Nonlinear dynamics of rods and tops - localisation and chaos

Gert van der Heijden, University College London. Part of the applied nonlinear dynamics seminar series.

After reviewing the close analogy between twisted rods (statics) and spinning tops (dynamics) we consider the problem of a conducting rod in a uniform magnetic field, motivated by electrodynamic space tethers. When expressed in body coordinates the equilibrium equations are found to sit in a hierarchy of non-canonical Hamiltonian systems involving an increasing number of vector fields. The systems are completely integrable, and generated by a Lax pair, in the case of a transversely isotropic rod. Remarkably, unlike in the non-magnetic case, extensibility of the rod destroys integrability, leading to a multiplicity of localised solutions and spatial chaos. A codimension-two Hamiltonian Hopf-Hopf bifurcation is found to act as organising centre for tether stability. The chaotic nature of the system is proved using a (Hamiltonian) Melnikov approach that highlights problems with similar Melnikov applications in rigid-body dynamics in the literature as well as a way around these problems.