I have the system of ODE $$x' = x(2y-1) \\ y' = y - x^2 -y^2$$ I am interested in the behavior around the critical point $(-1/2,1/2)$. The linearization has pure imaginary eigenvalues so I would like to know if $(-1/2,1/2)$ is a spiral point or a center. First I noticed that $$\frac{dy}{dx} = \frac{y - x^2 - y^2}{-x+2xy}$$ or rewriting it, $$x^2 + y^2 - y + (-x+2xy)\frac{dy}{dx}=0$$ is an exact ODE and has solutions given implicitly by $$-xy + xy^2 + \frac{x^3}{3} = c$$ I would think that since I have now solved the ODE I should be able to determine the precise nature of the critical point $(-1/2,1/2)$ but I am at a loss of what to do. Any hints would be much appreciated. I know that I could try to use a Lyapunov function but this seems like it would be overkill since I already solved the system.
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1Have you heard of first integrals? What happens when you differentiate $-xy + xy^2 + x^3/3$ w.r.t. $t$? – Evgeny Nov 06 '18 at 15:01
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@Evgeny I have not but I will look into this! – 1729 Nov 06 '18 at 15:02
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You can read why it is important here. – Evgeny Nov 06 '18 at 15:14
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@Evgeny thanks. Looks like an interesting write-up. I'll try to answer my own question when I have time to read through all this in the next few days. Unless someone else does before me. – 1729 Nov 06 '18 at 15:16
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@evgueny may I ask you a book or a good reference ont his subject ? Thanks in advance – user577215664 Nov 06 '18 at 20:03
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@Isham What subject do you mean exactly? – Evgeny Nov 06 '18 at 21:16
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@Evgeny Separatrix first integral for critical points ...id you know some good reference – user577215664 Nov 06 '18 at 21:24
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@Isham I think books by Wiggins (maybe, Meiss) should cover this stuff. The basics of this stuff were summarized in part of my older answer. – Evgeny Nov 07 '18 at 16:28
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@Evgeny Thanks a lot for the reference and I have read your old answer need to reread it Thanks again – user577215664 Nov 07 '18 at 16:36