My question is about the invariance of the x and y-axis of the following system of differential equations:
$$\begin{pmatrix} \dot{x}(t)\\ \dot{y}(t) \end{pmatrix} = \begin{pmatrix} 3x(t)\\ -2y(t)+x(t)^2 \end{pmatrix}$$
Now I want to know if the x-axis or the y-axis is invariant. My idea is to look at the dynamical flow as seen here: Flow of the system. Then we see that the flow is stable for the y-axis. So therefore if we have a set $C \subseteq \mathbb{R}^2$, we see that $y(t) \in C$ and that $y(\tau) \in C$ for all $\tau \geq t$. So therefore the the y-axis is invariant. While the x-axis is unstable and we can't be sure $x(t)$ will stay in the set. Therefore I think the y-axis is invariant, but the x-axis is not. Is it correct to see it this way? How can I show this mathematically, any material you can recommend that would improve my understanding?
Thanks for the help, much appreciated! :)
When $x=0$, $\dot{x} = 0$, so you stay on the $y$ axis. When $y = 0$ but $x \ne 0$, $\dot{x} \ne 0$, so you don't stay on the $x$ axis. Thus the $y$ axis is invariant but the $x$ axis is not.