# Math Help - Phase Space/ State Space/ Phase Portrait

1. ## Phase Space/ State Space/ Phase Portrait

I am having difficulty understanding the concepts of phase space, state space and phase portraits.

From what I've seen, a phase space is a vector field representing the behavior of points in a dynamical system.

A phase portrait is a set of curves in a phase space corresponding with a given initial condition.

I'm not sure what a state space is but it seems similar, if not the same, to phase space.

An example is given that $\frac{dx}{dt}=\sin{x}$. And I am given this graph corresponding to this equation.

Would this be a graph of the phase space? What would the phase portrait look like then? I'm pretty lost on how to construct a phase space/ phase portrait.

Thank you.

2. According to this definition
Phase space - Wikipedia, the free encyclopedia
"In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901[1], is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space."
In your graph not all possible states are shown but a curve with some initial conditions.

I think that your graph a phase portrait.
The definition of phase portrait is here
Phase portrait - Wikipedia, the free encyclopedia