I have a general question about local center manifolds at fixed points. In particular trying reduce a dynamical system to its center manifold. I have been reading Perko and wiggins.
Wiggins gives a few examples of planar systems with only complex conjugate eigenvalues, with zero real part. In these cases I have deduced that the center manifold has dimension 2 and is equal to the center subspace of the corresponding linear system, by using the center manifold theorem. In this case how would you write an expression for the center manifold as it is parameterized by points on the center subspace?
Also, in the vicinity of the fixed point, could you ignore the nonlinear terms and just use the linearized system to find the dynamics on the local center manifold?
Finally, You cannot solve the quasi-linear PDE as their are no stable or unstable subspace. Is there another method???