Abstract
Let \(F:{\mathbb R}^n\rightarrow {\mathbb R}^n\) be a \(C^1\) vector field with \(F(0)=0\). A center manifold for \(F\) at \(0\) is an invariant manifold containing \(0\) which is tangent to and of the same dimension as the center subspace of \(DF(0)\).
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Notes
- 1.
We use the Contraction Principle rather than the Implicit Function Theorem because this gives a solution for every \(y_0\in E_c\), thanks to the fact that \(M=\Vert f\Vert _{C^1}\) is small.
- 2.
The dimension of \({\mathcal H}^n_r\) is \(n\left( \begin{array}{c} r+n-1\\ r\end{array}\right) \).
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Sideris, T.C. (2013). Center Manifolds and Bifurcation Theory. In: Ordinary Differential Equations and Dynamical Systems. Atlantis Studies in Differential Equations, vol 2. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-021-8_9
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DOI: https://doi.org/10.2991/978-94-6239-021-8_9
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Publisher Name: Atlantis Press, Paris
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Online ISBN: 978-94-6239-021-8
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