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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク heteroclinic orbit ： ウィキペディア英語版 heteroclinic orbit In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit. Consider the continuous dynamical system described by the ODE ::$\backslash dot\; x=f(x)$ Suppose there are equilibria at $x=x\_0$ and $x=x\_1$, then a solution $\backslash phi(t)$ is a heteroclinic orbit from $x\_0$ to $x\_1$ if ::$\backslash phi(t)\backslash rightarrow\; x\_0\backslash quad\; \backslash mathrm\backslash quad\; t\backslash rightarrow-\backslash infty$ and ::$\backslash phi(t)\backslash rightarrow\; x\_1\backslash quad\; \backslash mathrm\backslash quad\; t\backslash rightarrow+\backslash infty$ This implies that the orbit is contained in the stable manifold of $x\_1$ and the unstable manifold of $x\_0$. ==Symbolic dynamics== By using the Markov partition, the long-time behaviour of hyperbolic system can be studied using the techniques of symbolic dynamics. In this case, a heteroclinic orbit has a particularly simple and clear representation. Suppose that $S=\backslash$ is a finite set of ''M'' symbols. The dynamics of a point ''x'' is then represented by a bi-infinite string of symbols :$\backslash sigma\; =\backslash \; \backslash \}$ A periodic point of the system is simply a recurring sequence of letters. A heteroclinic orbit is then the joining of two distinct periodic orbits. It may be written as :$p^\backslash omega\; s\_1\; s\_2\; \backslash cdots\; s\_n\; q^\backslash omega$ where $p=\; t\_1\; t\_2\; \backslash cdots\; t\_k$ is a sequence of symbols of length ''k'', (of course, $t\_i\backslash in\; S$), and $q\; =\; r\_1\; r\_2\; \backslash cdots\; r\_m$ is another sequence of symbols, of length ''m'' (likewise, $r\_i\backslash in\; S$). The notation $p^\backslash omega$ simply denotes the repetition of ''p'' an infinite number of times. Thus, a heteroclinic orbit can be understood as the transition from one periodic orbit to another. By contrast, a homoclinic orbit can be written as :$p^\backslash omega\; s\_1\; s\_2\; \backslash cdots\; s\_n\; p^\backslash omega$ with the intermediate sequence $s\_1\; s\_2\; \backslash cdots\; s\_n$ being non-empty, and, of course, not being ''p'', as otherwise, the orbit would simply be $p^\backslash omega$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「heteroclinic orbit」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.