= Solution
For the supplied <normal form>, the four local topological <phase portraits> are as follows. When $\mu_2<0$, only the origin is present and it is a <saddle equilibrium>. When $\mu_2>0,\mu_1<0$, the origin is attracting, between the two outer <saddle equilibria>; inward-facing <separatrices> enter its basin. When $\mu_2>0$ and $0<\mu_1<h(\mu_2)$, the origin is repelling and a stable <limit cycle> surrounds it, inside the separatrix boundary formed by the outer <saddle equilibria>. Finally, when $\mu_2>0,\mu_1>h(\mu_2)$, the origin remains repelling but the local <limit cycle> has disappeared, and trajectories leave the central region through the opened separatrix channels. Whether the origin is a <node> or a <focus> within one of these regions does not change this orbit topology.
On the connection curve, the two outer <saddle equilibria> are joined in both directions; the period of the disappearing <limit cycle> diverges. This is a <heteroclinic cycle>, not a pair of saddle <homoclinic orbits>. The sketches below show the four open regions of the supplied system; they are not presented as phase portraits of the inequivalent printed spherical equations.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-59-normal-form-portraits.png]
{title=Four phase portraits of the supplied reflection-symmetric cubic normal form}
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