Resonances in the Codimension-2 Bifurcations in the Couette–Taylor Problem

Abstract

The investigation of codimension-2 bifurcations, in particular in systems with cylindric symmetry, enables us to deduce new types of secondary regimes branching-off from the symmetric regimes. This investigation also allows us the unique possibility of a rigorous treatment of chaotic solutions to Navier–Stokes and other nonlinear PDE’s. The central manifold approach combined with the reduction to the normal form lead to the so-called amplitude systems. These ODE systems describe the nonlinear interaction between the neutral modes, and always include several nonlinear terms due to so-called intrinsic resonances. However, sometimes additional resonances appear. In this paper we present the complete list of all possible resonances in dynamic systems with cylindric symmetry and the corresponding forms of the amplitude equations. Further, we present the results of extensive numerical investigation of the resonant codimension-2 bifurcations in the Couette–Taylor problem, thus creating an intriguing subject for further investigation.