Chebyshev Spectral Methods For Computing Center Manifolds

We propose a numerical approach for computing center manifolds of equilibria in ordinary differential equations. Near the equilibria, the center manifolds are represented as graphs of functions satisfying certain partial differential equations (PDEs). We use a Chebyshev spectral method for solving the PDEs numerically to compute the center manifolds. We illustrate our approach for three examples: A two-dimensional system, the Henon-Heiles system (a two-degree-of-freedom Hamiltonian system) and a three-degree-of freedom Hamiltonian system which have one-, two-and four-dimensional center manifolds, respectively. The obtained results are compared with polynomial approximations and other numerical computations.