ABSTRACT
In this chapter we discuss some dynamical systems that are a paradigm for complexity. More specifically, we are concerned with systems where the future is not only determined by their present state, but by part of their history. Such systems can be formally described by seemingly simple difference-differential equations. They not only play an important role in applications, e.g those featuring nonlinear delayed feedback, but are also very suitable objects for a numerical and substantial analytical discussion giving insight into complex dynamics. This includes new types of bifurcation patterns, multi-stability of highly structured periodic orbits, and high dimensional strange attractors. The aim of the present paper is two-fold: first, to briefly give the state of the art in the field and problems that have remained unanswered until now and, second, to open the way to the discovery of new kinds of complex dynamics.
