Question: Let $\phi$ be the flow of $X$ a vector field of class $C^1$ defined on an open set of $\mathbb{R}^n$ and $\gamma=\{\phi(t,p);0 \leq t \leq T\}$ a periodic orbit. Show that the matrix $D_2(T,p)$ is similar to $D_2(T,q)$ for every $q \in \gamma$.
Attempt: Given $q \in \gamma$ there is some $t_0 \in [0,T]$ such that $q=\phi(t_0,p)$. Notice that using group property of the flow
$$
\phi(-t_0,\phi(T,\phi(t_0,x)))=\phi(-t_0,\phi(T+t_0,x))=\phi(-t_0+(T+t_0),x))=\phi(T,x).
$$
Differentiating the expression above on the point $p$ using the chain rule
$$
D_2 \phi(T,p)=D_2\phi(-t_0,\phi(T,\phi(t_0,p)))D_2\phi(T,\phi(t_0,p))D_2\phi(t_0,p)
$$
$$
D_2 \phi(T,p)=D_2\phi(-t_0,\phi(T,q))D_2\phi(T,q)D_2\phi(t_0,p).
$$
Then I tried to show that $D_2\phi(-t_0,\phi(T,q))=(D_2\phi(t_0,p))^{-1}$ and got stuck. Any hints on how to proceed from here are appreciated. If there is a mistake or some other way of proving it, please let me know. Thanks in advance ;)
Since no answer was given, I will post how I ended up answering the question and leave it open for others to find here.
First notice that using group property of the flow and that $\gamma$ is a periodic orbit $$ \phi(T,q)=\phi(T,\phi(t_0,p))=\phi(T+t_0,p)=\phi(t_0,p)=q. $$ So to prove that $D_2\phi(-t_0,\phi(T,q))=(D_2\phi(t_0,p))^{-1}$ is equivalent to prove that $D_2\phi(-t_0,q)=(D_2\phi(t_0,p))^{-1}$. But again using group property of the flow $$ x=\phi(0,x)=\phi(-t_0+t_0,x)=\phi(-t_0,\phi(t_0,x)). $$ Then differentiating the expression above on the point p using the chain rule $$ E=D_2\phi(-t_0,\phi(t_0,p))D_2\phi(t_0,p)=D_2\phi(-t_0,q)D_2\phi(t_0,p) $$ where $E$ is the identity matrix. Hence $D_2\phi(-t_0,q)=(D_2\phi(t_0,p))^{-1}$.