SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 61 , Issue 6 : 064511(2018) https://doi.org/10.1007/s11433-017-9161-8

## Entropy method of measuring and evaluating periodicity of quasi-periodic trajectories

• AcceptedJan 5, 2018
• PublishedApr 2, 2018
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### Acknowledgment

This work was supported by the National Natural Science Foundation for Distinguished Young Scholars of China (Grant No. 11525208), and the National Natural Science Foundation of China (Grant No. 11572166).

### Supplementary data

Appendix

In this Appendix the proof of eq. (25) is derived as follows: the Fourier transform of x3 is

 $x ^ 3 = π ∑ i = 1 m a i δ ( ω − ω i ) , (a1)$

 $p 3 ( ω ) = f 3 ( ω ) ∫ f 3 ( ω ) d ω = ( ∑ i a i δ ( ω − ω i ) ) 2 ∫ ( ∑ i a i δ ( ω − ω i ) ) 2 d ω . (a2)$

Since δ(ωωi1δ(ωωi2) ≡ 0,

 $p 3 ( ω ) = ∑ i a i 2 ⁢ [ δ ( ω − ω i ) ] 2 ∫ ∑ i a i 2 ⁢ [ δ ( ω − ω i ) ] 2 d ω = a 1 2 ⁢ [ δ ( ω − ω 1 ) ] 2 ∫ ∑ i a i 2 [ δ ( ω − ω i ) ] 2 d ω + a 2 2 [ δ ( ω − ω 2 ) ] 2 ∫ ∑ i a i 2 [ δ ( ω − ω i ) ] 2 d ω + … + a m 2 [ δ ( ω − ω m ) ] 2 ∫ ∑ i a i 2 [ δ ( ω − ω i ) ] 2 d ω . (a3)$

kk', we have

 $∫ δ ( ω − ω k ) d ω = ∫ δ ( ω − ω k ′ ) d ω , (a4)$

thus, the denominator above becomes

 $∫ ∑ i a i 2 [ δ ( ω − ω i ) ] 2 d ω = ∫ ∑ i a i 2 [ δ ( ω − ω k ) ] 2 d ω = ∑ i a i 2 ∫ [ δ ( ω − ω k ) ] 2 d ω = a ∫ [ δ ( ω − ω k ) ] 2 d ω , (a5)$

and then we have

 $p 3 ( ω ) = ∑ i a i 2 [ δ ( ω − ω i ) ] 2 ∫ ∑ i a i 2 [ δ ( ω − ω i ) ] 2 d ω = a 1 2 a [ δ ( ω − ω 1 ) ] 2 ∫ [ δ ( ω − ω 1 ) ] 2 d ω + a 2 2 a [ δ ( ω − ω 2 ) ] 2 ∫ [ δ ( ω − ω 2 ) ] 2 d ω + … + a m 2 a [ δ ( ω − ω m ) ] 2 ∫ [ δ ( ω − ω m ) ] 2 d ω . (a6)$

Due to the quality of the Dirac-delta functions,

 $S 3 = ∫ p 3 ( ω ) log p 3 ( ω ) d ω = ∫ a 1 2 a [ δ ( ω − ω 1 ) ] 2 ∫ [ δ ( ω − ω 1 ) ] 2 d ω log a 1 2 a [ δ ( ω − ω 1 ) ] 2 ∫ [ δ ( ω − ω 1 ) ] 2 d ω d ω + ∫ a 2 2 a [ δ ( ω − ω 2 ) ] 2 ∫ [ δ ( ω − ω 2 ) ] 2 d ω log a 2 2 a [ δ ( ω − ω 2 ) ] 2 ∫ [ δ ( ω − ω 2 ) ] 2 d ω d ω + ⋯ + ∫ a m 2 a [ δ ( ω − ω m ) ] 2 ∫ [ δ ( ω − ω m ) ] 2 d ω log a m 2 a [ δ ( ω − ω m ) ] 2 ∫ [ δ ( ω − ω m ) ] 2 d ω d ω . (a7)$

Since the bounds of integration are ±∞, ∀ i,

 $∫ a i 2 a [ δ ( ω − ω i ) ] 2 ∫ [ δ ( ω − ω i ) ] 2 d ω log a i 2 a [ δ ( ω − ω i ) ] 2 ∫ [ δ ( ω − ω i ) ] 2 d ω d ω = ∫ a i 2 a [ δ ( ω − ω 1 ) ] 2 ∫ [ δ ( ω − ω 1 ) ] 2 d ω log a i 2 a [ δ ( ω − ω 1 ) ] 2 ∫ [ δ ( ω − ω 1 ) ] 2 d ω d ω . (a8)$

Thus,

 $S 3 = ∫ p 3 ( ω ) log p 3 ( ω ) d ω = ∫ a 1 2 a p ( ω ) log a 1 2 a p ( ω ) d ω + ∫ a 2 2 a p ( ω ) log a 2 2 a p ( ω ) d ω + ⋯ + ∫ a m 2 a p ( ω ) log a m 2 a p ( ω ) d ω = ∫ p ( ω ) ∑ i [ a i 2 a log a i 2 a p ( ω ) ] d ω = ∫ p ( ω ) log ∏ i [ a i 2 a p ( ω ) ] a i 2 a d ω = ∫ p ( ω ) log ∏ i [ a i 2 a ] a i 2 a p ( ω ) d ω = ∫ p ( ω ) log p ( ω ) + ∑ i p ( ω ) a i 2 a log a i 2 a d ω , (a9)$

where

 $p ( ω ) = [ δ ( ω − ω 1 ) ] 2 ∫ [ δ ( ω − ω 1 ) ] 2 d ω .$

Similarly, we have

 $x ^ 4 = π ∑ j = 1 n b j δ ( ω − ϕ j ) , (a10)$

 $p 4 ( ω ) = b 1 2 b [ δ ( ω − ϕ 1 ) ] 2 ∫ [ δ ( ω − ϕ 1 ) ] 2 d ω + b 2 2 b [ δ ( ω − ϕ 2 ) ] 2 ∫ [ δ ( ω − ϕ 2 ) ] 2 d ω + … + b n 2 b [ δ ( ω − ϕ n ) ] 2 ∫ [ δ ( ω − ϕ n ) ] 2 d ω , (a11)$

and

 $S 4 = ∫ p 4 ( ω ) log p 4 ( ω ) d ω = ∫ p ( ω ) log p ( ω ) + ∑ j p ( ω ) b j 2 b log b j 2 b d ω . (a12)$

So the difference between entropies S3 and S4 is

 $S 3 − S 4 = ∫ ∑ i p ( ω ) a i 2 a log a i 2 a d ω − ∫ ∑ j p ( ω ) b j 2 b log b j 2 b d ω = [ ∑ i a i 2 a log a i 2 a − ∑ j b j 2 b log b j 2 b ] ∫ p ( ω ) d ω = ∑ i a i 2 a log a i 2 a − ∑ j b j 2 b log b j 2 b . (a13)$

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• Figure 1

(Color online) Shannon entropies of different probability distributions. (a) Most certain case. Event 23 always occurs and HS = 0; (b) most uncertain case. 50 events are equal probability and HS = log50 = 3.9120; (c) 50 events, each with random probabilities, and HS = 3.6855.

• Figure 2

(Color online) Sketch of frequency distribution and its associated entropy. (a) Sketch of p(ω) = 0.5, 1, and 2; (b) entropy vs. differentp(ω) with uniform distributed.

• Figure 3

(Color online) Entropies of x1 = cost(S1) and x2 = cost + cos2t(S2) vary with time. ∆S = log2 = 0.6931.

• Figure 4

(Color online) Entropy comparison of periodic trajectories x5 and x6 with DFT. ∆S = S5S6 = 0.5054.

• Figure 5

(Color online) Entropy variations of x9. Taking eq. (27) as a modification of entropy calculation can avoid the oscillation of entropy between periods.

• Figure 6

(Color online) Frequency maps in the y-direction and trajectories with an initial condition of y = −0.121, 0.2, and 0.25480176.

• Figure 7

(Color online) Comparison of information given by entropy method (a) and OFLI (b) about trajectories for E = 1/12, where the initial condition y varies from −0.35 to 0.45. The integration time is 100π.

• Figure 8

(Color online) Variation of trajectories in the vicinity of (a) y = 0.071596 and (b) y = 0.183407.

• Figure 9

(Color online) Map of entropy values computed with eq. (27) about trajectories for E = 1/12, where the initial condition y varies from−0.35 to 0.45. The integration time is 100π.

• Figure 10

(Color online) Entropy values calculated on a grid of 1401 × 1001 initial conditions uniformly distributed on the y-axis and the E-axis. The dark blue zone corresponded to the chaotic trajectories, and the bright yellow zone corresponded to the neighborhood of regular quasi-periodic or periodic at an integration time of 100π.

• Figure 11

(Color online) Frequency maps in y-direction and trajectories with initial conditions x = −0.09 (a), −0.22 (b), and −0.25065550 (c).

• Figure 12

(Color online) Entropy map in the y-direction and trajectories after an integration time of 150, with a set of 2001 initial condition of x ∈ [–0.49, –0.01] for $y=x˙=0$. $y˙$ is deduced from the Jacobi integral C = 4.

• Figure 13

(Color online) Variation of trajectories in the vicinity of (a) x = −0.126993 and (b) x = −0.351045.

• Figure 14

(Color online) Entropy-map after an integration time of 150 for a set of 2001 × 601 initial conditions x∈ [−0.49, −0.01] and C∈ [−1, 5] for $y=x˙=0$. $y˙$ is deduced from the Jacobi integral C.

• Figure 15

(Color online) OFLI-map after an integration time of 150 for a set of 2001 initial condition for x∈ [−0.49, −0.01], and C= 3 for $y=x˙=0$. $y˙$ is deduced from the Jacobi integral C.

• Figure 16

(Color online) Trajectory corresponding to the initial condition x = −0.25912 and C = 3 for $y=x˙=0$, which is the minimum point in Figure 15 and considered as a periodic motion in Fouchard’s Figure 13 [45] (region B). The integration time is 150 and $y˙$ is deduced from the Jacobi integral C.

• Figure 17

(Color online) Quasi-periodic trajectory in the vicinity of 243 Ida (t = 1.0 × 108 s).

• Figure 18

(Color online) Frequencies in the x-, y-, and z-directions and their powers transformed by DFT.

• Figure 19

(Color online) Eight trajectories in the vicinity of 6489 Golevka (t = 5.0 × 106 s).

• Figure 20

(Color online) Entropy variations for trajectories in Figure 19 at t.

• Figure 21

(Color online) Five quasi-periodic trajectories in the vicinity of 6489 Golevka (t = 2.0 × 106 s).

• Figure 22

(Color online) Entropy variations for trajectories in Figure 21 at t.

• Table 1   Effect of sample number and sampling time on the entropies after DFT
 Case 1 Case 2 Case 3 Case 4 Case 5 Sample number N 3.0 × 106 3.0 × 106 3.0 × 106 3.0 × 105 3.0 × 104 Sample time t (s) 3.0 × 104 3.0 × 105 3.0 × 106 3.0 × 104 3.0 × 104 S7 10.30895 12.61154 14.91412 10.30895 10.30895 S8 9.80855 12.11114 14.41372 9.80855 9.80855 S7−S8 0.5004 Theoretical difference 0.5004 log10 2.3026
• Table 2   Physical characteristics of 243 Ida and 6489 Golevka
 Asteroid 243 Ida 6489 Golevka Size (km) 57.8 × 30.5 × 22.6 0.685 × 0.489 × 0.572 Density (kg·m−3) 2.6 × 103 2.7 × 103 Mass (kg) 4.2 × 1016 2.10 × 1011 Rotational period (h) 4.63 6.026 Equilibrium position and stability x (km) y (km) z (km) Stability x (m) y (m) z (m) Stability 31.3969 5.9627 0.0340 U 564.128 −23.416 −2.882 U −2.1610 23.5734 0.0975 U −571.527 35.808 −6.081 LS −33.3563 4.8507 −1.0884 U 21.647 537.470 −1.060 U −1.4150 −25.4128 −0.3785 U 26.365 −546.646 −0.182 LS
• Table 3   Initial conditions for a sample trajectory near 243 Ida in
 x (km) y (km) z (km) vx (m/s) vy (m/s) vz (m/s) 89.4871 74.3733 −33.7861 25.49 −30.33 −5.59
• Table 4   Initial conditions of trajectories near 6489 Golevka in
 Figures x (m) y (m) z (m) vx (m/s) vy (m/s) vz (m/s) Figure 19(a) 195.311 −302.412 21.385 0.0964 0.0453 −0.0169 Figure 19(b) −230.376 −297.859 784.038 0.0307 0.0221 0.0189 Figure 19(c) −386.741 178.200 −38.018 −0.0593 −0.0449 0.0205 Figure 19(d) −193.016 −606.909 −71.613 −0.0373 0.0515 0.0149 Figure 19(e) 26.728 −652.815 −47.209 −0.0467 −0.0016 −0.0077 Figure 19(f) −512.004 16.620 −4.066 0.0112 −0.0265 0.0129 Figure 19(g) 51.862 −780.369 −3.724 −0.0821 −0.0028 −0.0003 Figure 19(h) −51.291 −517.164 −292.830 −0.0042 0.0188 0.0309
• Table 5   Initial conditions of quasi-periodic trajectories near 6489 Golevka in
 Figure x (m) y (m) z (m) vx (m/s) vy (m/s) vz (m/s) Figure 21(a) 912.385 46.392 −39.291 0.0391 −0.0123 −0.0384 Figure 21(b) −362.339 −135.785 −819.084 −0.0080 −0.0115 0.0085 Figure 21(c) 119.874 −164.922 −1232.626 −0.0291 −0.1050 0.0315 Figure 21(d) −141.194 358.973 −263.625 −0.0601 0.0096 0.0193 Figure 21(e) −230.376 −297.859 784.038 0.0307 0.0221 0.0189

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