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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

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  • ReceivedDec 7, 2017
  • AcceptedJan 5, 2018
  • PublishedApr 2, 2018
PACS numbers

Abstract


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

    S7S8

    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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