Models of Multiparameter Bifurcations in Boundary Value Problems for ODEs of the Fourth Order on Divergence of Elongated Plate in Supersonic Gas Flow

Abstract

At the application of bifurcation theory methods to nonlinear boundary value problems for ordinary differential equations of the fourth and higher order there usually arise technical difficulties, connected with determination of bifurcation manifolds, spectral investigation of the direct and conjugate linearized problems and the proof of their Fredholm property. For overcoming of this difficulty here the roots separation method is applied to the relevant characteristic equations with subsequent presentation of critical manifolds, that allows to investigate nonlinear problems in the precise statement. Such approach is applied here to two-point boundary value problem for the nonlinear ODE of the fourth order describing the buckling (divergence) of an elongated plate in a supersonic flow of gas, subjected to compressed or extended boundary stresses at the various boundary fastenings.