= Solution
Write $F=\kappa_{\rm fold}(\lambda)$, $G=\kappa_{\rm hom}(\lambda)$ and $H=-\lambda$ for $\lambda<0$, so $0<F<G<H$ near the double-zero point. The six open regions are:
* I: $\lambda>0,\kappa<0$. The origin is the only <equilibrium> and is attracting; there is no <periodic orbit>.
* II: $\lambda>0,\kappa>0$. The origin is repelling and is surrounded by an attracting <periodic orbit>, born at its <supercritical Hopf bifurcation>.
* III: $\lambda<0,\kappa<F$. The origin is a <saddle equilibrium> and the two nonzero <equilibria> are <sink equilibria>. Its <stable manifold> separates their basins; there is no <periodic orbit>.
* IV: $\lambda<0,F<\kappa<G$. There are two <sink equilibria> and the <saddle equilibrium>, an unstable <periodic orbit> surrounding all three, and a larger attracting <periodic orbit>. Initial conditions inside the unstable <periodic orbit> go to one of the <sink equilibria>, according to the <saddle equilibrium>'s basin boundary; those outside it approach the attracting <periodic orbit>.
* V: $\lambda<0,G<\kappa<H$. The <saddle equilibrium> and two <sink equilibria> remain, but each <sink equilibrium> is surrounded by its own unstable <periodic orbit>. A larger attracting <periodic orbit> surrounds the entire configuration. Each unstable inner <periodic orbit> bounds its <sink equilibrium>'s basin.
* VI: $\lambda<0,\kappa>H$. The two nonzero <equilibria> are repelling. The <saddle equilibrium>'s <stable manifold> still leads to the <saddle equilibrium>, while generic trajectories approach the attracting outer <periodic orbit>.
The changes III–IV, IV–V and V–VI are respectively a fold of <periodic orbits>, the double homoclinic event, and the two <subcritical Hopf bifurcations>. Thus the unstable <periodic orbits> from the <subcritical Hopf bifurcations>, the expanding <saddle equilibrium> passage, and the attracting <periodic orbit> from the other <Hopf bifurcation> fit into one consistent portrait.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-56-six-portraits.png]
{title=The six local phase-portrait types; solid cycles attract and dashed cycles repel}
Back to article page
