- 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))=0Corresponding 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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