Limit set

Types

• fixed points
• periodic orbits
• limit cycles
• attractors

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 \omega-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 X be a metric space, and let f:X\rightarrow X be a continuous function. The \omega-limit set of x\in X, denoted by \omega(x,f), is the set of cluster points of the forward orbit {f^n(x)}{n\in \mathbb{N}} of the iterated function f. 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): kn}},

where \overline{S} denotes the closure of set S. 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

:\omega(x,f) = \bigcap_{n=1}^\infty \overline{\bigcup_{k=n}^\infty {f^k(x)}}.

If f is a homeomorphism (that is, a bicontinuous bijection), then the \alpha-limit set is defined in a similar fashion, but for the backward orbit; i.e. \alpha(x,f)=\omega(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,\varphi) with flow \varphi:\mathbb{R}\times X\to X, a point x, we call a point y an \omega-limit point of x if there exists a sequence (t_n){n \in \mathbb{N}} in \mathbb{R} so that :\lim{n \to \infty} t_n = \infty :\lim_{n \to \infty} \varphi(t_n, x) = y .

For an orbit \gamma of (T,X,\varphi), we say that y is an \omega-limit point of \gamma, if it is an \omega-limit point of some point on the orbit.

Analogously we call y an \alpha-limit point of x if there exists a sequence (t_n){n \in \mathbb{N}} in \mathbb{R} so that :\lim{n \to \infty} t_n = -\infty :\lim_{n \to \infty} \varphi(t_n, x) = y .

For an orbit \gamma of (T,X,\varphi), we say that y is an \alpha-limit point of \gamma, if it is an \alpha-limit point of some point on the orbit.

The set of all \omega-limit points (\alpha-limit points) for a given orbit \gamma is called \omega-limit set (\alpha-limit set) for \gamma and denoted \lim_{\omega} \gamma (\lim_{\alpha} \gamma).

If the \omega-limit set (\alpha-limit set) is disjoint from the orbit \gamma, that is \lim_{\omega} \gamma \cap \gamma =\varnothing (\lim_{\alpha} \gamma \cap \gamma =\varnothing), we call \lim_{\omega} \gamma (\lim_{\alpha} \gamma) a ω-limit cycle (α-limit cycle).

Alternatively the limit sets can be defined as :\lim_\omega \gamma := \bigcap_{s\in \mathbb{R}}\overline{{\varphi(x,t):ts}} and :\lim_\alpha \gamma := \bigcap_{s\in \mathbb{R}}\overline{{\varphi(x,t):t

Examples

• For any periodic orbit \gamma of a dynamical system, \lim_{\omega} \gamma =\lim_{\alpha} \gamma =\gamma
• For any fixed point x_0 of a dynamical system, \lim_{\omega} x_0 =\lim_{\alpha} x_0 =x_0

Properties

• \lim_{\omega} \gamma and \lim_{\alpha} \gamma are closed
• if X is compact then \lim_{\omega} \gamma and \lim_{\alpha} \gamma are nonempty, compact and connected
• \lim_{\omega} \gamma and \lim_{\alpha} \gamma are \varphi-invariant, that is \varphi(\mathbb{R}\times\lim_{\omega} \gamma)=\lim_{\omega} \gamma and \varphi(\mathbb{R}\times\lim_{\alpha} \gamma)=\lim_{\alpha} \gamma

References

References

1. (1996). "Chaos, an introduction to dynamical systems". Springer.