Georgia Institute of Technology College of Sciences

Local invariant manifolds provide a natural way to separate center,
stable, and unstable dynamics near an invariant object and, at the
same time, to construct lower-dimensional models of the dynamics. In
this talk, I will present a unified numerical procedure for computing
high-order local invariant manifolds attached to fixed points and
reducible quasi-periodic invariant tori of Poincaré maps. The
fixed-point setting appears naturally as the zero-dimensional case of
the quasi-periodic formulation.

For a quasi-periodic skew-product, the normal variational dynamics
along the torus defines a linear cocycle over a rigid
rotation. Assuming reducibility, a Floquet transformation converts
this cocycle into a constant matrix, so that a spectral splitting can
be introduced as in the fixed-point case. The desired invariant
manifold is then represented as a graph over a selected spectral
subspace. Expanding this graph in Fourier–Taylor series leads, order
by order, to cohomological equations whose solvability is governed by
the interaction between the base rotation and the Floquet
spectrum. The same construction also provides the nonlinear dynamics
induced on the selected variables, yielding an explicit reduced model
together with a lifting to the full phase space.  

I will illustrate the method first on a (quasi-)periodically forced
pendulum, which provides a simple setting in which to visualize the
response torus, its center reduction, and the dynamics generated by
the reduced map. Time permitting, I will also discuss how the same
ideas arise in (quasi-)periodically forced models from celestial
mechanics.


Joint works with M. Barcelona, L. Domingo, and M. Jorba-Cusco.