Supercritical degenerate Hopf-bifurcation conjecture for car-following wavefronts

Let $V$ be a $C^2$-smooth function satisfying (OVF 1)--(OVF 4). For $h^{*}<h_H<\hat{h}$ and $d\in\mathbb{R}$, let $w^{c_1(h_H),d}$ be a constant-speed wavefront solution of the canonical delay differential equation lying on the first branch, and let $S\subset(-\infty,0]\times[0,\infty)$ be the bounded region introduced in the preceding stability analysis. Suppose that the associated stability parameters satisfy

$$(-h_H,h_H^2V'(c_1(h_H)))\in\partial S.$$

Supercritical degenerate Hopf-bifurcation conjecture. At the parameter value $h=h_H$ and solution $w=w^{c_1(h_H),d}$, the canonical equation, and hence the original car-following equation at parameter $h=h_H$ and solution $z=-c_1(h_H)t+d$, undergoes a degenerate Hopf bifurcation that is supercritical. The bifurcating solutions are not periodic, but their first derivatives are periodic. This conjecture predicts the precise nonlinear transition associated with the stability boundary of the first wavefront branch; the source says that it is to be addressed analytically in subsequent work.