Krdytkyu

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?

My impression is that, in dynamic systems theory, the two terms 'orbit' and 'trajectory' are often used interchangeably. They both seem to come from physics, e.g. orbits of planets, etc. Earlier, I had the impression that 'trajectory' was more commonly used when talking about continuous systems, and that 'orbit' was used for discrete ones. This distinction does not seem to be even nearly universally accepted however, and is probably more of a preference than anything else. Possibly this impression comes from the fact that you use 'orbit' in group theory, when a group is acting on elements of a set, see definition here. It therefore feels somewhat more natural to call the sequence emerging from a discrete system, orbit. Trajectory seems to be somewhat more common among physicists. I get a feeling that this is similar to the distinction (or non-distinction) between 'map' and 'function', see discussion here.

I find it hard to find any real distinction being made explicitly in dynamical systems literature (at least in the many books on my shelf). Three examples of the terms being explicitly used interchangeably by respected authorities are:

1. At the Scholarpedia article Dynamical Systems by Prof. James Meiss, Applied Mathematics University of Colorado, Boulder, CO, USA, under the "Evolution Rule" section, §2.

"The forward orbit or trajectory of a state $s$ is the time-ordered collection of states that follow from $s$ using the evolution rule."

2. In Robert C Hilborn's Chaos and Nonlinear Dynamics, An Introduction for Scientists and Engineers, Second Edition, p.20, §1.

The sequence of $x$-values generated by this iteration procedure will be called the trajectory or orbit in analogy to the sequence of position values for a planet or satellite taken at successive time intervals.

3. In Mario Martelli's Introduction to Discrete Dynamical Systems and Chaos, Definition 1.1.1, p.11, §3.

The evolution of the system starting from $x_0$, is given by the sequence: $(x_0, x_1=F(x_0), x_2=F^2(x_0), ..., F^n(x_o), ...)$.

Definition 1.1.1 [3]. The sequence ${x_0,x_1,ldots, x_n,ldots}$ is denoted by $O(x_0)$ and is called the orbit or trajectory of the system starting from $x_0$.

I want to make clear that I am a mathematician and not a physicist, therefore my impression is very much based on literature about pure and applied mathematics. It might be the case that physicists make more of a distinction between the two, even though I doubt it.

Another, perhaps more important distinction that should be made when talking about orbits/trajectories, is whether one means orbit/trajectory a as defined in 1.1.1, as a sequence, or as a set. Often this does not pose any problems, as it is clear from context. Personally, dealing mostly with discrete systems, I usually write ${f^n}_{ngeq 0}$, with set braces, for the unordered set, which could be finite for a periodic orbit, and $(f^n)_{ngeq 0}$ for the ordered infinite sequence. This is somewhat OT, but nevertheless important to have in mind when reading about these things.

I hope that this either suffice as an answer, or at least makes things a bit more clear.

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.