Period-doubling cascades for large perturbations of Henon families

TL;DR

This paper demonstrates that period-doubling cascades persist in a broad class of Henon family perturbations, classifies cascade periods, and counts their occurrences, extending the understanding of cascade phenomena.

Contribution

It extends the theory of period-doubling cascades to larger perturbations of Henon families and provides a classification and enumeration of cascades by period.

Findings

Cascades persist under large perturbations

Classification of cascades by period

Quantitative count of cascades per period

Abstract

The Henon family has been shown to have period-doubling cascades. We show here that the same occurs for a much larger class: Large perturbations do not destroy cascades. Furthermore, we can classify the period of a cascade in terms of the set of orbits it contains, and count the number of cascades of each period. This class of families extends a general theory explaining why cascades occur.