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Characterization of stickiness by means of recurrence
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Image of FIG. 1.
FIG. 1.

(Color) The phase portrait of the standard map [Eq. (1) ] with different perturbation values: (a) and (b) . The color is determined by the number of different return times. The value of 1 corresponds to periodic orbits.

Image of FIG. 2.
FIG. 2.

(Color online) The phase portrait of the standard map for . (a) The first 3000 iterations of three orbits are plotted with different colors: quasiperiodic (gray), sticky (dark gray), and filling chaotic (black). (b) The first 3000 iterations of the sticky orbit. (c) The first iterations of the sticky orbit.

Image of FIG. 3.
FIG. 3.

RPs of different trajectories consisting of 3000 iterations. (a) quasiperiodic orbit, (b) sticky orbit, (c) filling chaotic orbit, and (d) zooming of the black structure in (c).

Image of FIG. 4.
FIG. 4.

(Color online) Lyapunov exponents for three orbits with a dependence on the iteration time. Since the initial conditions for the sticky and filling orbits are in the same chaotic component, converge to the same value of 1.406, albeit after a long time.

Image of FIG. 5.
FIG. 5.

RQA measures with a dependence on time for the sticky orbit. The size of each window is points and there are 4500 points overlapped between two consecutive windows. The vertical dashed line corresponds to the transition time around . The length of the orbit is points.

Image of FIG. 6.
FIG. 6.

(Color online) Thresholded meta-recurrence plot of the sticky orbit. There exists a significant change in the density at the transition point denoted by the solid lines. The window size .

Image of FIG. 7.
FIG. 7.

(Color online) (a) The phase portrait of the parameter with initial values indicated by the black upward triangle point for iterations. Sticky regions I and II are colored with medium gray and dark gray, respectively. (b) Dependence of the with a running window of size with 4500 points overlapped between two consecutive windows. The small peak at is due to weaker stickiness in comparison with I and II. The series of the sticking events is denoted by . (c, d) Two major sticky regions in the phase plane.

Image of FIG. 8.
FIG. 8.

Cumulative distribution of sticking events of duration greater than . A single chaotic trajectory consisting of iterations has been used for the computation.


Generic image for table
Table I.

Three selected RP-based measures of complexity computed from trajectories shown in Figs. 3(a)–3(c) .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Characterization of stickiness by means of recurrence