Limit set - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position Limit sets are important because they can be used to understand the long term behavior of a dynamical system. Examples of limit sets include fixed points, periodic orbits, limit cycles and attractors. In Mathematics, in the study of Dynamical systems an orbit is a collection of points related by the Evolution function of the dynamical system In Mathematics, in the area of Dynamical systems, a limit-cycle on a plane or a Two-dimensional manifold An attractor is a set to which a Dynamical system evolves after a long enough time

In general limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincare-Bendixson theorem provides a simple characterization of all possible limit sets as a union of fixed points and periodic orbits. An attractor is a set to which a Dynamical system evolves after a long enough time In Mathematics, the Poincaré–Bendixson theorem is a statement about the long term behaviour of orbits of Continuous dynamical systems on the plane

1 Definition for iterated functions
2 Definition for flows
- 2.1 Examples
- 2.2 Properties
3 See also

Definition for iterated functions

Let $X$ be a metric space, and let $f:X\rightarrow X$ be a continuous function. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The $ω$ -limit set of $x\in X$ , denoted by $ω(x, f)$ , is the set of cluster points of the forward orbit $\{f^n(x)\}_{n\in \mathbb{N}}$ of the iterated function $f$ . In Mathematics, iterated functions are the objects of deep study in Computer science, Fractals and Dynamical systems An iterated function is Hence, $y\in \omega(x,f)$ if and only if there is a strictly increasing sequence of natural numbers $\{n_k\}_{k\in \mathbb{N}}$ such that $f^{n_k}(x)\rightarrow y$ as $k\rightarrow\infty$ . ↔ Another way to express this is

$\omega(x,f) = \bigcap_{n\in \mathbb{N}} \overline{\{f^k(x): k>n\}}.$

The points in the limit set are called recurrent points. In Mathematics, a recurrent point for function f is a point that is in the Limit set of the Iterated function f.

If $f$ is a homeomorphism (that is, a bicontinuous bijection), then the $α$ -limit set is defined in a similar fashion, but for the backward orbit; i. Topological equivalence redirects here see also Topological equivalence (dynamical systems. e. $α(x, f) = ω(x, f - 1)$ .

Both sets are $f$ -invariant, and if $X$ is compact, they are compact and nonempty.

Definition for flows

Given a real dynamical system (T, X, φ) with flow $\varphi:\mathbb{R}\times X\to X$ , 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 $(t_n)_{n \in \mathbb{N}}$ in R so that

$\lim_{n \to \infty} t_n = \infty$

$\lim_{n \to \infty} \varphi(t_n, x) = y$ . The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Mathematics, a flow formalizes in mathematical terms the general idea of "a variable that depends on time" that occurs very frequently in Engineering In Mathematics, in the study of Dynamical systems an orbit is a collection of points related by the Evolution function of the dynamical system

Analogously we call y an α-limit point if there exists a sequence $(t_n)_{n \in \mathbb{N}}$ in R so that

$\lim_{n \to \infty} t_n = -\infty$

$\lim_{n \to \infty} \varphi(t_n, x) = y$ .

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). In Mathematics, in the area of Dynamical systems, a limit-cycle on a plane or a Two-dimensional manifold In Mathematics, in the area of Dynamical systems, a limit-cycle on a plane or a Two-dimensional manifold

Alternatively the limit sets can be defined as

$\lim_\omega \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t>n\}}$

and

$\lim_\alpha \gamma := \bigcap_{n\in \mathbb{R}}\overline{\{\varphi(x,t):t<n\}}.$

Examples

For any periodic orbit γ of a dynamical system, lim_ω γ = lim_α = γ

Properties

lim_ω γ and lim_α γ are closed
if X is compact then lim_ω γ and lim_α γ are nonempty, compact and simply connected
lim_ω γ and lim_α γ are φ-invariant, that is φ(R × lim_ω γ) = lim_ω γ and φ(R × lim_α γ) = lim_α γ

This article incorporates material from Omega-limit set on PlanetMath, which is licensed under the GFDL. In Mathematics, in the study of Dynamical systems an orbit is a collection of points related by the Evolution function of the dynamical system In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Complex dynamics, the Julia set J(f\ of a Holomorphic function f\ informally consists of those points whose long-time behavior under In Mathematics, and in particular the study of Dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical In Mathematics, in the area of Dynamical systems, a limit-cycle on a plane or a Two-dimensional manifold In Mathematics, in the study of Iterated functions and Dynamical systems a periodic point of a function is a point which returns to itself after In those branches of Mathematics called Dynamical systems and Ergodic theory, the concept of a wandering set formalizes a certain idea of movement and PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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Contents

Definition for iterated functions

Definition for flows

Examples

Properties

See also