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

TL;DR

This paper investigates the 1:4 resonance in conservative cubic Hénon maps, revealing different bifurcation structures and degeneracies for maps with positive and negative cubic terms, including global bifurcations and island rotations.

Contribution

It provides a detailed analysis of the 1:4 resonance in both variants of the conservative cubic Hénon maps, highlighting distinct degeneracies and bifurcation phenomena.

Findings

For $ extbf{C}_-$, the Arnold degeneracy occurs with the first Birkhoff twist coefficient equal to the first resonant term.

For $ extbf{C}_+$, the resonant term can vanish, leading to non-symmetric points and pitchfork bifurcations.

Global bifurcations cause the 1:4 resonant chain of islands to rotate by $rac{ ext{pi}}{4}$.

Abstract

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 $\pi/4$. For both maps several bifurcations are detected and illustrated.