Cantori for symplectic maps near the anti-integrable limit

Abstract

N-mode truncations of the three-dimensional Navier-Stokes equations with periodic boundary conditions are considered. We show that the solution of truncated model exhibits, under some assumptions concerning the external force, general properties of symmetry and invariance. A particular seven-mode truncation is then derived and investigated in detail by making use of numerical tools typical of dynamical systems. An intricate phenomenon is found with the significant presence of two- and three-dimensional tori. Among different transitions the one from quasiperiodicity to chaos in a three-torus appears of particular interest.