A periodically forced flow displaying symmetry breaking via a three-tori gluing bifurcation and two-tori resonances

Abstract

The dynamics due to a periodic forcing (harmonic axial oscillations) in a Taylor–Couette apparatus of finite length is examined numerically in an axisymmetric subspace. The forcing delays the onset of centrifugal instability and introduces a Z 2 symmetry that involves both space and time. This paper examines the influence of this symmetry on the subsequent bifurcations and route to chaos in a one-dimensional path through parameter space as the centrifugal instability is enhanced. We have observed a well-known route to chaos via frequency locking and torus break-up on a two-tori branch once the Z 2 symmetry has been broken. However, this branch is not connected in a simple manner to the Z 2-invariant primary branch. An intermediate branch of three-tori solutions, exhibiting heteroclinic and homoclinic bifurcations, provides the connection. On this three-tori branch, a new gluing bifurcation of three-tori is seen to play a central role in the symmetry breaking process.