Hopf-Takens-Bogdanov interaction for a fluid-conveying tube

We investigate the dynamics after loss of stability of the downhanging configuration of a fluid conveying tube with a small end mass and an elastic support.

By varying the fluid flow rate and the stiffness and location of the elastic support, different degenerate bifurcation scenarios can be observed. In this article we investigate the bifurcating solution branches of the codimension 3 interaction between a Hopf bifurcation and a Bogdanov-Takens bifurcation.

A complete discussion of the primary and secondary solution branches was already given by W. F. Langford and K. Zhan. After reducing the system to the three-dimensional Normal Form equations we apply a numerical continuation procedure to locate the expected higher order bifurcation branches and detect more complicated dynamics, like Shilnikov orbits. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)