# Nonlinear Centers in Reversible systems

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!

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