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 $r^*_j > 0$ 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?

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?

The dynamics of the $j$th system: \begin{equation} \begin{split} \dot{\overline r}_j &= h_j (\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, $\zeta$, $\chi$, $\xi$, $R_\mathrm{Th}$ are all positive scalars.

Jacobian around $\theta^{*}_{j} = \frac{2 \pi j}{N}$ and $h_j(\overline r^*_j)=0$ (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?

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 $r^*_j > 0$ 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?