Limit Set - Definition For Flows

Definition For Flows

Given a real dynamical system (T, X, φ) with flow, a point x and an orbit γ through x, we call a point y an ω-limit point of γ if there exists a sequence in R so that

.

Analogously we call y an α-limit point if there exists a sequence in R so that

.

The set of all ω-limit points (α-limit points) for a given orbit γ is called ω-limit set (α-limit set) for γ and denoted lim_ω γ (lim_α γ).

If the ω-limit set (α-limit set) is disjunct from the orbit γ, that is lim_ω γ ∩ γ = ∅ (lim_α γ ∩ γ = ∅), we call lim_ω γ (lim_α γ) a ω-limit cycle (α-limit cycle).

Alternatively the limit sets can be defined as

and