Connecting period-doubling cascades to chaos

TL;DR

This paper explores the intricate relationship between period-doubling cascades and chaos in parameter-dependent maps, revealing that most cascades are connected to regular periodic orbits and are robust to perturbations.

Contribution

It demonstrates that nearly all regular periodic orbits at chaos are linked to cascades, simplifying the study of complex bifurcation structures in dynamical systems.

Findings

Most cascades are paired or solitary, each connected to regular periodic orbits.

Solitary cascades are robust to large perturbations.

Infinitely many cascades can be understood through regular periodic orbits at chaos.

Abstract

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value $\mu_2$ of the map at which there is chaos. We show that often virtually all (i.e., all but finitely many) ``regular'' periodic orbits at $\mu_2$ are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired -- connected to exactly one other cascade, or solitary -- connected to exactly one regular periodic orbit at $\mu_2$. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of $F(\mu_2, \cdot)$. Examples discussed include the forced-damped pendulum and the double-well Duffing equation.