stylistic:tone
Limit set of a trajectory

This notion is used in dynamical systems, to study limits of trajectories. Defining a trajectory to be a function γ: R → X {\displaystyle \gamma :\mathbb {R} \rightarrow X}, the point γ (t) {\displaystyle \gamma (t)} is thought of as the "position" of the trajectory at "time" t {\displaystyle t}. The limit set of a trajectory is defined as follows. To any sequence of increasing times { t n } {\displaystyle \{t_{n}\}}, there is an associated sequence of positions { x n } = { γ (t n) } {\displaystyle \{x_{n}\}=\{\gamma (t_{n})\}}. If x {\displaystyle x} is the limit set of the sequence { x n } {\displaystyle \{x_{n}\}} for any sequence of increasing times, then x {\displaystyle x} is a limit set of the trajectory.

Technically, this is the ω {\displaystyle \omega } -limit set. The corresponding limit set for sequences of decreasing time is called the α {\displaystyle \alpha } -limit set.

An illustrative example is the circle trajectory: γ (t) = (cos (t), sin (t)) {\displaystyle \gamma (t)=(\cos(t),\sin(t))}. This has no unique limit, but for each θ ∈ R {\displaystyle \theta \in \mathbb {R} }, the point (cos (θ), sin (θ)) {\displaystyle (\cos(\theta),\sin(\theta))} is a limit point, given by the sequence of times t n = θ + 2 π n {\displaystyle t_{n}=\theta +2\pi n}. But the limit points need not be attained on the trajectory. The trajectory γ (t) = t / (1 + t) (cos (t), sin (t)) {\displaystyle \gamma (t)=t/(1+t)(\cos(t),\sin(t))} also has the unit circle as its limit set.