# Iterates of the Poincaré map $T$ and it's correspondence to the periodic orbits?

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