Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 51 / 3 / d / Solution

At the saddle equilibrium $x=a=\sqrt{\lambda }$, put $T=\mu +a$. The two eigenvalues and saddle index areOn the nearby saddle-loop curve, $T=12a/7+O(a^2)>0$. Therefore $0<\delta <1$, with the more explicit expansionThe positive saddle quantity $m_{-}+m_{+}=T$ makes the loop repelling. In a transverse section its leading return map has the form $s\mapsto C{s}^{\delta }$; its derivative grows without bound as $s\downarrow 0$. This is the mechanism by which a positive saddle quantity makes a nearby saddle loop repelling. It agrees with the repelling periodic orbit coming from the subcritical Hopf bifurcation, rather than predicting an attracting cycle at the collision.