I saw that the Poincaré map is defined by the flow of the periodic system with the least period $T$. that is, $$P(x):=\phi_T(x)$$ is a Poincaré map with flow $\phi$ of time $T$.
but I think if we define it like this way, it is just $P(x)=x$ because the solution is periodic, $$P(x)=\phi_T(x)=\phi_0(x)=x.$$ did I see the right definition of Poincaré map? if I did, what do I understand wrong?
Edit: I read the wikipedia, but still don't get it. could you explain it more simply, like, using flow or something? with some examples if you don't mind?