Local and Global Bifurcation Analyses of a Spatial Cable Elastica

Abstract

This paper focuses on a boundary value problem governing the equilibrium of a slender cable subject to thrust, torsion, and gravity. In the absence of field (gravity) loading, this boundary value problem is integrable and admits periodic solutions describing planar and spatial equilibrium forms. A bifurcation analysis of the integrable problem reveals the conditions controlling local stability of periodic solutions and the existence of two limiting (bounding) homoclinic solutions. The addition of field (gravity) loading renders the boundary value problem nonintegrable. This effect is first investigated through perturbation of the limiting homoclinic solutions for weak gravity loading. Approximate existence conditions for aperiodic and spatially complex forms are determined using Melnikov’s method. The effect of field loading is then re-evaluated through numerical solution of the original problem. Spatially complex solutions are determined that might mimic the loops and tangles sometimes found in underwater cables.