2 + y2.

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