Introduction

Many phenomena from physics, biology, chemistry and economics are modeled by differential equations with parameters. When a nonlinear equation is established, its behavior/dynamics should be understood. In general, it is impossible to find a complete dynamics of a nonlinear differential equation. Hence at least, either periodic or irregular/chaotic solutions are tried to be shown. So a property of a desired solution of a nonlinear equation is given as a parameterized boundary value problem. Consequently, the task is transformed to a solvability of an abstract nonlinear equation with parameters on a certain functional space. When a family of solutions of the abstract equation is known for some parameters, the persistence or bifurcations of solutions from that family is studied as parameters are changing. There are several approaches to handle such nonlinear bifurcation problems. One of them is a topological degree method, which is rather powerful in cases when nonlinearities are not enough smooth. The aim of this book is to present several original bifurcation results achieved by the author using the topological degree theory. The scope of the results is rather broad from showing periodic and chaotic behavior of non-smooth mechanical systems through the existence of traveling waves for ordinary differential equations on infinite lattices up to study periodic oscillations of undamped abstract wave equations on Hilbert spaces with applications to nonlinear beam and string partial differential equations.