Let $f: E \rightarrow \mathbb{R}^n$ be a locally Lipschitz vector field for Initial Value Problem (IVP) $\dot{y} = f(y)$ with $y(0)=x$, where $E \subset \mathbb{R}^n$ is open.
Definition (flow): Let $\phi : \Omega \rightarrow E $ be the solution to the IVP $\dot{y} = f(y)$ with $y(0)=x$ where $\Omega=\{(t,x) \mid x \in E , t \in I_x\}$ and $I \in \mathbb{R}$ is an interval which includes $0$.
and
Definition (orbit through $x$): $ \Gamma_x = \{\phi_t(x) \mid t \in (\alpha_x,\beta_x)\} $
1)
My first question is how useful this latter definition. What I understand is that we know a part of trajectory which we do not know about the qualitative behaviour of the flow on it is available. In other words, we do not know how fast this specific part of trajectory is swept by the flow. (Is my understanding correct?)
2)
Second question: What are the constraints on $\alpha_x,\beta_x$? Should they be the upper and lower bound of the maximal interval or $(\alpha_x,\beta_x)$ is just a subset of the maximal interval which might not have $0$?
3)
Suppose $w \in \Gamma _x$, prove that $\Gamma _w = \Gamma _x$.
My try for $(3)$:
If I have understood $\Gamma_x$ correctly, it seems kind of obvious because when $w \in \Gamma _x$ means that $w$ can take all they value of $\phi_t(x)$. However, how can we show it rigorously.