Shilnikov bifurcation

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.