Period-doubling cascades galore

The appearance of numerous period-doubling cascades is among the most prominent features of {\bf parametrized maps}, that is, smooth one-parameter families of maps $F:R \times {\mathfrak M} \to {\mathfrak M}$, where ${\mathfrak M}$ is a smooth locally compact manifold without boundary, typically $R^N$. Each cascade has infinitely many period-doubling bifurcations, and it is typical to observe -- such as in all the examples we investigate here -- that whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic $F$. We illustrate this theory with several examples. We show that there is a close connection between the transition through infinitely many cascades and the creation of a horseshoe.