# Period doubling on Möbius strip

I have a hard time understanding a simple argument for the following. Quite often I find in papers that limit cycles before and after period-doubling bifurcations lie on the Möbius strip in the state-space. But anything similar I can not find in the Wiggins nor Guckenheimer/Holmes.

But still, what does it even mean? How can I understand it, like that the center manifold (associated with the original limit cycle) forms the Möbius strip in geometrical interpretation?

• During a period-doubling bifurcation, an eigenvalue of the periodic orbit passes through -1, hence it is associated with a non-orientable manifold (Möbius strip). Can you tell from which book the picture is from? Jul 5 at 5:41