From my textbook (Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields). Consider the following system \begin{equation} \left( \begin{array}{c} \dot{u} \\ \dot{v} \\ \dot{w} \end{array}\right)= \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & -(1+\sigma) & 0 \\ 0 & 0 & -\beta \end{array} \right)\left( \begin{array}{c} u \\ v \\ w \end{array}\right)+ \left( \begin{array}{ccc} -\frac{\sigma}{1+\sigma} (u+\sigma v)w \\ \\ \frac{1}{1+\sigma}(u+\sigma v)w \\ \\ (u+\sigma v)(u-v) \end{array}\right) \end{equation}
My textbook says: "u axis is not invariant because the equation for $\dot w$ includes the term $u^2$". It is not clear to me, maybe because I'm not English speaking. But I have made the following reasoning. If $v=w=0$ we have:
\begin{equation} \left( \begin{array}{c} \dot{u} \\ \dot{v} \\ \dot{w} \end{array}\right)= \left( \begin{array}{c} 0 \\ 0 \\ u^2 \end{array}\right) \end{equation} from which
\begin{equation} \left( \begin{array}{c} u \\ v \\ w \end{array}\right)= \left( \begin{array}{c} u_0 \\ v_0 \\ u_0^2 t \end{array}\right) \end{equation}
If $u_0=1$ and $v_0=0$ then: $(u,v,w)^T=(1,0,t)^T$ that does not belong to the u axis for $t\neq 0$.
What do you think of my proceedings? Is my procedure right? I made a mistake?
Thank you very much