SILNIKOV-HOPF BIFURCATION

We derive a 2-parameter Poincare map modeling a 3-dimensional vector flow near the codimension-2 point defined by a Sil'nikov homoclinic orbit to a fixed point undergoing a Hopf bifurcation. In one parameter regime the map reproduces Glendinning and Sparrow's results for the Sil'nikov bifurcation while in another regime it provides new results for a small Hopf cycle with a global reinjection mechanism. The map is analyzed finding primary homoclinic tangencies to the small Hopf cycle. Parameter dependence of periodic orbits associated with these tangencies is described. Global behavior of the tube-like unstable manifold of the Hopf cycle is studied in depth, analytically and numerically, locating its simplest subsidiary tangencies.