Rigorous proof of global bifurcations in the generic inversion-symmetry case

Prove that the global bifurcation observed when the system crosses the symmetry corresponding to the 2:2 resonance always occurs in the generic inversion-symmetry cases for the values of the coupling parameter at which it occurs in the 2:2 resonance.

Background

The catastrophe-map analysis describes global bifurcations associated with crossing the symmetry \beta_3=\delta_3=0. For parameter ranges in which the corresponding 2:2 resonance admits such a global bifurcation, the authors expect the phenomenon to persist after symmetry breaking to the generic inversion-symmetry case. They explicitly state that a rigorous proof has not been supplied.

References

We expect the global bifurcation to always appear in the generic cases for the values of $\alpha$ that make it possible for the $2:2$ resonance. However, a rigorous proof is beyond the scope of this work.

— Perturbed isotropic oscillators with inversion symmetry  (2610.13743 - Marchesiello et al., 9 Oct 2026) in Section 6.2, “$\alpha<-1$”