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Theorem: The fixed point origin $**X^{*}=0**$ is a linear center for the continuously differentiable system

$\dot{x}=f\left ( x,y \right )$

$\dot{y}=g\left ( x,y \right )$

and suppose that the system is reversible. Then, sufficiently close to the origin, all trajectories are closed curves.

The idea of the proof provided by Strogaz in 'Non-linear dynamic and chaos' is not very clear to me. He started off with the claim that any trajectories 'close' to the origin swirls about the origin. Where does this claim follows from? I can recall that all trajectories are closed orbits (closed curve) in the neighbourhood of a maxima/ minima but this would stipulate that the fixed point origin given in the hypothesis of the theorem is either a maxima or minima.

Some explanation would be appreciated.

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  • This illustration might be helpful to answer the question what is the benefit of reversibility in situation similar to yours. So, you just have to prove that trajectory intersects the line of fixed points of involution twice. Usually this follows from the way how nullclines are arranged around equilibrium. – Evgeny Apr 27 '17 at 17:24

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