Unfolding Complexity: Hereditory Dynamical Systems - New Bifurcation Schemes and High Dimensional Chaos

We discuss some dynamical systems which are paradigmatic for complexity. The aim of the present paper is twofold: First, to state briefly the state of the art and the problems unanswered until now and, second, to open the way for the discovery of new kinds of complex dynamics. Systems where the future is not only determined by their present state but by part of their history and which formally can be described by seemingly simple difference -differential equations not only play an important role in the applications e. g. of nonlinear delayed feedback but are very suitable for numerical and substantial analytical discussion and insight of complex dynamics. This includes new types of bifurcation patterns, multi–stability of highly structured periodic orbits and high dimensional strange strange attractors. Subject classes: 34K13, 34K23, 39A11 ∗To appear in Nonlinear Dynamics and chaos : Where do we go from here? (J. Hogan et al. eds.), IoP Publ. (in press)