Center Manifold Theory - Computing Center Manifolds

Delve into the intricacies of center manifold theory for continuous dynamical systems in this comprehensive lecture. Explore the Taylor series approximation of center manifolds and analyze the dynamics restricted to these manifolds to determine equilibrium point stability. Examine a 2D example illustrating the limitations of center subspace approximation, and study a 3D example requiring linear coordinate transformation. Learn to compute center manifolds, understand the tangency condition, and grasp why Galerkin approximations may fail. Gain valuable insights into nonlinear dynamics, bifurcations, and the analysis of equilibrium points with stable and center directions in ordinary differential equations.

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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.