Suppose the fixed point $P_{1}$ of the Poincaré map $T$ corresponds to a periodic orbit of the continuous system of the period $\tau$.

Then the iterates of the map $T$ that is say the $n$th iterate $T^n$, now the fixed point $P_{n}$ would correspond to a periodic orbit of period $n \tau$?

Is this true, any examples or counter examples, also how do I prove this any reference/hints?

  • Yes, it corresponds to a periodic orbit, but why should it relate to $n\tau$? You can easily find counterexamples using Markov systems of hyperbolic flows, although you need at least dimension $3$. – John B Aug 7 at 21:38
  • Summarizing @JohnB, the answer in most cases is "no". There are two cases where the answer is "approximately yes". The first case is when you consider a system $\dot{x} = \mathcal{F}(x, t), \; \mathcal{F}(x, t) = \mathcal{F}(x, t + \tau)$ and study its stroboscopic map. An orbit of period $n \tau$ would correspond to a periodic point of period $n$. However, orbits of period $\frac{p}{q}\tau$ also correspond to period-$q$ points. The second case is when a periodic point is born through a bifurcation of a fixed point of Poincaré map. In that case the period of this orbit is $\approx n \tau$. – Evgeny Aug 8 at 15:17

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.