What is the defininition of phase Space in simple terms? I’m currently studying a Differential Equations Course, and I’m focusing on a chapter called “Systems of ODE”, the text gives a definition of phase space, but I can’t seem to visualise or understand what it is graphically. Could someone break it down?
Here is the definition of phase space I’ve been given:

For an autonomous $N$-th order ODE system $dx/dt = f(x)$, $x$, $f \in \mathbb R^N$, the phase space is $\text{dom }(f) \subset \mathbb R^N$, that is, the sub-set of the $N$-dimensional space, where the right-hand sides of the system are defined.

Thank you
 A: As it has been presented by you, the phase space $D$ is the domain of $f$. Usually is an open subset of $\mathbb{R}^N$.
I interpret it in the following way. Every point in $D$ is a state of a dynamical system. Suppose you want to study a simulation of a physical process. You select $N$ variables to be tracked during the simulation and all the possible combinations of admisible values constitute the set of all states: the phase space $D$.
Then you have to write down a dynamical law for this system, which says how the variables change with time: this is the function $f$ in your question. It is an "arrow" in every point of the phase space (state) that tells you in which direction are you moving from this state when times goes on.
Example:
Imagine two variables $R,F$ depending on time, representing the amount of rabbits and foxes in a forest. All the possible combinations of values constitute the phase space, in this case
$$
D=\{(x,y)\in R^2:x\geq 0,y\geq 0\}
$$
You can consider a dynamical law (the Lotka-Volterra equations, for example) that tells you how the system change.
I want to put an image but the server is down. Here you are a link.
