Inner Angles Made of Consecutive Three Points on ...

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Procedia IUTAM 5 (2012) 292 – 295

IUTAM Symposium on 50 years of Chaos: Applied and Theoretical

Inner Angles Made of Consecutive Three Points on a Circle for Chaotic and Random Series Ryo Takahashia*, Etsushi Namedab, Ken Umenoc a Graduate School of Engineering, Kyoto University, Katsura, Nishikyo, Kyoto 615-8510, Japan Flucto-Order Functions Research Team, Advanced Science Institute, RIKEN, Hanyang Univ. Fusion Technology Center 5F, 17 Haendang-dong, Seongdong-gu, Seoul 133-791, Korea c National Institute of Information and Communications Technology, 4-2-1 Nukui-Kitamachi, Koganei-shi, Tokyo 184-8795, Japan b

Abstract Inner angle of triangle on a circle made of consecutive three points is investigated. The quantity is known to show differences between chaotic series and random series analytically. In the paper, the inner angle properties for several series are calculated by numerical simulation. The chaotic series by the Bernoulli map, uniform random number series and normal random number series are used concretely. The inner angles for these series are compared. The formulas can be conformed numerically. In addition, it is found that the inner angles show little difference between uniform random series and normal one.

©2012 2012Published Published Elsevier Peer-review under responsibility of Takashi Hikihara and Tsutomu Kambe © by by Elsevier Ltd.Ltd. Selection and/or Peer-review under responsibility of Takashi Hikihara and Tsutomu Kambe Keywords: solvable chaos; Bernoulli map

2210-9838 © 2012 Published by Elsevier Ltd. Selection and/or Peer-review under responsibility of Takashi Hikihara and Tsutomu Kambe doi:10.1016/j.piutam.2012.06.041

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Ryo Takahashi et al. / Procedia IUTAM 5 (2012) 292 – 295

Xi+1A

Xi+2 '

Xi-1 1.3

Bernoulli

Xi

O

O'

Inner angle / rad

Random Num. (uniform and normal)

1.2

1.1

2

4

6

8

10

12

14 16

18

20

p

Fig. 1: An inner angle which we focus on here is obtained like this figure. In focusing on the consecutive three points, namely Xi-1, Xi and Xi+1, the objective inner angle in this step is D. The inner angle D’ in nest step is obtained by the next consecutive points, namely Xi, Xi+1 and Xi+2.

Fig. 2: Inner angle as functions of p.

1. Introduction These days, many properties of chaos are found. Solvable chaos models help us to find new characteristic of chaos[1-3]. These models also help us to evaluate performance of applications utilizing chaos[4-8]. However, it is still difficult to distinguish chaotic data and pure random data in observed data. Even now, researches on methods to judge whether the obtained series are chaos or pure random attracts many researchers' interests. Recently, an inner angle of triangle on a circle made of consecutive three points is found to show the difference between chaotic series and random series[9]. In the paper, the inner angle method proposed in [9] is summarized and conformed by numerical simulations. We focus on the chaotic series obtained by the Bernoulli map. Also, the uniform random number series and normal number series are focused on and compared. 2. Inner angle of triangle on a circle made of consecutive three points The inner angle which we focus on here is defined as follows. After obtaining the series data {Xi} which are normalized and inflated to be within [0,1], we plot points consecutively on the circle whose circumferential length is unity one. A length of segment in the circle starting at point O' in Fig. 1 corresponds to the series data. For example, the data 0 and 0.25 are plotted on the points O' and A in Fig. 1, respectively. Then, we make a triangle by using consecutive three points, namely 'Xi-1XiXi+1. The angle which we focus on here is the inner angle at the middle point of consecutive three points, namely D = ‘ Xi. Figure 1 shows an example. We investigate an averaged value of this angle. Here, we set an angle between the segments OXi-1 and OXi+1 as T. This value takes a positive value in a counterclockwise direction from OXi-1 to OXi+1. By using the angle T, the angle D can be represented as follows:

D = (2S - T) / 2, Xi-1 o Xi o Xi+1 T / 2, Xi-1 o Xi+1 o Xi

(1)

where these expressions about D consist in the case that these points are in order Xi-1 o Xi o Xi+1 and Xi1 o Xi+1 o Xi counterclockwise, respectively. Considering the order of the points Xi-1, Xi and Xi+1 and these magnitudes, these cases can be classified into six cases. Tables 1 and 2 show the values of T and D

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Ryo Takahashi et al. / Procedia IUTAM 5 (2012) 292 – 295

in these six cases. In the numerical simulation, the relation among the consecutive three points is classified into these six cases. Then, using the formulas in tables, the inner angle is calculated. It is already known analytically that the expected inner angle for the chaotic series generated by the Bernoulli map and the uniform random number series are obtained as = S (2 + 1 / p) / 6, Bernoulli map = S / 3, uniform random

(2)

Respectively[9]. Here the quantity p is the order of Bernoulli map defined in Eq. (3) shown below. Table 1. Xi-1 o Xi o Xi+1

T

D

Xi-1 < Xi < Xi+1

2S ( Xi+1 - Xi-1 )

S ( 1 + Xi-1 - Xi+1 )

Xi+1 < Xi-1 < Xi

2S ( Xi+1 + 1 - Xi-1 )

S ( Xi-1 - Xi+1 )

Xi < Xi+1 < Xi-1

2S ( Xi+1 + 1 - Xi-1 )

S ( Xi-1 - Xi+1 )

T

D

Xi-1 < Xi+1 < Xi

2S ( Xi+1 - Xi-1 )

S ( Xi+1 - Xi-1 )

Xi < Xi-1 < Xi+1

2S ( Xi+1 - Xi-1 )

S ( Xi+1A - Xi-1 )

Xi+1 < Xi < Xi-1

2S ( Xi+1 + 1 - Xi-1 )

S ( 1 - Xi-1 + Xi+1 )

Table 2. Xi-1 o Xi+1 o Xi

3. Numerical simulation of inner angle for chaotic and random series Here, the Bernoulli map B is used to generate chaotic series. As is well known, the map is defined as follows: Bp(x) = px – ¬px¼ , 0 d x d 1

(3)

where p t 2, p  N is the order. By the map, the chaotic series {Xp,j} which are within (0,1) are obtained by Xp,j+1 = Bp(Xp,j). The series by Bernoulli map is plotted on a circle whose circumferential length is unity 1 according to the procedure explained in Sec. 2. We also focus on the uniform random number and normal one. These series are normalized and inflated to set within [0,1]. The uniform random number are generated within [0,1] uniform distribution. On the other hand, since the distribution for the normal random number is Gaussian, normalization is necessary. After obtaining the normal random number series {Xj}, the series are divided by the maximum value in {|Xj|}, and 0.5 is added to normalized data. Applying these procedures mentioned above, the averaged inner angles for these series were calculated numerically. In the numerical simulation, 10 samples of series for each series whose length is 106 are used to calculate averaged inner angles. Figure 2 shows the averaged inner angle as functions of the order p for the chaotic series and pure random ones. Since the inner angles for the random number series are

Ryo Takahashi et al. / Procedia IUTAM 5 (2012) 292 – 295

independent of p, the averaged values obtained numerically are used. The averaged inner angle for the uniform random number series and normal one were 1.046825 and 1.047239, respectively. It is found that the angles for the random number series are almost the same each other in spite of the difference in the distribution. In addition, the angles for the chaotic series approach the one for the random number as p increases. 4. Conclusion In the paper, the differences about the inner angle among the chaotic series and random number ones have been investigated. The angles for the chaotic series are obviously different from the ones for the random number series. In addition, in spite of the different distribution of the uniform and normal random number, the angles are almost the same. The property about the inner angle can be expected to have a potential for judging whether the series is chaotic or pure random.

Acknowledgements This work was supported in part by Kyoto University, Grant-in-Aid for Young Scientists (Startup), 2011.

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