We consider the problem of a long thin weightless rod constrained to lie on a cylinder while being held by end tension and twisting moment. Applications of this problem are found, for instance, in the buckling of drill strings inside a cylindrical hole. In a previous paper the general geometrically-exact formulation was given and the case of a rod of isotropic cross-section analysed in detail. It was shown that in that case the static equilibrium equations are completely integrable and can be reduced to those of a one-degree-of-freedom oscillator whose non-trivial fixed points correspond to helical solutions of the rod. A critical load was found where the rod coils up into a helix. Here the anisotropic case is studied. It is shown that the equations are no longer integrable and give rise to spatial chaos with infinitely many multiply looped localised solutions. Helices become slightly modulated. We study the bifurcations of the simplest `one-looped' solution and a representative multi-loop as the aspect ratio of the rod's cross-section is varied. It is shown how the anisotropy unfolds the `coiling bifurcation'. The resulting bifurcation behaviour is shown to have strong similarities with cellular buckling in structures with a `stiffening-restiffening' nonlinearity, which can be interpreted in terms of a so-called `Maxwell' effective failure load.
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