Investigation of the ABC flow instability with application of centre manifold reduction

Abstract

We demonstrate that application of a generalized centre manifold is advantageous when low-dimensional reductions of continuous dynamical systems of hydrodynamic type are considered. The centre manifold theorem that we use differs from the standard one in the following way. We define a centre manifold as an invariant manifold, tangent to the eigenspace of the linearization of the mapping defining a dynamical system, where the eigenspace is associated with eigenvalues with small but not necessarily zero real parts (usually only eigenvalues with zero real parts are employed). As a result, several bifurcations following the onset of instability of the trivial steady state are reproduced in the eight-dimensional system that we obtain, whilst in the six-dimensional system constructed with application of the conventional centre manifold theorem only the first bifurcation of the steady state is reproduced correctly.