= Positive saddle quantity makes a nearby saddle loop repelling
{title2=$\delta=-m_-/m_+<1$}
At a planar <saddle equilibrium> with $m_-<0<m_+$, positive saddle quantity $m_-+m_+>0$ is equivalent to <saddle index> $\delta<1$. The local passage followed by the regular return has leading transverse map $s\mapsto C s^\delta$, $C>0$. Its derivative diverges as $s\downarrow0$, so a nearby periodic orbit created by the saddle-loop splitting is repelling. This local mechanism agrees with the growth of the unstable periodic orbit born at a <subcritical Hopf bifurcation> in the <quadratic Bogdanov–Takens unfolding>.
Back to article page
