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Regular approximations to chaotic maps
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View: Figures


Image of FIG. 1.
FIG. 1.

The characteristic function approximation for Arnold’s Cat map with regularization parameter in grayscale with 256 levels (a) and with 10 levels (b). It is seen that level lines are highly irregular. The biggest spike (in black) corresponds to the unstable stationary point (0,0), and the lines visible in (a) plot approximate its stable and unstable manifolds. For smaller the detail becomes finer, but the general character is similar.

Image of FIG. 2.
FIG. 2.

The characteristic function cross section for the Standard Map. The regularization parameter value : (a) , ; (b) , ; and (c) , . The relatively flat regions for , , and correspond to mostly chaotic trajectories in the map [see Fig. 4(b) ].

Image of FIG. 3.
FIG. 3.

Some regular trajectories for (lines), and their approximation with characteristic function (points). The split trajectories correspond to level lines, which pass through saddle points.

Image of FIG. 4.
FIG. 4.

(a) The amplified region of a Standard Map with level lines of characteristic function for . The calculation is made for , and . (b) The pattern of trajectories in the same region of a map. (c) The superposition of (a) and (b).

Image of FIG. 5.
FIG. 5.

(a) The level lines of a characteristic function for a developed chaos, . (b) The same as in (a), with a single chaotic trajectory plotted.

Image of FIG. 6.
FIG. 6.

(a) The characteristic function approximation (grayscale, 256 levels) and the attractor region (line) for the map with dissipation Eqs. (21) and (22) , in a square , . The map parameter values are , . (b) The same as in Fig. 6(a) , with level lines for characteristic function and some additional information plotted. The black lines now include unstable manifold of stationary point (0,0), and stable manifolds for stationary point (2.305…, 2.305…) (big circle), and the orbit of period 4 (small circles).

Image of FIG. 7.
FIG. 7.

The example of split invariant surface for perturbed volume-preserving map in 3D equations (23)–(25) . The small parameter values are , . The parameter .

Image of FIG. 8.
FIG. 8.

The cross section for the 3D map equations (23)–(25) (thin), and the regions between close level lines (thick) for characteristic function in the same cross section. For the map, initial points were taken, and values were plotted for orbit points with value in a small interval around zero.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Regular approximations to chaotic maps