Consider the following one-parameter systems of differential equations in polar form: (a) r = r(r + )2,

Consider the following one-parameter systems of differential equations in polar form:

(a) ˙r = μr(r + μ)2, ˙ θ = 1;

(b) ˙r = r(μ − r)(μ − 2r), ˙ θ = −1;

(c) ˙r = r(μ2 − r2), ˙ θ = 1.

Plot phase portraits for μ < 0, μ = 0, and μ > 0 in each case. Sketch the corresponding bifurcation diagrams.

142 6. Bifurcation Theory