On local and global aspects of the 1:4 resonance in the conservative cubic Hénon maps

We study the 1:4 resonance for the conservative cubic Hénon maps C± with positive and negative cubic terms. These maps show up different bifurcation structures both for fixed points with eigenvalues ±i and for 4-periodic orbits. While for C–, the 1:4 resonance unfolding has the so-called Arnold degeneracy [the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient], the map C+ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by π/4. For both maps, several bifurcations are detected and illustrated.