= Solution
At the <saddle equilibrium> $x=a=\sqrt\lambda$, put $T=\mu+a$. The two <eigenvalues> and <saddle index> are
$$
m_\pm={T\pm\sqrt{T^2+8a}\over2},\qquad
\boxed{\delta=-{m_-\over m_+}={\sqrt{T^2+8a}-T\over\sqrt{T^2+8a}+T}.}
$$
On the nearby saddle-loop curve, $T=12a/7+O(a^2)>0$. Therefore $0<\delta<1$, with the more explicit expansion
$$
\boxed{\delta=1-{6\sqrt2\over7}\lambda^{1/4}+O(\lambda^{1/2}).}
$$
The positive saddle quantity $m_-+m_+=T$ makes the loop repelling. In a transverse section its leading return map has the form $s\mapsto Cs^\delta$; its derivative grows without bound as $s\downarrow0$. 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.
Back to article page
