Chapter 1: Bifurcational and Chaotic Dynamics of Simple Structural Members: Literature Review

There are numerous papers and books devoted to dynamics of structural members, i.e. beams, plates, panels and shells. The aim of this chapter is to overview the literature and state-of-the art research devoted to dynamics of the aforementioned structural members and their interplay with bifurcation and chaotic phenomena. This is a novel challenging research track, and hence not many papers and books are published in this topic. In general, from a mathematical point of view, the book deals with nonlinear PDEs and the developed methods for analysis of their solutions with respect to stability, bifurcations, buckling, as well as regular and chaotic dynamics of the modeled continuous objects. On the other hand, the problem is always reduced (though by different ways) to a study of a set of large amount of nonlinear ODEs. In the case of a few first-order nonlinear ODEs, they may govern dynamics of simple nonlinear autonomous and non-autonomous oscillators (two or three first-order ODEs) or dynamics of coupled oscillators. This is why in the first four chapters a background about bifurcational and chaotic dynamics of difference and differential equations has been given to introduce the reader with basic knowledge devoted to dynamics of lumped mechanical systems and beyond. This effort allows for a smooth transition from nonlinear dynamics exhibited by simple dynamical systems to that of simple continuous systems, which can be also understood as 1D or 2D chains of nonlinear oscillators.