Ordering conjecture for multiplying bifurcations of the symmetric periodic orbit

Let $\Gamma^{*}$ be the single-loop $R^{*}$-symmetric periodic orbit in the zero-energy level, with nontrivial Floquet multipliers $\xi_{\pm}$, and let $\beta_2$ denote the system parameter. Write $\mathbf{{B}_{k}}$ for the period-$k$ multiplying bifurcations and $\mathbf{{\widehat{B}}_{k}}$ for the symmetry-breaking period-$k$ multiplying bifurcations. The symbols $R_{1}$ and $R_{2}$ denote the relevant symmetries of the bifurcating orbits; $\mathbf{{T}_{3}}$, $\mathbf{SN}$ and $\mathbf{HH}$ denote, respectively, the transcritical-type triple-loop bifurcation, the saddle-node point and the Hamiltonian-Hopf bifurcation.

Ordering conjecture.

1. The period-$k$ multiplying bifurcations $\mathbf{{B}_{k}}$ occur at the values of $\beta_2$ where

$$\xi_{\pm}=e^{\pm 2\pi\frac{k-2}{k}}$$

for all odd $k\geq1$.

2. The symmetry-breaking period-$k$ multiplying bifurcations $\mathbf{{\widehat{B}}_{k}}$ occur at the values of $\beta_2$ where

$$\xi_{\pm}=e^{\pm 2\pi\frac{k-1}{k}}$$

for all odd $k\geq1$.

3. The corresponding bifurcating $R^{*}$-symmetric, $R_{1}$-symmetric and $R_{2}$-symmetric $k$-loop orbits exist below and up the respective $\beta_2$-value, except for the transcritical-type case $\mathbf{{B}_{3}}=\mathbf{{T}_{3}}$, where the triple-loop orbits exist up to $\mathbf{SN}$ at $\beta_2\approx0.649089$.

4. Since $\frac{k-2}{k}$ and $\frac{k-1}{k}$ tend to $1$ as $k\to\infty$, both sequences $\mathbf{{B}_{k}}$ and $\mathbf{{\widehat{B}}_{k}}$ approach $\mathbf{HH}$ and hence the associated value $\beta_2\approx0.8164$.