Orbits and periodicity of points, the basics

Question

Im trying to put into practice what I've learn from the basics of dynamical systems and Im having some troubles with the proposed problems from G.Teschl ODE & Dynamical systems

Let $\phi$ be the flow of some first order autonomous system. I'm asked to prove:

1. Show that if T satisfies $\phi(T,x)=x$, the same is true for any integer multiple of T. Moreover, $T=nT(x)$ for some $n \in \mathbb{Z}$.

2. Show that a point x is fixed if and only if $T(x)=0$

3. Show that x is periodic if and only if $\gamma_{+}(x) \cap \gamma_{-}(x) \neq \emptyset$ in which case $\gamma_{+}(x)= \gamma_{-}(x)$ and $\phi(t+T(x),x)=\phi(t,x)$ for all $t\in \mathbb{R}$. In particular, the period is the same for all points in the same orbit.

where $\gamma_{+}(x)=\{\phi(t,x):t_{x}\in (0,t_{x}^{+})\}, \gamma_{-}(x)=\{\phi(t,x):t_{x}\in (t_{x}^{-},0)\}$ are the forward and backwards orbit respectively, $T(x)=inf\{T>0: \phi(T,x)=x\}$ and a $y$ is a fixed point for $\dot x = f(x)$ if $f(y)=0$.

My attempt:

For 1) is not much of a problem cause, by propoerties of the flow we have that if $\phi(T,x)=x$ then $\phi(2T,x)=\phi(T,\phi(T,x))=\phi(T,x)=x$ then by induction we have that $\phi(nT,x)=\phi(T,x)=x$. And by the definition given above, $nT(x)=inf\{nT>0: \phi(T,x)=x\}$ we have that and since $\phi(nT,x)=\phi(T,x)$ for continuity of the flow $nT=T$, then $T=nT(x)$.

For 2) I've been trying to use the fact that if x is a fixed point then $\gamma(x)=\gamma_{+}(x) \cup \{x\} \cup \gamma_{-}(x)=x$ then I tried to relate it to \phi but Its been unsuccessful.

For 3) I know that if x is periodic then $\phi(t+T(x),x)=\phi(t,x)$, $t \in (0,t_{x}^{+})$ but then for the backwards case I got lost.

I understand this problems are easy but I been having problems with the proof techinques so, any help would be really appreciated. Thanks so much in advance guys <3

Answer 1

1. Your finishing argument is not convincing. You have to show the existence of such an $n$, not start from it being given. You could apply that if no such $n$ exists, then there is some period smaller than $T(x)$, which is a contradiction.

2. $T(x)=0$ implies that there are periods $T_n$ with limit $T_n\to 0$ for $n\to\infty$, thus there is a dense set with $ϕ(kT_n,x)=x$. Then apply continuity.

3. $γ_+(x)∩γ_-(x)≠∅$ means that there are $t_-<0<t_+$ with $ϕ(t_-,x)=ϕ(t_+,x)$, which implies that $T(x)\ge 0$ and all that this implies.