The cascades route to chaos

TL;DR

This paper demonstrates that in certain dynamical systems, the transition to chaos involves infinitely many period-doubling cascades, many of which appear in pairs connected by unstable orbits, with implications for understanding complex chaotic behavior.

Contribution

It proves that infinitely many cascades occur during the transition to chaos in 1D and 2D systems, and characterizes their pairing and stability properties.

Findings

Infinitely many cascades are present in the transition from no chaos to chaos.

Many cascades appear in pairs connected by unstable periodic orbits.

Paired cascades can be created or destroyed by perturbations, unpaired cascades are conserved.

Abstract

The presence of a period-doubling cascade in dynamical systems that depend on a parameter is one of the basic routes to chaos. It is rarely mentioned that there are virtually always infinitely many cascades whenever there is one. We report that for one- and two-dimensional phase space, in the transition from no chaos to chaos -- as a parameter is varied -- there must be infinitely many cascades under some mild hypotheses. Our meaning of chaos includes the case of chaotic sets which are not attractors. Numerical studies indicate that this result applies to the forced-damped pendulum and the forced Duffing equations, viewing the solutions once each period of the forcing. We further show that in many cases cascades appear in pairs connected (in joint parameter-state space) by an unstable periodic orbit. Paired cascades can be destroyed or created by perturbations, whereas unpaired cascades are conserved under even significant perturbations.