- Center manifold
Let

:$dot\{\; extbf\{x\; =\; f(\; extbf\{x\})$

be a

dynamical system withequilibrium point ::$extbf\{x\}^\{*\}\; =\; 0$

The linearization of the system at the equilibrium point is:

:$dot\{\; extbf\{x\; =\; A\; extbf\{x\}$

The linearized system has the following sets of eigenspaces, which are

invariant subspace s of $dot\{\; extbf\{x\; =\; A\; extbf\{x\}$:$extbf\{E\}\_\{s\}$

**: set of stable eigenspaces**which is defined by the eigenvectors corresponding to the eigenvalues $extbf\{Re\}(lambda\_\{i\}(A))<0$

$extbf\{E\}\_\{u\}$**: set of unstable eigenspaces**which is defined by the eigenvectors corresponding to the eigenvalues $extbf\{Re\}(lambda\_\{i\}(A))>0$

$extbf\{E\}\_\{c\}$**: center eigenspace**which is defined by the eigenvectors corresponding to the eigenvalue $extbf\{Re\}(lambda\_\{i\}(A))=0$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 center eigenspace**$extbf\{E\}\_\{c\}$**.****Center manifold and the analysis of nonlinear systems**As the stability of the equilibrium correlates with the "stability" of its manifolds, the existence of a center manifold brings up the question about the dynamics on the center manifold. This is analyzed by the

Center manifold reduction , which, in combination with some system-parameter μ, leads to the concepts of bifurcations.**ee also***

Stable manifold

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