Efficient Computation of Invariant Tori in Volume-Preserving Maps

Abstract

In this paper we implement a numerical algorithm to compute codimension-one tori in three-dimensional, volume-preserving maps. A torus is defined by its conjugacy to rigid rotation, which is in turn given by its Fourier series. The algorithm employs a quasi-Newton scheme to find the Fourier coefficients of a truncation of the series. This technique is based upon the theory developed in the accompanying article by Blass and de la Llave. It is guaranteed to converge assuming the torus exists, the initial estimate is suitably close, and the map satisfies certain nondegeneracy conditions. We demonstrate that the growth of the largest singular value of the derivative of the conjugacy predicts the threshold for the destruction of the torus. We use these singular values to examine the mechanics of the breakup of the tori, making comparisons to Aubry-Mather and anti-integrability theory when possible.

References in corpus (5)

• A lecture on the classical KAM theorem
• Renormalization and destruction of $1/{\gamma }^2$ tori in the standard nontwist map
• Breakup of Shearless Meanders and "Outer" Tori in the Standard Nontwist Map
• Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps
• The Destruction of Tori in Volume-Preserving Maps

Cited by in corpus (5)

• Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps
• Diffusion and Drift in Volume-Preserving Maps
• Barriers to Transport and Mixing in Volume-Preserving Maps with Nonzero Flux
• A new method to compute periodic orbits in general symplectic maps
• Nonlinear Transport in the Stochastic Standard Map