Analysis of bifurcation solutions to nonlinear wave equations with Corner singularities

Haiffa Muhsan  Buite; Hussein Khashan  Kadhim; Ayed E. Hashoosh

doi:10.5269/bspm.76334

Haiffa Muhsan Buite
Hussein Khashan Kadhim
Ayed E. Hashoosh University of Thi-Qar

Abstract

We investigate bifurcation solutions of a wave ODE with new nonlinear parts that involves elastic beams on elastic bases with the new nonlinear part: $-z^{2}\left(z z^{\prime \prime}+\frac{3}{2}\left(z^{\prime}\right)^{2}\right)$ by employing the local LyapunovSchmidt 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).

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