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.

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 possible limit sets as a union of fixed points and periodic orbits.

Let ${\displaystyle X}$ be a metric space, and let ${\displaystyle f:X\rightarrow X}$ be a continuous function. The ${\displaystyle \omega }$-limit set of ${\displaystyle x\in X}$, denoted by ${\displaystyle \omega (x,f)}$, is the set of cluster points of the forward orbit ${\displaystyle \{f^{n}(x)\}_{n\in \mathbb {N} }}$ of the iterated function ${\displaystyle f}$. Hence, ${\displaystyle y\in \omega (x,f)}$ if and only if there is a strictly increasing sequence of natural numbers ${\displaystyle \{n_{k}\}_{k\in \mathbb {N} }}$ such that ${\displaystyle f^{n_{k}}(x)\rightarrow y}$ as ${\displaystyle k\rightarrow \infty }$. Another way to express this is

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