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Show that the system, $$x'=-x-y+x(x^2+2y^2)$$ $$y'=x-y+y(x^2+2y^2)$$ has at least one peridic solution.

I know that I need to use the Poincare Bendixon Theorem, but I'm not to sure how to find the trapping region. When my teacher did an example in class he basically did a proof by picture and made it seem like all arrows were pointed inward within the trapping region. I'm wondering how would I find and rigorously prove that a trapping region really has all arrows pointed inward? Any help is appreciated, thanks!

EDIT: I believe I have figured out the trapping region to be the ellipse $x^2+2y^2=1$ and I believe I have proven it by looking at cases depending on which quadrant the coordinates is in.

Now my question is, how do I deal with the fixed point $(0,0)$ within the trapping region?

TAPLON
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1 Answers1

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Because of the form of the linear part of the vector field it seems advisable to explore the dynamic of the Euclidean radius. For simplicity of computation, use $E=\frac12r^2=\frac12(x^2+y^2)$ to get $$ \frac{d}{dt}E=x\dot x+y\dot y=-2E+2E(x^2+2y^2) $$ so that $$ 2E(2E-1)\le\dot E\le 2E(4E-1) $$ This means that $\dot E$ is negative for $0< E< \frac14$ and positive for $E>\frac12$. Looking in the direction of negative times, the time reversed ODE, this means that the annulus $\frac14<E<\frac12$, $\sqrt{\frac12}<r<1$, is invariant in that time direction and thus has to contain a limit cycle which is also a periodic solution.

diagram showing limit cycle some numerical solutions, the circles have radii $0.6$ to $1.1$ by $0.1$.

Lutz Lehmann
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  • So it is possible to have a trapping region even with an unstable origin as long as we never cross the perodic orbit? – Number4 Dec 09 '18 at 12:12
  • Yes? Note that the trapping region in question was for the limit cycle itself, not the origin. The dynamic of $E$ describes how the radius of a solution behaves, the differential inequalities thus reduce a 2-dimensional dynamic to a one-dimensional. Large radii get reduced, small positive radii get increased, which gives the above annulus as trapping region for the limit cycle. – Lutz Lehmann Dec 09 '18 at 12:24
  • Is a trapping region by definition related to some attracator? This is not my impression I tought it was just a set. – Number4 Dec 09 '18 at 12:32