A neutral homoclinic bifurcation in a 3D map

Abstract

We investigate the structure of Arnol’d tongues passing through a quasi-periodic saddle-node bifurcation in a 3D map. Due to resonances, the bifurcation set has a complicated structure. Using numerical continuation in MatcontM and Lyapunov exponents we explore the tongues and the quasi-periodic saddle-node bifurcation set. The bifurcation set emerges from two Chenciner bifurcations. Both sets terminate in a homoclinic bifurcation of a neutral saddle cycle of period 3. It is similar to a case for vector fields, but the first report of such a codim-2 homoclinic tangency bifurcation in a map. We also show how these manifolds oscillate around the saddle near the tangency, for real and complex multipliers.