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 \(DF(0)\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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 \(n\left( \begin{array}{c} r+n-1\\ r\end{array}\right) \).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 Atlantis Press and the authors
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.2991/978-94-6239-021-8_9
Published:
Publisher Name: Atlantis Press, Paris
Print ISBN: 978-94-6239-020-1
Online ISBN: 978-94-6239-021-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)