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For the system :

$x'=a(1-x^2-y^2)x -y\\ y'=x+a(1-x^2-y^2)y$

It is required to prove that if $\bar{w}=(x,y)$ is a solution of the system and $||\bar{w}(t)||<1$ for some $t$, then $||\bar{w}(t)||<1$ for each $t$.

The goal of this is to find the maximal interval of existence (which is $(-\infty, \infty))$. Now I understand how to prove the maximal interval of existence from the above, I just don't understand how to show that the norm is bounded.

I will be very grateful even for a small hint as to how to solve the problem.

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    $\begingroup$ Welcome. In order for other users to be able to give you the help you need, you should edit your post and tell something about your background and attempts to solve the problem. $\endgroup$ – Amitai Yuval May 17 '17 at 15:09
  • $\begingroup$ As @AmitaiYuval has correctly stated, it is required that you show your current progress, and the background to the question. Therefore, I have flagged it. $\endgroup$ – VortexYT May 17 '17 at 15:12
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Write $N:=\|(x,y)\|^2=x^2+y^2.$ Then $$\dot{N}=2x\dot{x}+2y\dot{y}=2a\left(1-x^2-y^2\right)\left(x^2+y^2\right)=2a(1-N)N.$$ In other words, the squared norm satisfies another ODE, which is independent of $x$ and $y$.

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  • $\begingroup$ Thanks, it would've taken me a long time before I thought of that, the rest of the proof is easy. $\endgroup$ – user2132672 May 17 '17 at 15:46

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