A "saddle-node" bifurcation scenario for birth or destruction of a Smale-Williams solenoid

TL;DR

This paper investigates how a hyperbolic chaotic attractor, specifically the Smale-Williams solenoid, can form or be destroyed through saddle-node bifurcations involving periodic orbits, in a system of coupled oscillators.

Contribution

It introduces a saddle-node bifurcation scenario for the birth or destruction of a Smale-Williams solenoid in a non-autonomous oscillator system.

Findings

Bifurcation occurs within a narrow parameter interval.

Periodic orbits form a skeleton undergoing saddle-node bifurcations.

The attractor's basin boundary involves a non-attracting invariant set.

Abstract

Formation or destruction of hyperbolic chaotic attractor under parameter variation is considered with an example represented by Smale--Williams solenoid in stroboscopic Poincar\'{e} map of two alternately excited non-autonomous van der Pol oscillators. The transition occupies a narrow but finite parameter interval and progresses in such way that periodic orbits constituting a "skeleton" of the attractor undergo saddle-node bifurcation events involving partner orbits from the attractor and from a non-attracting invariant set, which forms together with its stable manifold a basin boundary of the attractor.