Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 59 / 1 / e / 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.