omega limit set

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

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

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

单词	OmegaLimitSet1
释义	omega limit set Let $\\Phi(t,x)$ be the flow of the differential equation $x^{\\prime}=f(x)$ , where $f\\in C^{k}(M,\\mathbb{R}^{n})$ , with $k\\geq 1$ and $M$ an open subset of $\\mathbb{R}^{n}$ .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 $t_{n}\\to\\infty$ with $\\Phi(t_{n},x)=y$ . Similarly, the alpha limit set of $x$ , denoted $\\alpha(x)$ , is the set of points $y\\in M$ such that there exists a sequence $t_{n}\\to-\\infty$ with $\\Phi(t_{n},x)=y$ . Note that the definition is the same for more general dynamical systems.
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