Homoclinic 2003-07-29 11:42:48 2003-07-29 12:03:17 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Per}{\operatorname{Per}} If $X$ is a topological space and $f$ is a flow on $X$ or an homeomorphism mapping $X$ to itself, we say that $x\in X$ is an homoclinic point (or homoclinic intersection) if it belongs to both the stable and unstable sets of some fixed or periodic point $p$; i.e. $$x\in W^s(f,p)\cap W^u(f,p).$$ The orbit of an homoclinic point is called an homoclinic orbit.