TITLE: Orbit , trajectory, dynamical system
QUESTION [6 upvotes]: The orbit of $φ$ through $x_0$ is the set $O(x_0) \equiv \{φ_t(x_0) : −∞ < t < ∞\}$. This is also called the trajectory through $x_0$. Then, what is the difference between an orbit and a trajectory?

REPLY [7 votes]: I admit this might be late, but for future reference:
An orbit is the set of points of the manifold as you defined above. A trajectory is a function which has the orbit as image set.