The dynamics of the $j$th system: 
\begin{equation}
\begin{split}
\dot{\overline r}_j &= h (\overline r_j) 
\,\, -  \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \zeta_k \cos (\overline \theta_{jk}+\delta)  ,\\
\dot{\overline \theta}_j &=  \frac{ \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}}{\overline r_j}\sum^N_{k=1}  \zeta_k \sin (\overline \theta_{jk} + \delta),
\end{split}
\end{equation}
where $\overline \theta_{jk}:= \overline \theta_j - \overline \theta_k$, $h_j (\overline r_j)$ is a nonlinear function, $\zeta$, $\chi$, $\xi$,  $R_\mathrm{Th}$, $\omega_\mathrm{sw}$ and $\delta$ are all positive scalars. 

$h(x)$ is a nonlinear function given by $a x - b x^3$ with $a, b > 0$.

Jacobian around $\theta^{*}_{j} = \frac{2 \pi j}{N}$ and $\overline r^*_j > 
0$ such that $h(\overline r^*_j)=0$ (splay state equilibrium) features $N-2$ zero eigenvalues. 

Is there any hope of using normal form analysis or center manifold theory in such cases to establish stability?