Question: Is the origin a nonlinear center for the system

$$x'=-y-x^2$$ $$y'=x$$

We have a theorem that states if the system is reversible orbits close to the origin are closed. It seems that this system is not reversible and so I was wondering if the converse holds. Does this imply that the center is not a nonlinear center? any help is appreciated, thanks!

  • 1
    $\begingroup$ Have you plotted the system in phase space? If it looks like a center it is very likely that the converse isnot true. $\endgroup$ – MrYouMath Mar 13 '18 at 20:11
  • $\begingroup$ Yeah, It doesn't seem to be reversible $\endgroup$ – Justin Stevenson Mar 13 '18 at 20:14
  • $\begingroup$ I mean does it look like a center? $\endgroup$ – MrYouMath Mar 13 '18 at 20:28
  • $\begingroup$ Yeah it looks like a center $\endgroup$ – Justin Stevenson Mar 13 '18 at 20:29
  • $\begingroup$ The eigenvalues of the linearization at $(0,0)$ are $\pm i$, so yes you've got a center there. $\endgroup$ – dbx Mar 13 '18 at 21:06

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