On codimension-three bifurcations in the motion of articulated tubes conveying a fluid

Abstract

We consider the planar motion of two articulated tubes conveying a fluid near a double degeneracy in the linearized dynamical equations. In particular, the interaction between Hopf and pitchfork (symmetric saddle-node) bifurcations, and the effects of asymmetries on that interaction, are studied in detail. The fourth order system of nonlinear equations is reduced to a third order system using center manifold theory. The method of averaging provides the normal form which can be considered as a planar vector field with three important parameters, one each for the Hopf bifurcation, the pitchfork bifurcation and the asymmetry. This second order system is shown to possess twenty-three open sets in the parameter space, each with qualitatively different phase portraits. Some of these portraits are presented, along with bifurcation analysis, and the physical implications of the results are discussed.