Bifurcation Theory for a Rod with Small Bending Stiffness

In this paper we study the equilibrium states of a conducting rod with small bending stiffness in a magnetic field. The magnetic field is produced by current flowing in a pair of infinitely long parallel wires. The line between the supports of the rod is in the plane of the wires and equidistant from them. The rod is clamped at both ends. We consider planar deformations of the rod. We prove a bifurcation theorem describing the set of equilibrium states. Our analysis of this problem brings together two important theories in modern applied mathematics; bifurcation theory and the theory of singular perturbations for systems of nonlinear ordinary differential equations