Analysis of bifurcation solutions to nonlinear wave equations with Corner singularities

Résumé

We investigate bifurcation solutions of a wave ODE with new nonlinear parts that involves elastic beams on elastic bases with the new nonlinear part:  by employing the local Lyapunov-Schmidt method. The key function that corresponds to the functional of the ODE is identified. Subsequently, we demonstrate that the key function is equivalent to a smooth function of fifth degree. This study focuses on the singularities at the corners of the fifth-degree function to conduct a bifurcation analysis of ordinary differential equation solutions, employing real analysis, functional analysis, and catastrophe theory. This study aims to establish the parametric equation of the bifurcation set (caustic) and provide a geometric interpretation, along with an analysis of the bifurcation propagation of critical points (singularities).