Homoclinic complexity in the localised buckling of an extensible conducting rod in a uniform magnetic field

Caifa Guo, G.H.M. van der Heijden & Hong Cai

We study the localised buckling of an extensible conducting rod subjected to end loads and placed in a uniform magnetic field. The trivial straight but twisted rod is described by a fixed point of a four-dimensional Hamiltonian system of equations previously shown to be chaotic. Localised solutions are given by homoclinic orbits to this fixed point and we explore the spatial complexity of localised rod configurations by means of shooting and parameter continuation methods that exploit the reversibility of the system of equations. Unlike in localisation studies of non-magnetic rods we find that for certain parameter values multiple Hamiltonian-Hopf bifurcations occur. Where these collide as parameters are varied, solutions exhibit delocalisation-relocalisation behaviour. Our results predict buckling instabilities and post-buckling behaviour of rods under combined mechanical and magnetic loads, which is relevant for electrodynamic space tethers and potentially for conducting nanowires in future electromechanical devices.

keywords: extensible elastic rod, Lorentz force, magnetic buckling, Hamiltonian-Hopf bifurcation, homoclinic orbits, spatial localisation

Physica D 284, 42-52 (2014)