  This work provides a systematic recipe for computing accurate high order
Fourier expansions of quasiperiodic invariant circles in area preserving maps.
The recipe requires only a finite data set sampled from the quasiperiodic
circle. Our approach, being based on the parameterization method, uses a Newton
scheme to iteratively solve a conjugacy equation describing the invariant
circle. A critical step in properly formulating the conjugacy equation is to
determine the rotation number of the quasiperiodic subsystem. For this we
exploit a the weighted Birkhoff averaging method. This approach facilities
accurate computation of the rotation number given nothing but the already
mentioned orbit data.
  The weighted Birkhoff averages also facilitate the computation of other
integral observables like Fourier coefficients of the parameterization of the
invariant circle. Since the parameterization method is based on a Newton
scheme, we only need to approximate a small number of Fourier coefficients with
low accuracy to find a good enough initial approximation so that Newton
converges. Moreover, the Fourier coefficients may be computed independently, so
we can sample the higher modes to guess the decay rate of the Fourier
coefficients. This allows us to choose, a-priori, an appropriate number of
modes in the truncation. We illustrate the utility of the approach for explicit
example systems including the area preserving Henon map and the standard map.
We present example computations for invariant circles with period as low as 1
and up to more than 100. We also employ a numerical continuation scheme to
compute large numbers of quasiperiodic circles in these systems. During the
continuation we monitor the Sobolev norm of the Parameterization to
automatically detect the breakdown of the family.
