# Invariance of the x and y axis

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.