On local and global aspects of the 1:4 resonance in the conservative cubic H\'enon maps

We study the 1:4 resonance for the conservative cubic H'enon maps ${\mathbf{C}}_{\pm }$ with positive and negative cubic term. These maps show up different bifurcation structures both for fixed points with eigenvalues $\pm i$ and for 4-periodic orbits. While for ${\mathbf{C}}_{-}$ 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 ${\mathbf{C}}_{+}$ 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 $\pi \mathrm{/}4$. For both maps several bifurcations are detected and illustrated.