I'm studying on my own Dynamicals Systems and having difficulties to undersrtand ODE defined on a manifold $M$.
Firstly, let $X: \Omega \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} $ a vector field and consider the equation
$\dfrac{d \varphi(t)}{dt}= X(\varphi(t))$,
a solution for this ODE is a path $\varphi(t): I \subset \mathbb{R} \rightarrow \Omega$, such that $\frac{d \varphi(t)}{dt}= X(\varphi(t))$, for all $t \in I$.
Then, I have the following doubts
1- If $M$ is a $m$-dimensional manifold $(m\neq n)$, then, is the vector field defined by $X: M \rightarrow TM$, which associes each $p \in M$ to the vector $X(p) \in T_{p}M$? This make sense for me, once that, the space $T_{p}M$ has more algebraic and topologic properties. However, take $M= S^{1}$ for example, for each $p \in S^{1}$ the tangent space $T_{p}M$ is equal to the orthogonal complement of $p$, that is, $T_{p}M = [p]^{\perp}$. How to construct a vector field with this property? How to construct vector fields for more general manifolds?
2- Has the solution $\varphi(t)$ lie in the manifold $M$, for all $t$? If yes, then is every solution on $S^{1}$ "like" $(\sin (\pi t), \cos(\pi t))$?
3- How to draw phase portraits on manifolds? I would like to see examples of two vector fields $X,Y:S^{1}\rightarrow TS^{1}*(?)*$, when $X$ and $Y$ are conjugated topologically and when they are not.
I would like to apologize if something pointed out here is too obvious or standart. Any help will be aprecciated.