Validating a 2-mode asymptotic expansion in computational bifurcation theory (Before introducing HDN for differential operation)

For error-free computation of high-derivatives of mathematical functions used in engineering applications, hyper-dual numbers (HDN) are receiving much attention in computational mechanics. Differently from classical finite differences, HDN provides a practically exact evaluation of higher-derivatives, such as the first and second derivatives of stiffness matrix with respect to nodal degrees-of-freedom (dof). As a preliminary step for introducing HDN in stability problems, the present paper formulates the theoretical basis of a 2-mode asymptotic bifurcation theory and examines its versatility on simple bench models. All obtained results in numerical examples well predict the stability behavior and agree with existing analytical solutions.