PlanetMath: omega limit set

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omega limit set

(Definition)

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

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

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

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

"omega limit set" is owned by mathcam. [ full author list (2) | owner history (1) ]

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See Also: limit cycle

Other names:

$\omega$ -limit set, $\alpha$ -limit set

Also defines:

alpha limit set

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Cross-references: dynamical systems, sequence, points, open subset, differential equation, flow
There are 3 references to this entry.

This is version 2 of omega limit set, born on 2002-12-23, modified 2005-03-25.
Object id is 3818, canonical name is OmegaLimitSet.
Accessed 6971 times total.

Classification:

AMS MSC:	34C05 (Ordinary differential equations :: Qualitative theory :: Location of integral curves, singular points, limit cycles)
	37B99 (Dynamical systems and ergodic theory :: Topological dynamics :: Miscellaneous)

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