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Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation
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10.1063/1.2748619
/content/aip/journal/chaos/17/2/10.1063/1.2748619
http://aip.metastore.ingenta.com/content/aip/journal/chaos/17/2/10.1063/1.2748619

Figures

Image of FIG. 1.
FIG. 1.

Probability distribution of time- transversal Lyapunov exponents for maps (a) , and (b) , , and different values of the bifurcation parameter . Solid lines: distributions evaluated using typical chaotic trajectories with points. Dashed lines: distributions computed using all unstable period- orbits.

Image of FIG. 2.
FIG. 2.

Fraction of positive time- transversal Lyapunov exponents for maps (a) , and (b) , as a function of the bifurcation parameter . Solid lines: typical chaotic orbits; dashed lines: fraction of transversely unstable period- orbits. The insets in both figures show blowouts of the regions near the onset of UDV for both maps.

Image of FIG. 3.
FIG. 3.

Critical values of the bifurcation parameter for which the (a) first and (b) last period- orbits lose transversal stability. Circles and crosses refer to models and , respectively. The lines in (b) are least-squares fits with slopes 0.35 and 0.26 for models and , respectively.

Image of FIG. 4.
FIG. 4.

Contrast measure vs bifurcation parameter for unstable orbits with period (a) (model ); (b) (model ).

Image of FIG. 5.
FIG. 5.

Average number of iterations for a typical trajectory of model (circles) and (crosses) to be trapped in a vicinity of width of the chaotic set vs the difference between the bifurcation parameter and its critical value . The solid lines are least-squares fits, cf. Eq. (30), with slopes and for model , and and for model .

Image of FIG. 6.
FIG. 6.

Evolution of the transversal coordinate of a trajectory off but very close to the chaotic set of models (a) and (b) , with .

Image of FIG. 7.
FIG. 7.

Average interburst interval for models and with , . Solid lines represent least-squares fit with slopes and , respectively.

Image of FIG. 8.
FIG. 8.

Evolution of the log-shadowing distances for the same time series depicted in Figs. 6(a) and 6(b).

Image of FIG. 9.
FIG. 9.

Probability distribution of log-shadowing distances for the map (a) and (b) , for different values of and with addition of random kicks of strength along the transversal direction.

Tables

Generic image for table
Table I.

Model .

Generic image for table
Table II.

Model .

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/content/aip/journal/chaos/17/2/10.1063/1.2748619
2007-06-27
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation
http://aip.metastore.ingenta.com/content/aip/journal/chaos/17/2/10.1063/1.2748619
10.1063/1.2748619
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