Saddle index ($\delta =-{\lambda }_{-}/{\lambda }_{+}$)

For a hyperbolic equilibrium point with one unstable eigenvalue ${\lambda }_{+}>0$ and weakest stable decay rate ${\lambda }_{-}<0$, the saddle index is $\delta =-{\lambda }_{-}/{\lambda }_{+}$. For a stable complex pair, use its real part. An incoming distance $x$ from the stable manifold takes time $T={\lambda }_{+}^{-1}\log (h/|x|)$ to reach an outgoing section at unstable coordinate magnitude $h$. A stable coordinate is multiplied by $e^{{\lambda }_{-}T}=(|x|/h{)}^{\delta }$, giving a power-law local passage. A smooth nondegenerate reinjection consequently has derivative tending to zero when $\delta >1$ and an unbounded leading derivative when $\delta <1$. This determines local contraction or expansion near a homoclinic orbit; existence of a particular attractor also depends on the global return geometry.

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 $s\downarrow 0$, 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.