stable manifold theorem

Let E be an open subset of n containing the origin, let fC1(E), and let ϕt be the flow of the nonlinear system x=f(x).

Suppose that f(x0)=0 and that Df(x0) 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 ES of the linear system x=Df(x)x at x0 such that for all t0, ϕt(S)S and for all yS,


and there exists an n-k dimensional differentiable manifold U tangent to the unstable subspaceMathworldPlanetmath EU of x=Df(x)x at x0 such that for all t0, ϕt(U)U and for all yU,

Title stable manifold theorem
Canonical name StableManifoldTheorem
Date of creation 2013-03-22 12:57:17
Last modified on 2013-03-22 12:57:17
Owner jarino (552)
Last modified by jarino (552)
Numerical id 4
Author jarino (552)
Entry type Theorem
Classification msc 34C99