A “saddle-node” bifurcation scenario for birth or destruction of a Smale–Williams solenoid

Formation or destruction of hyperbolic chaotic attractor under parameter variation is considered with an example represented by Smale–Williams solenoid in stroboscopic Poincaré 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.