ODE Theory: Are centers and linear centers the same for reversible systems?

everyone!

I'm trying to prove that a linear center of a planar system IS a center when the system is reversible (invariant under the change of variables $t\mapsto -t$ and $y \mapsto -y$). I fooled around with the setup for quite a while and came to the somewhat dubious conclusion that this may not be true unless the rest point lies on the $x$-axis. Is that true? Or should it be true regardless of the rest point's location (taking into account, of course, the mirror equilibrium)?

Thanks in advance if you can provide any assistance!

1 Answer

I think it is implicitly assumed that the center in question lies on $y=0$. The notion of reversibility you stated appears to treat $y$ as the velocity function (this is why its sign is reversed when the direction of time is reversed). If this is indeed the nature of the system ($\dot x=y$), then equilibria can occur only when $y=0$.

Otherwise, one has a counterexample in the form of two spirals: stable and unstable, symmetric about the $x$-axis, both linearize to a center.