Center manifold

Let

${\displaystyle {\dot {\textbf {x}}}=f({\textbf {x}})}$

be a dynamical system with equilibrium point:

${\displaystyle {\textbf {x}}^{*}=0}$

The linearization of the system at the equilibrium point is:

${\displaystyle {\dot {\textbf {x}}}=A{\textbf {x}}}$

The linearized system has the following sets of eigenspaces, which are invariant subspaces of ${\displaystyle {\dot {\textbf {x}}}=A{\textbf {x}}}$:

${\displaystyle {\textbf {E}}_{s}}$  : set of stable eigenspaces which is defined by the eigenvectors corresponding to the eigenvalues ${\displaystyle {\textbf {Re}}(\lambda _{i}(A))<0}$
${\displaystyle {\textbf {E}}_{u}}$  : set of unstable eigenspaces which is defined by the eigenvectors corresponding to the eigenvalues ${\displaystyle {\textbf {Re}}(\lambda _{i}(A))>0}$
${\displaystyle {\textbf {E}}_{c}}$  : set of center eigenspaces which is defined by the eigenvectors corresponding to the eigenvalues ${\displaystyle {\textbf {Re}}(\lambda _{i}(A))=0}$

Corresponding to the linearized system, the nonlinear system has invariant manifolds, which are some kind of "invariant subspaces" for nonlinear systems. These invariant manifolds are tangent to the eigenspaces at the equilibrium point.

Now the center manifold is the invariant subspace which is tangent to the set of center eigenspaces.