= Shilnikov bifurcation
{c}
A Shilnikov bifurcation is a homoclinic connection to a saddle-focus in three dimensions, with one real unstable eigenvalue $\lambda_+>0$ and a stable complex pair $\lambda_-\pm i\omega$. The eigenvalue ratio $\delta=-\lambda_-/\lambda_+$ distinguishes a contracting return for $\delta>1$ from the oscillatory expanding return associated with positive saddle value when $\delta<1$. Generic local and global maps yield infinitely many saddle periodic orbits at a connection in the latter case.
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