stable manifold theorem
Let E be an open subset of ℝn containing the origin, let f∈C1(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 t≥0, ϕt(S)⊂S and for all y∈S,
lim |
and there exists an dimensional differentiable manifold tangent to the unstable subspace of at such that for all
, and for all ,
Title | stable manifold theorem |
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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 |