= Saddle index
{title2=$\delta=-\lambda_-/\lambda_+$}
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.
Back to article page
