Center Manifold Theory - Computing Center Manifolds

This lecture develops center manifold theory for continuous dynamical systems with stable and center directions. It computes local manifolds using Taylor expansions and tangency conditions, then analyzes the vector field restricted to them through two- and three-dimensional examples.

Syllabus

► Jump to center manifold theory computations: .
Center Manifold Theory introduction.
Motivation from linear vector fields with block diagonal matrix D=diag{A,B} where A has only eigenvalues of zero real part and B is a matrix having only eigenvalues of negative real part. We need to focus on exp(A*t) to know the stability of the equilibrium..
Nonlinear case, expanding about an equilibrium point. Need to know the nonlinear vector field along the center manifold..
Center manifold theory computation.
Approximate the center manifold locally as a function and do a Taylor series expansion to obtain it.
Vector field on the center manifold.
the tangency condition, main computational 'workhouse'.
2D example: two-dimensional system where stability of the origin is not obvious.
Why not do a tangent space (Galerkin) approximation for center manifold dynamics?.
3D example with 2D center manifold.