Instability of Dual Eigenvalue Fourth-Order Systems

Investigations of the postbuckling behavior of an elastic structure at a twofold branching point generally involve a potential of the cubic type. However, when the cubic part vanishes identically (for example, when there is symmetry in both active coordinates), the potential becomes of the quartic type. Here, quartic potentials in normalized coordinates are considered and formulas for limit points and bifurcations on imperfect paths are given in terms of trigonometric polynomials. These are used in certain structural examples to show that a theorem of Ho is invalid unless extra conditions are placed on the potential. They are also used in a proof of the modified Ho theorem.