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

Abstract

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.

Hyperbolic Chaos and Other Phenomena of Complex Dynamics Depending on Parameters in a Nonautonomous System of Two Alternately Activated Oscillators

“…hyperbolic attractor that we discussed earlier [Isaeva et al, 2012[Isaeva et al, , 2013. 1 A richer assortment of dynamical behavior is observed in the right-hand half of the parameter plane diagram, at positive values of h. In the lower part of the chart, each of two individual non-autonomous van der Pol oscillators constituting the system manifests quasi-periodic oscillations that can be synchronized in some parameter regions due to the coupling between the subsystems and the external driving.…”

“…In the right top corner of this "black square" the trivial attractor without oscillations coexists with the Smale -Williams attractor that can arise with initial conditions of sufficiently large amplitude. Mechanism of formation of the Smale -Williams attractor corresponds to the above mentioned "saddle-node" scenario [Isaeva et al, 2012[Isaeva et al, , 2013.…”

“…Shil'nikov & Turaev [1997] have advanced a scenario of formation of the Smale -Williams attractor in a kind of the socalled blue sky catastrophe; an explicit example was suggested later [Kuznetsov, 2010]. One more scenario for the onset or destruction of the Smale -Williams attractor [Isaeva et al, 2012[Isaeva et al, , 2013 is associated with a collision of two chaotic invariant sets, an attracting solenoid and a non-attracting one.…”

Bernoulli Mapping with Holeanda Saddle-Node Scenario of the Birth of Hyperbolic Smale–Williams Attractor

“…Table 4: The sequence of bifurcation points of cycles with codes R{RL} N of the map (12) at ε = 0, b = 0.3. It was shown in [3] that the considered scenario is realized in the system of two coupled van der Pol oscillators [5] x = ω 0 u, u = (h + a cos 2πt/T − x 2 )u − ω 0 x + (εy/ω 0 ) cos ω 0 t,…”

“…The saddle-node scenario of the birth/destruction of the Smale-Williams solenoid in a dynamical system under variation of its parameters [1,2] assumes a situation where at first the hyperbolic chaotic attractor coexists with another attractor (possibly an infinitely distant one), and its destruction occurs as a result of a collision/fusion with a saddle chaotic set lying on a common boundary of their basins of attraction [3,4]. In the extended space of phase variables and the control parameter, this scenario can be regarded as a bifurcation "return point" at which the attracting solenoid loses stability and turns into a saddle set (see scheme at Fig.1).…”