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Theory of intermittency applied to classical pathological cases
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10.1063/1.4813857
/content/aip/journal/chaos/23/3/10.1063/1.4813857
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Figures

FIG. 1.

Sketch of the map , exhibiting anomalous type-III intermittency. Thick arrow illustrates mapping of points from the chaotic region (around the maximum of ()) into the region with practically zero tangent of (). Then thin arrow indicates the following reinjection of these points into the laminar region.

FIG. 2.

Analysis of the anomalous Laugesen type-III intermittency (map , : , and the laminar interval ). (a) Assessment of the RPD by numerical simulation. Dots correspond to () evaluated by and dashed line corresponds to the least squares fit. The dashed line with slope corresponds to the uniform RPD. (b) Numerical RPD. Dashed curve corresponds to with the parameters found in (a). (c) Probability density of the length of the laminar phase. Dashed line corresponds to .

FIG. 3.

Characteristic relations of the averaged length of the laminar phase ε for the map , . Circles and triangles show numerical data. For , the solid line has slope in agreement (within 6% of relative error) with the analytical value 0.948 given by . For , the horizontal dashed line shows the asymptotic behavior of , with .

FIG. 4.

Second iteration of the map (31) demonstrating the Pikovsky type-II intermittency. (a) Non-overlapping case with a gap between two reinjection intervals. Arrows show two routs of reinjection into two disjoin intervals and for the upper laminar region. Dots mark positions of the fixed points. There are two chaotic attractors in the map. (b) Slightly overlapping case. Reinjection intervals and overlap. There exists single chaotic attractor. (c) Time evolution of the map corresponding to the case (b). Bottom subplot shows zoomed trajectory with two laminar phases near two unstable fixed points ( ).

FIG. 5.

Analysis of the Pikovsky intermittency in the non-overlapping (top row, , two chaotic attractors) and slightly overlapping (bottom row, , single chaotic attractor) cases. Results are shown for the second iteration of the map . (a) and (d) Numerical data (dots) for two branches of () computed using for reinjections in the intervals and . Dashed gray lines show the corresponding least mean square fits, which then used to plot and . Dashed line with slope corresponds to the uniform RPD. (b) and (e) RPDs for and . Numerical data (dots) and pdfs evaluated by (dashed curves). (c) and (f) Probability density of the length of laminar phase for the interval (for the pdf is similar). Dashed curve corresponds to .

FIG. 6.

Fitting the RPD model to short data sets. (a) Mean and standard error of least squares estimation of averaged over 100 independent experiments using Eq. . Horizontal dashed line marks the exact value of . The inset illustrates a representative example of least squares fitting of 30 data points. The data and straight line underestimate the exact value of . (b) Mean and standard error for estimated by using the modified scheme . There exists no significant bias in the estimate even for relatively short data sets. The inset illustrates typical distributions of the error obtained for ordinary least squares using Eq. (dots) and modified (triangles) methods.

FIG. 7.

Characteristic relation of the average length of the laminar phase ε. Dots correspond to numerical data, whereas the curve marked as slight overlap refers to numerical integration of Eq. using as RPD. The asymptotic behavior is given by dashed line (marked by ) with the slope −0.5 ( ). The straight line (marked by ) with the slope −0.5 ( ) matches the numerical data for the strongly overlapping case considered in Ref. .

FIG. 8.

RPD for the Pikovsky's map in the strongly overlapping case. Dots correspond to numerical simulations and the curve is obtained by Eq. with the fitted values for the reinjection on : . The corresponding values for are . In this case, .

/content/aip/journal/chaos/23/3/10.1063/1.4813857
2013-07-17
2014-04-16

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