Weighted Birkhoff Averages and the Parameterization Method

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 1766-1804, September 2024.
Abstract. This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles (and systems of such 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 of [A. Haro and R. de la Llave, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 142–207; A. Haro and R. de la Llave, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261–1300; A. Haro and R. de la Llave, J. Differential Equations, 228 (2006), pp. 530–579], uses a Newton scheme to iteratively solve a conjugacy equation describing the invariant circle (or systems of circles). A critical step in properly formulating the conjugacy equation is to determine the rotation number of the quasiperiodic subsystem. For this we exploit the weighted Birkhoff averaging method of [S. Das et al., Nonlinearity, 30 (2017), pp. 4111–4140; S. Das et al., The Foundations of Chaos Revisited, Springer, Cham, 2016, pp. 103–118; S. Das and J. A. Yorke, Nonlinearity, 31 (2018), pp. 491–501]. 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 (say, a few correct digits) 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 Hénon map and the standard map (polynomial and trigonometric nonlinearity respectively). We present example computations for invariant circles and for systems of invariant circles with as many as 120 components. We also employ a numerical continuation scheme (where the rotation number is the continuation parameter) to compute large numbers of quasiperiodic circles in these systems. During the continuation we monitor the Sobolev norm of the parameterization, as explained in [R. Calleja and R. de la Llave, Nonlinearity, 23 (2010), pp. 2029–2058], to automatically detect the breakdown of the family.