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?