Stable Manifold Theorem

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Latest revision as of 03:56, 28 May 2008

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Let $E$ be an open subset of $\mathbb{R}^{n}$ , which contains the origin, let $\textbf{f}\in C^{1}(E)$ , and let $φ t$ be the flow of the nonliear system $\dot{\textbf{x}} = \textbf{f}(\textbf{x})$ . Suppose $\textbf{f}(\textbf{0})=\textbf{0}$ and that $D\textbf{f}(\textbf{0})$ has $k$ eigenvalues with negative real part and $n - k$ eigenvalues with positive real part. Then there exists a $k -$ dimensional differentiable manifold $S$ tangent to the sable subspace $E s$ of the linear system $\dot{\textbf{x}} =\textbf{Ax}$ at $\textbf{0}$ such that for all $t\geq 0$ , $\phi_{t}(S) \sub S$ and for all $\textbf{x}_{0} \in S$ , $\lim_{t\to\infty} \phi_{t}(\textbf{x}_{0}) = \textbf{0}$ ; and there exists an $n - k$ dimensional differentiable manifold $U$ tangent to the unstable subspace $E u$ of the linear system at $\textbf{0}$ such that for all $t\leq 0$ , $\phi_{t}(U)\sub U$ and for all $\textbf{x}_{0}\in U$ , $\lim_{t\to -\infty} \phi_{t}(\textbf{x}_{0})=\textbf{0}$

References

Differential Equations and Dynamical Systems : Lawrence Perko, page 109, theorem 2, third edition

@@ Line 4: / Line 4: @@
 {{End Hierarchy}}
-Let $E$ be an open subset of $\mathbb{R}^n$ containing the origin, let
+Let <math>E</math> be an open subset of <math>\mathbb{R}^{n}</math>, which contains the origin, let <math>\textbf{f}\in C^{1}(E)</math>, and let <math>\phi_{t} </math> be the flow of the nonliear system <math>\dot{\textbf{x}} = \textbf{f}(\textbf{x})</math>. Suppose <math>\textbf{f}(\textbf{0})=\textbf{0}</math> and that <math>D\textbf{f}(\textbf{0})</math> has <math>k</math> eigenvalues with negative real part and <math>n-k</math> eigenvalues with positive real part. Then there exists a <math>k-</math>dimensional differentiable manifold <math>S</math> tangent to the sable subspace <math>E^{s}</math> of the linear system <math>\dot{\textbf{x}} =\textbf{Ax}</math> at <math>\textbf{0}</math>such that for all <math>t\geq 0</math>, <math>\phi_{t}(S) \sub S</math> and for all <math>\textbf{x}_{0} \in S</math>, <math>\lim_{t\to\infty} \phi_{t}(\textbf{x}_{0}) = \textbf{0}</math>; and there exists an <math>n-k</math> dimensional differentiable manifold <math>U</math> tangent to the unstable subspace <math>E^{u}</math>of the linear system at <math>\textbf{0}</math> such that for all <math>t\leq 0</math>, <math>\phi_{t}(U)\sub U</math> and for all <math>\textbf{x}_{0}\in U</math>, <math>\lim_{t\to -\infty} \phi_{t}(\textbf{x}_{0})=\textbf{0}</math>
-$f\in C^1(E)$, and let $\phi_t$ be the flow of the nonlinear system $x'=f(x)$.
-Suppose that $f(x_0)=0$ and that $Df(x_0)$ has $k$ eigenvalues with negative real part and $n-k$ eigenvalues with positive real part. Then there exists a $k$-dimensional differentiable manifold $S$ tangent to the stable subspace $E^S$ of the linear system $x'=Df(x)x$ at $x_0$ such that for all $t\geq 0$, $\phi_t(S)\subset S$ and for all $y\in S$,
-\[
+==References==
-\lim_{t\to\infty}\phi_t(y)=x_0
+[http://books.google.com/books?id=A7fvvz9Puf8C&dq=differential+equations+and+dynamical+systems+perko&pg=PP1&ots=_DkccInx4B&sig=K27_M_77UOxgOjIJp41B-zz__4I&hl=en&prev=http://www.google.com/search%3Fhl%3Den%26client%3Dsafari%26rls%3Den%26q%3Ddifferential%2Bequations%2Band%2Bdynamical%2Bsystems%2BPerko%26btnG%3DSearch&sa=X&oi=print&ct=title&cad=one-book-with-thumbnail Differential Equations and Dynamical Systems] : Lawrence Perko, page 109, theorem 2, third edition
-\]
-and there exists an $n-k$ dimensional differentiable manifold $U$ tangent to the unstable subspace $E^U$ of $x'=Df(x)x$ at $x_0$ such that for all
-$t\leq 0$, $\phi_t(U)\subset U$ and for all $y\in U$,
-\[
-\lim_{t\to -\infty}\phi_t(y)=x_0.
-\]

Stable Manifold Theorem

Latest revision as of 03:56, 28 May 2008

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