Trajectory

Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the flow of the autonomous system $x' = f(x)$. Let $x_0 \in U$, and let $I(x_0)$ be the maximal interval of existence of the solution with $x(0) = x_0$. The trajectory, or orbit, of $x_0$ is the set $$\mathcal{O}(x_0) = \{\varphi_t(x_0) : t \in I(x_0)\} \subseteq U.$$ The positive half-trajectory is $\mathcal{O}^+(x_0) = \{\varphi_t(x_0) : t \in I(x_0),\ t \ge 0\}$, and the negative half-trajectory uses $t \le 0$.

Remarks

Two trajectories are either equal or disjoint. So the trajectories form a partition of $U$. A trajectory is a single point exactly when $x_0$ is an equilibrium point. It is a closed curve when the solution is periodic and not constant. Some books use the word trajectory for the solution curve $t \mapsto \varphi_t(x_0)$ itself, and the word orbit for its image.

Explanations

L1 human

The trajectory is the path that the state follows in the phase space. It forgets the time: it records where the state goes, not when.