HOMOCLINIC ORBITS IN THE DYNAMICS OF ARTICULATED PIPES CONVEYING FLUID

A model is considered representing an elastically jointed pair of articulated pipes conveying fluid. The motion is described by a four-component system of autonomous ordinary differential equations. Numerical techniques are used to investigate changes in the dynamics as two parameters are varied. These parameters represent the fluid flow-rate and a form of symmetry-breaking. Evidence is found that the global bifurcation picture is surprisingly complicated, involving chaos and two types of homoclinic behaviour: namely, Sil'nikov homoclinic orbits to a saddle-focus stationary point, and homoclinic tangencies to periodic orbits. Local theory respective to each type of homoclinicity is reviewed and compared with the numerical results. Periodic orbits are found that consist of a large-scale oscillation followed by several small-scale ones. Global branches of some of these orbits are shown to form isolas, whereas others originate in period-doubling or homoclinic bifurcations. All the branches feature characteristic sequences of period-doubling and saddle-node bifurcations that are termed towers. Codimension-two bifurcations are discovered between orbits approaching homoclinicity and those forming towers. It is conjectured how a series of such bifurcations may underlie the apparent sudden termination of a locus of homoclinic orbits. Finally, the relevance of the results to the behaviour of other systems is discussed.