Stable Manifold Theorem
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Let E be an open subset of , which contains the origin, let , and let φt be the flow of the nonliear system . Suppose and that 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 Es of the linear system at such that for all , and for all , ; and there exists an n − k dimensional differentiable manifold U tangent to the unstable subspace Euof the linear system at such that for all , and for all ,
References
Differential Equations and Dynamical Systems : Lawrence Perko, page 109, theorem 2, third edition