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The experimental localization of Aubry–Mather sets using regularization techniques inspired by viscosity theory
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10.1063/1.2756264
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1 Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, via Trieste 63, 35121 Padova, Italy
Chaos 17, 033107 (2007)
/content/aip/journal/chaos/17/3/10.1063/1.2756264
http://aip.metastore.ingenta.com/content/aip/journal/chaos/17/3/10.1063/1.2756264
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## Figures

FIG. 1.

Numerical computation of for the preliminary steps, using both the norm (upper curve) and the norm (lower curve). In both cases, the minimum value of is obtained for .

FIG. 2.

On the left: we report the numerical computation of for different values of . The minimum is found for . On the right: numerical computation of (continuous line) and (dashed line) versus , obtained for . The norm is reduced up to order for , and after this value the precision of the numerical computation evidently affects the computations.

FIG. 3.

On the top we represent vs ; on the middle we represent (part of the plot of) vs . The density of the points indicate that a good value for is , which corresponds to the limit distance of . On the bottom: representation in the phase–space of the 42 204 approximate points of the cantorus that satisfy (30).

FIG. 4.

On the top: representation in the phase-space of 1 947 069 approximate points of the cantorus. On the bottom: zoom of the 1 947 069 points of the cantorus and a zoom of the 42 204 points shifted of in the action coordinate (to compare the two sets). It is evident a very good correspondence of the two sets, with no significant enlargements of the part of the cantorus passing from the original set to the one obtained with 100 iterations.

FIG. 5.

On the top: representation in the phase–space of a set of 1 622 521 approximate points of the cantorus. On the bottom: zoom of the 1 622 521 points of the cantorus and a zoom of the 54 305 points shifted of in the action coordinate (to compare the two sets). It is evident a very good correspondence of the two sets, with no significant enlargements of the part of the cantorus passing from the original set to the one obtained with 100 iterations.

/content/aip/journal/chaos/17/3/10.1063/1.2756264
2007-08-17
2014-04-23

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