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A “saddle-node” bifurcation scenario for birth or destruction of a Smale–Williams solenoid
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10.1063/1.4766590
/content/aip/journal/chaos/22/4/10.1063/1.4766590
http://aip.metastore.ingenta.com/content/aip/journal/chaos/22/4/10.1063/1.4766590

Figures

Image of FIG. 1.
FIG. 1.

The Lyapunov exponents of the system (1) versus parameter h: for the chaotic attractor (black curves) and for the trivial attractor, the fixed point at the origin (gray line). Parameter values are , T = 6, , a = 6.5.

Image of FIG. 2.
FIG. 2.

Constructing the amplitude stroboscopic map at , a = 6.5, T = 6, for different values of parameter h: (a) the original diagram accounting all accumulated computational data, and (b) the idealized plot excluding the widening by the phase averaging.

Image of FIG. 3.
FIG. 3.

Diagrams illustrating dynamics of phase on successive stages of excitation of the first oscillator on the attractor A (black) and on the non-attracting invariant set S (gray).

Image of FIG. 4.
FIG. 4.

(a) Periodic solutions of the equations (1) with period plotted in the extended phase parameter space at , a = 6.5, T = 6, . (b) Superimposed graphs of radial (amplitude) component of the same set of solutions versus parameter h; this is actually a longitudinal slice of (a).

Image of FIG. 5.
FIG. 5.

Attractor A and non-attracting invariant set S in the stroboscopic Poincaré section shown on the plane of variables of the first oscillator (a) and in three-dimensional plot (b) accounting a variable of the second oscillator at , a = 6.5, T = 6, , and h = −1.15. The gray color on the panel (a) indicates roughly the basin of attraction of the stable fixed point in the origin.

Image of FIG. 6.
FIG. 6.

A plot of radial component of the model map (A1) (a) and disposition of attractors A, O and of a non-attractive invariant set S on the plane of complex variable z (b).

Image of FIG. 7.
FIG. 7.

Mutual disposition of attractors A and O and of a non-attractive invariant set (S) of the map (A3) on the plane of complex variable z for different values of parameter R at .

Tables

Generic image for table
Table I.

Bifurcation values of parameter h for the unstable periodic orbits of the system (1) at , a = 6.5, T = 6, .

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/content/aip/journal/chaos/22/4/10.1063/1.4766590
2012-11-26
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A “saddle-node” bifurcation scenario for birth or destruction of a Smale–Williams solenoid
http://aip.metastore.ingenta.com/content/aip/journal/chaos/22/4/10.1063/1.4766590
10.1063/1.4766590
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