Methods of Bifurcation Theory in Multiparameter Problems of Hydroaeroelasticity

Applying bifurcation theory methods to nonlinear boundary value problems for ordinary differential equations (ODEs) of the fourth and higher orders encounters technical difficulties related to studying the spectrum of direct and adjoint linearized problems and constructing Green functions (i.e., proving the spectral problems to be Fredholm and determining the manifolds of bifurcation points). In order to overcome these difficulties, methods for separating the roots of relevant characteristic equations have been proposed, with the subsequent representation of the bifurcation manifolds in terms of these roots; this enables investigation of nonlinear problems in their rigorous statement. Such an approach is considered using the example of a two-point boundary value problem for a fourth-order ODE in which a statically bent pipeline section is described as a flexible elastic hollow rod, with a liquid flowing inside it, that is compressed or stretched by external boundary conditions with a free sliding left end and fixedly mounted right ends.