Using Erdős's methods to study Yorke's problems

Abstract

In this paper, we study the possible bifurcations of periodic orbits by analyzing them as graphs. In detail, we construct a collection of graphs which are an idealized versions of bifurcation diagrams and color them using the Mallet-Yorke Orbit Index (and the Lefschetz Fixed Point Theorem). The aforementioned allows to study the genericity of routes to chaos, as well as to gain insight into their possible complexity. In particular, our results can be interpreted as saying that there is no upper bound on the possible complexity of routes to chaos in high dimensional systems.

Revised based on referee suggestions