I have a 4-dimensional system of the nonlinear ordinary differential equations with state variables $x,y,z$, and $w$. Also, the last equation is quite complicated due to the presence of the exponential term. At some equilibrium points $E_1=(0,0,z_1,w_1)$ and $E_2=(0,y_2,z_2,w_2)$ , the Jacobian ($J$) is not defined, cause, the entry $J_{41}$ is a product of $\frac{1}{x^2}$.
So, I think the Jacobian method (or, local linearization method) fails to give the local stability of those equilibrium points. Now, how do I investigate the local stability of $E_1$ and $E_2$?
Any ideas or help will be greatly appreciated.