Bifurcation Analysis of Nonlinear Fourth Order Differential Equation

In this book, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams on elastic foundations with small perturbation by using local method of Lyapunov–Schmidt. Two problems have been studied in this book; the first is to find bifurcation solutions of a boundary value problem and we showed that the bifurcation equation corresponding to the boundary value problem is given by a nonlinear system of two equations. Also, we found the parameters equation of the discriminate set of the specified problem as well as the bifurcation diagram. The second problem is to find bifurcation periodic solutions of an equation, and we showed that the bifurcation equation corresponding to the equation is given by a nonlinear system of four equations. In polar coordinate system, we showed that the bifurcation equation is given by a nonlinear system of two cubic equations.