I am currently working on the system:
\begin{align} \dot x = y, \ \ \dot y = -x-yz, \ \ \dot z = -xz + 7x^2 \end{align}
The critical points of this system are the whole of z-axis and that can be easily calculated. The positive part of z-axis is an attractor manifold based on the eigenvalues of the system at a critical point $(0,0,z_0)$ which are: \begin{align*} \lambda_1 = \frac{-z_0}{2} - \sqrt{\frac{z_0^2}{4} - 1} = \frac{- z_0}{2} -\frac{1}{2} \sqrt{z_0^2 -4}\\ \lambda_2 = \frac{-z_0}{2} + \sqrt{\frac{z_0^2}{4} - 1} = \frac{-z_0}{2} +\frac{1}{2} \sqrt{z_0^2 -4} \\ \lambda_3 =0 \end{align*}
I am actually trying to find the limit $\lim_{t \to \infty}z(t)$. For every initial point I get different limit based on the graph but I would like to know how to find analytically this limit.