Multiple bifurcation of rotating pipe flow

Viscous rotating Hagen-Poiseuille flow is shown to contain a multiple bifurcation point at which two Hopf bifurcations coalesce. The double Hopf bifurcation has nonsemisimple 1:1 resonance. The full problem in the neighborhood of this point may be reduced to a four-dimensional system of weakly nonlinear amplitude equations using an expansion of the center manifold type. An efficient numerical procedure to find the coupling coefficients in the amplitude equations is employed which avoids the explicit need to compute adjoint eigenfunctions. Numerical investigation of the amplitude equations has lead to the discovery of several new states of the fluid motion in this system, including stable rotating traveling and modulated traveling waves, a Feigenbaum period-doubling cascade, and chaotic trajectories. Homoclinic loops are found leading to dynamics which may be associated with bursting events in the physical problem under consideration.This work was supported by the Air Force Office of Scientific Research under Contracts AFOSR-89-0346, monitored by Dr. L. Sakell, and AFOSR-89-0226, monitored by Dr. J. McMichael. A. Mahalov is supported by an IBM Watson Fellowship.