Construction of the asymptotics of solutions of nonlinear boundary value problems for fourth order differential equation with two bifurcation parameters

Methods of many-parameter bifurcation theory are illustrated by the example of nonlinear boundary value problem of aeroelasticity. Bending forms of a thin elongated plate subjected to small normal load on elastic foundation and flowing around by supersonic flow of a gas in dimensionless variables are described by ODE of 4-th order with two bifurcational (spectral) parameters: Mach number $M$ and small normal load $\varepsilon_0q$. By bifurcation and catastrophe theory methods the bending forms computations are fulfilled for the boundary conditions $B$ (the left edge is free, the right one is rigidly fixed). Technical difficulties arising at the investigation of the linearized eigenvalue problem are overcome with the aid of the bifurcation curves representation through the roots of the relevant characteristic equation. Fredholm property of the linearized problem is proved with the aid of relevant Green function construction.