Shilnikov's Saddle-Node Bifurcation Dedicated to Leonid Shilnikov on his birthday.

homoclimc orbit In 1969 Shilnikov described a bifurcation for families of ordinan ' differential equations involving n ~ 7 trajectOIjes hI-asymptotic to a non-hyperbolic statIOnary pomt. At nearoy parameter values the ~stem lias trajectories in correspondence with the full shift on n s~bols. We investigate a simple (piecewise smooth) example witli an infiriite number of homoclinic loops. We also p~esent a smooth example which shows how Sfulnikov's mechanism is related to the Lorenz bifurcation by con~ide~g the llll(oldinR of a p'reviously unstudiea codlIDensIOn two bIfurcatIOn pomt