Limit set

In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system.
Types.
In general, limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all nonempty, compact formula_1-limit sets that contain at most finitely many fixed points as a fixed point, a periodic orbit, or a union of fixed points and homoclinic or heteroclinic orbits connecting those fixed points.
Definition for iterated functions.
Let formula_2 be a metric space, and let formula_3 be a continuous function. The formula_1-limit set of formula_5, denoted by formula_6, is the set of cluster points of the forward orbit formula_7 of the iterated function formula_8. Hence, formula_9 if and only if there is a strictly increasing sequence of natural numbers formula_10 such that formula_11 as formula_12. Another way to express this is
where formula_14 denotes the "closure" of set formula_15. The closure is here needed, since we have not assumed that the underlying metric space of interest to be a complete metric space. The points in the limit set are non-wandering (but may not be recurrent points). This may also be formulated as the outer limit (limsup) of a sequence of sets, such that 
If formula_8 is a homeomorphism (that is, a bicontinuous bijection), then the formula_18-limit set is defined in a similar fashion, but for the backward orbit; "i.e." formula_19. 
Both sets are formula_8-invariant, and if formula_2 is compact, they are compact and nonempty.
Definition for flows.
Given a real dynamical system ("T", "X", φ) with flow formula_22, a point "x", we call a point "y" an ω-limit point of "x" if there exists a sequence formula_23 in R so that
For an orbit γ of ("T", "X", φ), we say that "y" is an ω-limit point of γ, if it is an ω-limit point of some point on the orbit. 
Analogously we call "y" an α-limit point of "x" if there exists a sequence formula_23 in R so that
For an orbit γ of ("T", "X", φ), we say that "y" is an α-limit point of γ, if it is an α-limit point of some point on the orbit. 
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 limω γ (limα γ).
If the ω-limit set (α-limit set) is disjoint from the orbit γ, that is limω γ ∩ γ = ∅ (limα γ ∩ γ = ∅), we call limω γ (limα γ) a ω-limit cycle (α-limit cycle). 
Alternatively the limit sets can be defined as
and 