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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! :)

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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.

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  • $\begingroup$ Thanks! Well, that was a lot simpler than I expected, but makes sense. Can you recommend a good book about dynamical systems? Or invariance of systems, sets, etc? :) $\endgroup$ – Chris Oct 31 '19 at 18:25

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