What does $\dot{x}_1=x_2$ mean? If I have a three dimensional ODE for $x_1,x_2,x_3$ and for the first equation I have
$\dot{x}_1=x_2$
what does this mean?
That the change of $x_1$ is in the direction of $x_2$?
Background: 
In this paper it is said on page 436 (Case I.) that since $u'=v$, we have that $S_{\epsilon}\cap M_{\theta}$ has only one common point. Why does this hold?
 A: (I'm using primes instead of dots, since MathJax renders dots so lightly in my browser.) Suppose generally that your system has the form
\begin{align*}
x'_{1} &= F_{1}(x_{1}, x_{2}, x_{3}), \\
x'_{2} &= F_{2}(x_{1}, x_{2}, x_{3}), \\
x'_{3} &= F_{3}(x_{1}, x_{2}, x_{3}).
\end{align*}
Geometrically, this system defines a vector field, an association of the vector
$$
\mathbf{F}(x_{1}, x_{2}, x_{3})
  = \bigl(F_{1}(x_{1}, x_{2}, x_{3}), F_{2}(x_{1}, x_{2}, x_{3}), F_{3}(x_{1}, x_{2}, x_{3})\bigr)
$$
to each point of space (or to each point in the region where the ODE is defined).
A soluton of the ODE system, a.k.a., a flow line of the vector field $\mathbf{F}$, is a path $\mathbf{x}(t)$ (often interpreted as the position of a point particle at time $t$) whose velocity at each instant is the value of the vector field at the particle's location:
$$
\mathbf{x}'(t) = \mathbf{F}\bigl(\mathbf{x}(t)\bigr).
$$
The equation $x'_{1} = x_{2}$ says the $x_{1}$-component of the velocity of an arbitrary solution is equal to the $x_{2}$-position of the particle at each instant. (If $x_{2} \neq 0$, the particle therefore crosses each plane $x_{1} = \text{const}$ at most once; haven't read your linked article carefully enough to know if that's the gist of the argument, however.)
In my experience, extracting any satisfying intuition from this piece of qualitative information is tricky: As the particle moves, its position changes, which causes the velocity to change, which affects the position...in an endless cycle of feedback. (Based on other questions you've asked, I hope it's not presumptuous to suspect you're trying to overcome a similar issue.)
Instead, plot the vector field $\mathbf{F}$ (or imagine the vector field $\mathbf{F}$ existing throughout space as a static piece of information), then think of a solution as the trajectory of a point particle that "follows the flow".
Here are a couple of (possibly CPU-intensive) interactive visualization links that run in a web browser:


*

*A wind map of the earth, using (more or less) real-time data;

*The Lorenz differential equation.
A: If $x_1$ is a single variable function that depends on $t$, we have
$$\dot{x_1}=\frac{d}{dt}x_1=x_2$$
If $x_1$ is a multivariable function with that depends on $t$, we have
$$\dot{x_1}=\frac{\partial}{\partial t}x_1=x_2$$
Usually Newton's notation for differentiation is used to denote derivatives with respect to time. So if $x_1$ represents displacement, then $x_2$ is the instantaneous velocity at any time $t$. Until you and more context to your post, I cannot assist any further.
A: First off we have three function $x_1$, $x_2$ and $x_3$.
Since the dot notation is used it is safe to assume that these are functions of time, i.e. $x_i$ could be written more obviously as $x_i(t)$, for $i = 1,2,3$. Note that the meaning of $\dot{x}$ is normally $\frac{dx}{dt}$.
The first equation is then simply saying that the second function is the derivative of the first, or more explicitly $\frac{dx_1}{dt} = x_2$, this might make more sense if we turn it round, so $x_2(t) = \frac{dx_1(t)}{dt}$.
Consider the concrete example where $x_1(t) = 3t^2$, the the above is just saying that $\frac{dx_1}{dt} = 6t = x_2$, so $x_2 = 6t$.
Having seen the question you linked in the comments section the notation of the paper is actually $x_1 ' = x_2$. This notation indicates that we are dealing with a function of one variable and taking the derivative with respect to that variable. Essentially we don't necessarily have a function of time, but the above still all holds, it's just that $t$ would be a variable that doesn't necessarily have a specific meaning attached to it.
In the specific question you cited we do seem to have a function of time. 
