Phase diagram for empirical data The behaviour of partial or ordinary differential equations can be studied/visualized with phase diagrams. How would one plot such diagrams for empirical data, which are suspected to be governed by differential equations?
Suppose I have e.g. some time-series data. Should I plot the original series $x_n$ against first differences $\dot{x} = x_{n+1}-x_{n}$, or do something else?
 A: I assume that with "phase diagram" you mean the "phase portrait" of your system (the plot of typical trajectories in the "state space").
Therefore, if the state space of your system is the space of the points $(x,\dot{x})$, then the answer to your question is YES.
Since you have $(t^q_i,x^q_i)$, where $q$ is the label of a specific trajectory, you have to plot (for all the $q$ trajectories) the points $(x^q_i,v^q_i)$, where 
$$
v_i^q = f( ...x_{i-1}^q,x_i^q,x_{i+1}^q... ; ...t_{i-1}^q,t_i^q,t_{i+1}^q... )
$$
is a certain discrete approximation of the velocity along the trajectory $q$. There are some dots, indicating that the number of arguments depends on the "stencil" you want to use to calculate the derivative.
For example, simple possibilities are:
$$
v_i^q \approx \frac{x_{i+1}^q-x_{i}^q}{t_{i+1}^q-t_{i}^q} \approx \frac{x_{i}^q-x_{i-1}^q}{t_{i}^q-t_{i-1}^q} \approx \frac{x_{i+1}^q-x_{i-1}^q}{t_{i+1}^q-t_{i-1}^q}
$$
Of course you can consider higher-order methods to calculate the derivative (this is why in the definition of $f$ there are some dots).
