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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.
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 -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 be a metric space, and let be a continuous function. The -limit set of , denoted by , is the set of cluster points of the forward orbit of the iterated function . Hence, if and only if there is a strictly increasing sequence of natural numbers such that as . Another way to express this is
where denotes the closure of set . 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 is a homeomorphism (that is, a bicontinuous bijection), then the -limit set is defined in a similar fashion, but for the backward orbit; i.e. .
Both sets are -invariant, and if is compact, they are compact and nonempty.
Definition for flows
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 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).
Alternatively the limit sets can be defined as
- For any periodic orbit γ of a dynamical system, limω γ = limα γ = γ
- For any fixed point of a dynamical system, limω = limα =
- limω γ and limα γ are closed
- if X is compact then limω γ and limα γ are nonempty, compact and connected
- limω γ and limα γ are φ-invariant, that is φ(R × limω γ) = limω γ and φ(R × limα γ) = limα γ
- Alligood, Kathleen T.; Sauer, Tim D.; Yorke, James A. (1996). Chaos, an introduction to dynamical systems. Springer.
- Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.