omega limit set

Let $\mathrm{\Phi }(t,x)$ be the flow of the differential equation $x^{\prime }=f(x)$, where $f\in C^k(M,{ℝ}^n)$, with $k\ge 1$ and $M$ an open subset of ${ℝ}^n$. Consider $x\in M$.

The omega limit set of $x$, denoted $\omega (x)$, is the set of points $y\in M$ such that there exists a sequence $t_n\to \mathrm{\infty }$ with $\mathrm{\Phi }(t_n,x)=y$.

Similarly, the alpha limit set of $x$, denoted $\alpha (x)$, is the set of points $y\in M$ such that there exists a sequence $t_n\to -\mathrm{\infty }$ with $\mathrm{\Phi }(t_n,x)=y$.

Note that the definition is the same for more general dynamical systems.

Title	omega limit set
Canonical name	OmegaLimitSet
Date of creation	2013-03-22 13:18:42
Last modified on	2013-03-22 13:18:42
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	5
Author	mathcam (2727)
Entry type	Definition
Classification	msc 37B99
Classification	msc 34C05
Synonym	$\omega$-limit set
Synonym	$\alpha$-limit set
Related topic	LimitCycle
Defines	alpha limit set