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Chaotic dynamics in the Friedmann equation
Chaos, Solitons and Fractals 24 (2005) 407–422 www.elsevier.com/locate/chaos
Chaotic dynamics in the Friedmann equation Yosuke Tanaka b
a,*
, Yuzi Mizuno b, Tatsuhiko Kado
c
a Department of Physics, Faculty of Engineering, Kyushu Kyoritsu University, Kita-kyushu 807, Japan Department of Electrical, Electronic and Information Engineering, Faculty of Engineering, Kyushu Kyoritsu University, Kita-kyushu 807, Japan c Department of Mechanical Engineering, Faculty of Engineering, Kyushu Kyoritsu University, Kita-kyushu 807, Japan
Accepted 14 September 2004
Abstract We have studied relativistic equations and chaotic behaviors of the gravitational field on the basis of general relativity and chaotic dynamics. The Friedmann equation [the space component] shows chaotic behaviors in case of the _ inflation ðG=G > 0Þ and open (f = 1) universe. There occurs non-chaotic behaviors in other cases _ ðG=G 5 0; f ¼ 0; f ¼ þ1Þ. We have shown the following properties of the Friedmann chaos; (1) the sensitive dependence of solutions on parameters, (2) the self-similarity of solutions in the x–_x plane and the x–q plane. Numerical calculations were carried out with the use of the microsoft EXCEL. We have also discussed the self-similarity and the hierarchy structure of the universe on the basis of E infinity theory. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Recently, several authors have studied the metric tensors and the non-linear properties of the gravitational field on the basis of general relativity and non-linear theory [1–4]. Especially, El Naschie has studied the solitonic solution of the gravitational field equation with the use of the Eguchi–Hanson metric [1]. Rugh has analyzed the extended Robertson– Walker metric and discussed the chaos in the Einstein equation [3]. We have studied the compact Robertson–Walker metric and shown the Friedmann equation demonstrates the chaotic behaviors in the case the inflation universe _ ðG=G > 0Þ [4]. In this paper, we discuss the chaotic behaviors of the Friedmann equation in case of the open universe (f = 1). According to the general theory of relativity, the gravitational field equation is given as follows [5–7]: Gik þ jT ik þ kgik ¼ 0; j¼
8pK ¼ 1:86 1027 ½cm=g; c2
*
Corresponding author. Tel.: +81 93 693 3202; fax: +81 93 603 8186. E-mail address: [email protected] (Y. Tanaka).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.034
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
where the symbols Gik and Tik denote the Einstein tensor and the energy-momentum tensor, respectively. The third term is the cosmological term, and the symbols k and gik show the universal constant and the metric tensor of the fourdimensional continuum, respectively [8,9]. The above equation is called as ‘‘the Einstein equation’’ in this paper. From chaos theoretical point of view, the non-linear property of relativistic field equation is interesting and may introduce chaotic behaviors under certain conditions. In Section 2, following the text book of Einstein [7], we study the Friedmann model and show the chaotic behaviors of the gravitational field. In Section 3, we give numerical results and discuss the chaotic properties in the Friedmann equation. In Section 4, we give some discussions on the related problems. Summary is given in Section 5.
2. The Friedmann model Several authors discussed the Friedmann equation [3,4,7]. Here, we review briefly the Friedmann model of the universe [7]. 2.1. The Friedmann equation Friedmann found the solution of the Einstein equation in 1922. The metric is given as follows: ds2 ¼ ðdx4 Þ2 G2 A2 ½ðdx1 Þ2 þ ðdx2 Þ2 þ ðdx3 Þ2 ; 1 ; ðf ¼ þ1; 0; 1Þ: A¼ 1 þ 4f r2 The function G = G(t) satisfies the following equations: 1 € þ ðGÞ _ 2 g þ jp k ¼ 0; ff þ 2GG G2 3 _ 2 g þ jq þ k ¼ 0: 2 ff þ ðGÞ G Here, the symbols p and q show the pressure and the density, respectively. These equations are the space component and the time component of the Einstein equation, respectively. By eliminating the constant k from these equations, we have € G_ 2 f j G 2 þ ðp þ qÞ ¼ 0: 2 G G G Also, by eliminating the parameter f, we obtain € j 1 G þ ð3p þ qÞ k ¼ 0: 3 G 6 Among these four equations, two of them are independent, and we discuss the properties of three equations except for the second equation [the time component of the Einstein equation]. For convenience, we use the variable x instead of G; x¼
1 : G
With the use of this relation, the above three equations are transformed into the following form [4]: €x þ a
x_ 2 þ bx þ cx3 ¼ d F ðp; q; xÞ; x
where the function F(p,q,x) is defined as follows. 8 9 ½Case 1 > > < px; = F ðp; q; xÞ ¼ ðp þ qÞx; ½Case 2 : > > : ; ð3p þ qÞx; ½Case 3
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
409
Table 1 Values of coefficients a, b, c and d in the Friedmann equationa
Case 1 Case 2 Case 3 a
a
b
c
d
Comment
52
k 2
2f
1 2
0
f 0
k 2 k 2 k 6
F(p, q, x) = px F(p, q, x) = (p + q)x F(p, q, x) = (3p + q)x
k 3
The Friedmann equation is given as follows:
€x þ a
ð_xÞ2 þ bx þ cx3 ¼ d F ðp; q; xÞ: x
See text for details.
The coefficients a, b, c and d are listed in Table 1. At this stage, we have two independent field equations [the space component and the time component of the Einstein equation] and three unknowns (p, q and x = 1/G). To obtain the solutions of these equations, we must have one more equation. Here, we use the conservation law q_ ¼ 3
G_ ðp þ qÞ; G
which is obtained by the covariant derivative T ik;i ¼ 0;
ði; k ¼ 1; 2; 3; 4Þ:
Then, we have the standard form of equations. 8 > < x_ ¼ y; 2 y_ ¼ a yx bx cx3 þ d F ðp; q; xÞ; > : q_ ¼ 3 yx ðp þ qÞ: _ the standard form of ‘‘the Friedmann equation’’ in this paper. We summarize some _ qÞ We call these equations ð_x; y; chaotic equations and expansion rate of phase space volume in Table 2 [10–15]. As is well known, Lorenz introduced the two-hole type attractor [14], and Ro¨ssler derived the one-hole type attractor [15]. The expansion rate of phase space volume is given as follows:
Table 2 Nonlinear equation and expansion rate of phase space volume Nonlinear equation (1) Lorenz equationa 8 < x_ ¼ rðx yÞ y_ ¼ y xz þ rx : z_ ¼ xy bz (2) Ro¨ssler equationb 8 < x_ ¼ ðy zÞ y_ ¼ x þ ay : z_ ¼ b þ zðx lÞ (3) Friedmann equation 8 < x_ ¼ y 2 y_ ¼ a yx bx cx3 þ dF ðp; q; xÞ : q_ ¼ 3 yx ðp þ qÞ a b
Taken from Ref. [14]. Taken from Ref. [15].
Expansion rate
V_ V
¼ r 1 b
V_ V
¼aþxl
V_ V
¼ ð3 2aÞ xx_
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V_ y ¼ ð3 2aÞ ; V x G_ ¼ ð3 2aÞ : G _ Therefore, there may occur chaos in case of the inflation universe ðG=G > 0Þ, and there may occur no chaos in case of _ the flat and the contraction universe ðG=G 5 0Þ [4]. 2.2. Properties of the Friedmann equation Here, we discuss the approximate solutions of the Friedmann equation and give comments on the properties of these solutions. 2.2.1. The approximate solution in the x–_x plane We consider the Friedmann equation in Case 1 (a = 5/2, b = k/2, c = f/2, d = j/2). The equation is written as follows: €x þ a
x_ 2 þ cx3 ¼ d p x; x
where we set k = 0 in the Friedmann model. By integrating the above equation, we have
Table 3 The numerical solutions of the Friedmann equation (Case 1) t
x
y
q
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.5 0.508 0.51628 0.524853 0.533735 0.542939 0.552483 0.562383 0.572658 0.583327 0.594412 0.605935 0.617919 0.630391 0.643377 0.656908 0.671014 0.685729 0.70109 0.717135 0.733907 0.751451 0.769815 0.789052 0.809217 0.830373 0.852585 0.875924 0.900466 0.926295 0.9535
0.2 0.207 0.214337 0.222032 0.230108 0.238588 0.247501 0.256873 0.266736 0.277122 0.288067 0.29961 1 0.311793 0.324661 0.338263 0.352651 0.367884 0.384023 0.401138 0.419301 0.438592 0.459098 0.480915 0.504144 0.528896 0.555294 0.583468 0.613561 0.645726 0.68013 0.716951
0.2 0.224 0.249622 0.277004 0.306295 0.337662 0.371287 0.407374 0.446146 0.487851 0.532766 0.581195 0.633481 0.690004 0.751188 0.817509 0.889499 0.967756 1.052953 1.145846 1.24729 1.358251 1.479824 1.61325 1.759941 1.921503 2.099773 2.296848 2.515131 2.757379 3.026763
For convenience, only the first 30 lines are shown.
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
x_ 2 þ 2a
Z
411
x_ 3 c dx þ x4 ¼ d p x2 þ const; x 2
where R 3 we simply assume the pressure p is constant. The variable x and x_ satisfy the following condition because of the ð_x =xÞdx term: If x ! 0; then x_ ! 0: c =0.5
3.5 3 2.5 2
x
1.5 1 0.5 0
0
500
1000
1500
2000
2500
3000
t
Fig. 1. The numerical solutions of the Friedmann equation [Case 1]. The figure is shown in the t–x plane for c = +0.5. The other parameters are fixed and given as follows: a = 2.5, b = 0, d = 0.25, p = 0.3 and x0 = 0.5. Only in the case of c = +0.5, there occurs chaos in the Friedmann equation [Case 1].
c=0.5 3 2 1
.
x 0 -1 -2 -3 -4 0
500
1000
1500
2000
2500
3000
t
Fig. 2. The numerical solutions of the Friedmann equation [Case 1] The figure is shown in the t–_x plane for c = +0.5. The other parameters are fixed and given as follows: a = 2.5, b = 0, d = 0.25, p = 0.3 and x0 = 0.2. Only in case c = +0.5, there occurs chaos in the Friedmann equation [Case 1].
c=0.5 120 100 80
ρ
60 40 20 0 -20 0
500
1000
1500
2000
250 0
3000
t
Fig. 3. The numerical solutions of the Friedmann equation [Case 1]. The figure is shown in the t–q plane for c = +0.5. The other parameters are fixed and given as follows: a = 2.5, b = 0, p = 0.3 and q0 = 0.2. Only in case c = +0.5, there occurs chaos in the Friedmann equation [Case 1].
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In the next section, this property is shown to be approximately satisfied by the numerical solution of the Friedmann equation. (i) The vibrational mode (f = 1; c = +0.5). 3 For the positive value R of x, the factor x_ takes both positive and negative values in case of the vibrational mode. Therefore, the integral ð_x3 =xÞdx is very small and almost zero for the one cycle, and we have c x_ 2 d p x2 þ x4 ’ constðtÞ: 2 From this equation, we obtain 2 c 2 d p x ’ constðtÞ: x_ 2 þ 2 c This solution has the approximate symmetry under the following replacements: x ! ðxÞ and=or x_ ! ð_xÞ: In the next section, this property is also shown to be approximately satisfied by the numerical solution of the Friedmann equation. (ii) The exponential mode (f = 0, +1; c = 0, 0.5).
. x
c=-0.5
5 4 3 2 1 0 -1
-1 0 -2
x 1
2
3
4
2
3
4
3
4
-3 -4 -5
. x
c=0
5 4 3 2 1 0 -1
-1 0 -2
x 1
-3 -4 -5
c=0.5
5 4 3
-1
2 1 0 -1 0
x 1
2
-2 -3 -4 -5
Fig. 4. The parameter c dependence of solutions of the Friedmann equation [Case 1]. The figures are shown in the x–_x plane for c = +0.5, 0 and 0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0, d = 0.25, p = 0.3, x0 = 0.5 and x_ 0 ¼ 0:2. Only in case c = +0.5 (f =1), there occurs chaos in the Friedmann equation [Case 1].
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
413
The solution is written as follows: c x_ 2 ¼ const: þ d p x2 x4 2a 2
Z
x_ 3 dx; x
R In case of the exponential mode, the factor x_ 3 takes only the positive value, and the integral ð_x3 =xÞdx increases as the variable x_ increases. Therefore, from above equation, the variable x_ increases as the variable x increases because of negative values of coefficients a and c (in Table 1). The exponential mode of solutions is shown in the next section. 2.2.2. The approximate solution in the x–q plane The density q satisfies the following equation: q_ x_ ’3 ; q x in case the pressure p is small and negligible. From this equation, we have the approximate solution q ’ cx3 ;
ρ
c=-0.5
14 12 10 8 6 4 2 0 -1
x 0
-2
1
ρ
2
3
4
2
3
4
3
4
c=0
14 12 10 8 6 4 2 0 -1
-2
x 0
1
ρ
c=0.5
14 12 10 8 6 4 2 0 -1
-2
x 0
1
2
Fig. 5. The parameter c dependence of solutions of the Friedmann equation [Case 1]. The figures are shown in the x–q plane for c = +0.5, 0 and 0.5, respectively. The other parameters are fixed as follows: a = 2.5, b = 0, d = 0.25, p = 0.3, x0 = 0.5 and q0 = 0.2. Only in case c = +0.5 (f =1), there occurs chaos in the Friedmann equation [Case 1].
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
where the factor c is the numerical constant. As is shown in the next section, the trajectory in the x–q plane is described approximately by this equation. (i) The vibrational mode (f = 1; c = +0.5). In the next section, we show that the numerical results satisfy approximately the above relation. The violation of the symmetry may be attributed to the inhomogeneous term (p 5 0) and the non-linear terms ð_x2 and x3 Þ. (ii) The exponential mode (f = 0, +1; c = 0, 0.5). In case of the large value of x, the numerical results satisfy approximately the above relation (q / x3), which are shown in the next section.
3. Numerical results We have analyzed the Friedmann equation with the use of the microsoft EXCEL and obtained the chaotic behaviors for the open universe (f = 1) in Case 1. Following the name of ‘‘the Lorenz Chaos’’ [14] and that of ‘‘the Ro¨ssler Chaos’’ [15], we may call these chaotic behaviors ‘‘the Friedmann chaos’’. 3.1. Results of Case 1 The numerical results of Case 1 are given in Table 3, where only the first 30 lines are shown for convenience. The full length of numerical results is 5994 lines in our calculation. The unit of time (t), space (x) and density (q) is arbitrary. We show the numerical solutions of the Friedmann equation [Case 1] in Figs. 1–3, where the figures are demonstrated in the t–x plane, t–_x plane and t–q plane, respectively, for c = +0.5(f = 1). The other parameters are fixed and given in each figures. These figures show the typical non-linear vibrational modes. Only in case c = +0.5(f = 1), there occurs chaos in the Friedmann equation [Case 1], as far as we use the parameters listed in Table 1. We show the parameter c dependence of solutions in the Friedmann equation [Case 1] in Figs. 4 and 5, where the figures are shown in the x–_x plane and x–q plane for c = +0.5, 0 and 0.5, respectively. The other parameters are fixed and given in each figures. Only in case c = +0.5(f = 1), there occurs chaos in the Friedmann equation [Case 1], as stated before.
. x
-1
5 4 3 2 1 0 -1 0 -2 -3 -4 -5
x 1
. x
-1
. x
p=0.28
2
3
-1
4
5 4 3 2 1 0 -1 0 -2 -3 -4 -5
. x
p=0.32
5
5
4 3
4 3
2 1 0
2 1 0
-1 0 -2
x 1
2
3
4
-1
-1 0 -2
-3 -4
-3 -4
-5
-5
p=0.30
x 1
2
3
4
2
3
4
p=0.34
x 1
Fig. 6. Chaotic behaviors of the Friedmann equation [Case 1]. Figures are shown in the x–_x plane. The parameter p is changed from 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0, c = +0.5, d = 0.25, x0 = 0.5 and x_ 0 ¼ 0:2.
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
ρ
ρ
p=0.28
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0 -1
-2
x 0
1
ρ
2
3
4
-2
12
12
10
10
8
8
6
6
4
4
2
2
-2
3
4
2
3
4
3
4
0
x 2
1
p=0.34
14
1
x 0
ρ
14
0
p=0.30
0 -1
p=0.32
0 -1
415
-1
-2
x 0
1
2
Fig. 7. Chaotic behaviors of the Friedmann equation [Case 1]. Figures are shown in the x–q plane. The parameter p is changed from 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0, c = +0.5, d = 0.25, x0 = 0.5 and q0 = 0.2.
Also, we show the chaotic behaviors in the Friedmann equation [Case 1] in Fig. 6 and 7, where the figures are demonstrated in the x–_x plane and the x–q plane for p = 0.28,0.30,0.32 and 0.34, respectively. The other parameters are fixed and given in each figures. These figures show the self-similarity, which is one of the typical properties of chaotic behaviors. 3.2. Results of Case 2 The numerical results of Case 2 are given in Table 4, where only the first 30 lines are shown for convenience. The full length of numerical results is 5994 lines in our calculation. The unit of time (t), space (x) and density (q) is arbitrary. We show the numerical solutions of the Friedmann equation [Case 2] in Figs. 8–10, where the figures are demonstrated in the t–x plane, the t–_x plane and the t–q plane, respectively, for c = 1.0(f = 1.0). The other parameters are fixed and given in each figures. These figures show the critical vibrational modes, and there occurs no chaos in Case 2, as far as we use the parameters listed in Table 1. We show the parameter c dependence of solutions in the Friedmann equation [Case 2] in Figs. 11 and 12, where the figures are shown in the x–_x plane and the x–q plane for c = +1.0, 0 and 1.0, respectively. The other parameters are fixed and given in each figures. From these figures, we see that there occurs no chaos in Case 2, even if in case c = +1.0(f = 1.0), as far as we use the parameters listed in Table 1. Also, we show the numerical solutions in the Friedmann equation [Case 2] in Figs. 13 and 14, where the figures are shown in the x–_x plane and the x–q plane for p = 0.28, 0.30, 0.32 and 0.34, respectively. The other parameters are fixed and given in each figures. From these figures, we see also that there occurs no chaos in Case 2, even if in case c = +1.0(f = 1.0), as stated before.
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
Table 4 The numerical solutions of the Friedmann equation (Case 2) t
x
y
P
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.5 0.508 0.516028 0.52408 0.53215 0.540235 0.54833 0.55643 0.564529 0.572623 0.580708 0.588777 0.596825 0.604848 0.61284 0.620796 0.62871 0.636579 0.644395 0.652155 0.659854 0.667485 0.675046 0.682529 0.689932 0.69725 0.704477 0.71161 0.718645 0.725578 0.732404
0.2 0.2007 0.20129 0.201766 0.202127 0.202368 0.202489 0.202487 0.202359 0.202105 0.201723 0.201211 0.20057 0.199799 0.198898 0.197867 0.196706 0.195417 0.194001 0.192459 0.190792 0.189004 0.187097 0.185072 0.182934 0.180686 0.17833 0.175871 0.173312 0.170658 0.167913
0.2 0.224 0.248843 0.274533 0.301076 0.328473 0.356724 0.385826 0.415775 0.446563 0.478183 0.510621 0.543864 0.577895 0.612695 0.648241 0.684509 0.721472 0.759101 0.797363 0.836224 0.875648 0.915595 0.956025 0.996895 1.038159 1.079772 1.121685 1.163848 1.206212 1.248724
For convenience, only the first 30 lines are shown.
c=1.0
1 0.9 0.8 0.7 0.6
x 0.5 0.4 0.3 0.2 0.1 0 0
500
1000
1500
2000
2500
3000
t
Fig. 8. The numerical solutions of the Friedmann equation [Case 2]. The figure is shown in the t–x plane for c = +1. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3 and x0 = 0.5. There occurs no chaos even if c = 1.0 in the Friedmann equation [Case 2].
3.3. Results of Case 3 As discussed in the previous section (Section 2), the results of Case 3 show the exponential mode because of b = 0 and c = 0. The results are trivial and not shown. Needless to say, there occurs no chaos in Case 3.
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
417
c=1.0 0.25 0. 2 0.15
. x
0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0
500
1000
1500
2000
2500
3000
t
Fig. 9. The numerical solutions of the Friedmann equation [Case 2]. The figure is shown in the t–_x plane for c = +1. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3 and x_ 0 ¼ 0:2. There occurs no chaos even if c = 1.0 in the Friedmann equation [Case 2].
c=1.0
3 2.5 2
ρ 1.5 1 0.5 0 - 0.5 0
500
1000
1500
2000
2500
3000
t
Fig. 10. The numerical solutions of the Friedmann equation [Case 2]. The figure is shown in thet–q plane for c = +1. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3 and q0 = 0.2. There occurs no chaos even if c = 1.0 in the Friedmann equation [Case 2].
3.4. Properties of the Friedmann chaos Here, we discuss properties of the Friedmann chaos. (1) The sensitive dependence of solutions on the parameters. From Fig. 4, we see there occurs chaos in case c = +0.5 (f = 1.0). In case c = 0 (f = 0) and c = 0.5 (f = +1.0), there occurs no chaos, as far as we use the parameters listed in Table 1. These facts show that the trajectories in the x–_x plane demonstrate the sensitive dependence on the parameter c. Also, the trajectories in the x–q plane show the sensitive dependence on the parameter c, as in Fig. 5. As is well known, chaotic equations have the sensitive dependence on parameters. (2) The self-similarity of solutions. As demonstrated in Fig. 6, the trajectories in the x–_x plane show the self-similarity, which is the general property of chaotic motions. Also, the trajectories in the x–q plane show the self-similarity, as in Fig. 7. These properties (1) and (2) are general characters of chaotic behaviors and discussed by authors [10–13].
4. Discussions Here, we give some discussions on the related problems.
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
x
- 0.5
1 0.8 0.6 0.4 0.2 0 - 0.2 0 - 0.4 - 0.6 - 0.8 -1
. x
- 0.5
1 0.8 0.6 0.4 0.2 0 - 0.2 0 - 0.4 - 0.6 - 0.8 -1
. x
- 0.5
1 0.8 0.6 0.4 0.2 0 - 0.2 0 - 0.4 - 0.6 - 0.8 -1
c=-1
x 0.5
1
1.5
1
1.5
c=0
x 0.5
c=1
x 0.5
1
1.5
Fig. 11. The parameter c dependence of solutions of the Friedmann equation [Case 2]. The figures are shown in the x–_x plane for c = +1, 0 and 1.0, respectively. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3, x0 = 0.5 and x_ 0 ¼ 0:2.
4.1. Computer simulations In Case 2, we have chaotic diagrams, if we change the values of parameters in the Friedmann equation. For example, using the parameter a = 0.3 instead of a = 1.0, we have chaotic behaviors in the x–_x plane and the x–q plane. However, these are not realistic figures but computational simulations, and the figures are not shown here. 4.2. A new theory of space–time structure Recently, El Naschie has studied a new theory of space–time structure on the basis of E infinity theory [16–19]. According to this theory, the Haussdorff dimension of the universe is given as follows: hDimðe1 ÞiH ¼
1 X
n/n ;
n¼0 3
¼/ ; ¼ 4 þ /3 ; where the symbol / denotes the golden ratio. The dimension of the universe turns out to be fractal (4 + /3 = 4.236), and the symbol / satisfies the relations
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
ρ
419
c=-1.0
1 0.8 0.6 0.4 0.2 0 - 0.2 0
- 0.5
x 0.5
1
1.5
1
1.5
1
1.5
- 0.4 - 0.6 - 0.8 -1
ρ
c=0
1 0.8 0.6 0.4 0.2 0 - 0.2 0
- 0.5
x 0.5
- 0.4 - 0.6 - 0.8 -1
ρ
c=1.0
1
0.8 0.6 0.4 0.2 0 - 0.2 0
- 0.5
x 0.5
- 0.4 - 0.6 - 0.8 -1
Fig. 12. The parameter c dependence of solutions of the Friedmann equation [Case 2]. The figures are shown in the x–q plane for c = +1, 0 and 1.0, respectively. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3, x0 = 0.5 and q0 = 0.2.
/¼ ¼
1 ; 1þ/ 1 : 1 þ 1þ1 1 1þ
In general, the symbol / satisfies the following relations: /n ¼ ¼
1 fn þ ð1Þnþ1 /n
;
1 fn þ ð1Þnþ1 f
;
1 n þð1Þ
nþ1
1 fn þ
where {fn} denotes the Fibonacci sequence with f1 = 1 and f2 = 3. For example, in case n = 3, we have
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
x
- 0.5
x
p=0.28
1
1
0.8 0.6
0.8
0.4 0.2
0.4
0.6
0.2
0 - 0.2 0
0.5
1
- 0.5
1.5
x
0
x
0
- 0.2
- 0.4 - 0.6
- 0.4
- 0.8 -1
- 0.8
1
1.5
1
1.5
-1
p=0.32
x
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
p=0.34
0.2
x
0 - 0.2
0.5
- 0.6
x
- 0.5
p=0.30
0
0.5
1
1.5
x
0 - 0.5
- 0.2
- 0.4
- 0.4
- 0.6
- 0.6
- 0.8
- 0.8
-1
-1
0
0.5
Fig. 13. The numerical solutions of the Friedmann equation [Case 2]. Figures are shown in the x–_x plane. The parameter p is changed from 0.28 to 0.34. The other parameters are fixed and given as follows: a = 1.0, b = 0, c = +1.0, d = 0.25, x0 = 0.5 and x_ 0 ¼ 0:2.
ρ
p=0.28
1
ρ 1
0.8
0.8
0.6
0.6
0.4
0.4 0.2
0.2 0 - 0.5
-0.2 0
0 .5
1
- 0.5
1.5
-0.2 0 -0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
p=0.32
ρ
1
1
0.8 0.6
0.8
0.4 0.2
0.4
-0.2 0 -0.4 -0.6 -0.8 -1
x 0.5
1
1.5
p=0.34
0.6
0.2
0 -0.5
0
x
-0.4
ρ
p=0.30
x 0.5
1
1.5
0 - 0.5
-0.2 0
x 0.5
1
1.5
-0.4 -0.6 -0.8 -1
Fig. 14. The numerical solutions of the Friedmann equation [Case 2]. Figures are shown in the x–q plane. The parameter p is changed from 0.28 to 0.34. The other parameters are fixed and given as follows: a = 1.0, b = 0, c = +1.0, d = 0.25, x0 = 0.5 and q0 = 0.2.
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
421
1 ; 4 þ /3 1 : ¼ 4 þ 4þ1 1
/3 ¼
4þ
These relations reflect the self-similarity of the universe. Indeed, as discussed in the previous section, the Friedmann chaos shows the self-similarity in the x–_x plane and the x–q plane (Fig. 6 and 7). E infinity theory is suitable for describing the self-similarity and the hierarchy properties of the universe, such as the mass spectrum of high energy particle physics [17,20]. The fractal dimension of chaotic attractors, such as the Lorenz attractor and the Friedmann attractor may be described well by E infinity theory. This problem will be discussed elsewhere.
5. Summary Finally, we summarize our work as follows. We have studied relativistic equations and chaotic motions of the universe on the basis of general relativity and chaotic dynamics. The Friedmann equation [the space component] shows _ the chaotic behaviors in case of the inflation ðG=G > 0Þ and open (f = 1) universe. In other cases _ ðG=G 5 0; f ¼ 0; f ¼ þ1Þ, there occurs no chaos in the Friedmann equation, as far as we use the parameters listed in Table 1. The Friedmann chaos has following properties: (1) The sensitive dependence of solutions on the parameters. (2) The self-similarity of solutions in the x–_x plane and the x–q plane. These characters are general properties of chaotic behaviors, such as the Lorenz chaos [14] and the Ro¨ssler chaos [15]. Especially, the self-similar characters of these chaos correspond to the mathematical properties of E infinity theory [17,18]. Acknowledgments We would like to thank the Information Processing Center of Kyushu Kyoritsu University for using the microsoft EXCEL, which is very powerful for analyzing linear and non-linear equations. Thanks are also to Ms. H. Shiroshita for typing the manuscript of this paper.
Appendix A Here, we show the difference equations for numerical calculations of the Friedmann equation. (1) The differential equations. The differential equations of the Friedmann model, which is mentioned in the text, are given as follows. 8 dX ¼ Y; > < dt 2 dY ¼ a YX bX cX 3 þ d F ðp; q; X Þ; dt > : dq ¼ 3 XY ðp þ qÞ; dt where the function F is defined in the following form. 8 > < pX ; F ðp; q; X Þ ¼ ðp þ qÞX ; > : ð3p þ qÞX ;
½Case 1; ½Case 2; ½Case 3:
(2) The difference equations. The difference equations, which correspond to the above differential equations, are written in the following form.
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
8 X 2 ¼ X 1 þ DtðY 1 Þ; > > n o > < Y2 Y 2 ¼ Y 1 þ Dt a X 11 bX 1 cX 31 þ d F ðp; q1 ; X 1 Þ ; > n o > > : q ¼ q þ Dt 3 Y 1 ðp þ q Þ : 2 1 1 X1 where the function F is given as follows. 8 ½Case 1; > < pX 1 ; F ðp; q1 ; X 1 Þ ¼ ðp þ q1 ÞX 1 ; ½Case 2; > : ð3p þ q1 ÞX 1 ; ½Case 3: (3) Numerical calculations. With the use of the microsoft EXCEL (Office XP Professional), we have carried out the numerical calculations of the Friedmann equation. The EXCEL is powerful especially for analyzing non-linear equations.
References [1] El Naschie MS. Gravitational instantons in Hilbert space and the mass of high energy elementary particles. Chaos, Solitons & Fractals 2004;20:917–23. [2] Ôt Hooft G. A physical interpretation of gravitational instantons. Nucl Phys B 1989;315:512–27. [3] Rugh SE. Chaos in the Einstein equations. In: El Naschie MS, Ro¨ssler OE, Prigogine I, editors. Quantum mechanics diffusion and chaotic fractals. Oxford: Pergamon; 1995. p. 161–82. [4] Tanaka Y. Relativistic field theory and chaotic dynamics. Chaos, Solitons & Fractals 2005;23:33–41. [5] Einstein A. The principle of relativity, translated by Perrett W. and Jeffery GB. New York: Dover Publications; 1952. p. 107–64. [6] Weyl H. The principle of relativity, translated by Perrett W. and Jeffery GB. New York: Dover Publications; 1952. p. 199–216. [7] Einstein A. The meaning of relativity. 5th ed.. New Jersey: Princeton University Press; 1955. Appendix. [8] Einstein A. The principle of relativity, translated by Perrett W. and Jeffery GB. New York: Dover Publications; 1952. p. 177–88. [9] Bergmann PG. Introduction to the theory of relativity. New York: Dover Publications; 1976. p. 198–210. [10] Gutzwiller MC. Chaos in classical and quantum mechanics. New York: Springer-Verlag; 1990. [11] Rasband SN. Chaotic dynamics of nonlinear systems. New York: John Wiley & Sons; 1990. p. 196–8. [12] Ott E. Chaos in dynamical systems. Cambridge: Cambridge University Press; 1993. p. 339–44. [13] Mori H, Kuramoto Y. Dissipative structure and chaos. Berlin: Springer-Verlag; 1998. p. 245–7. [14] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 1963;20:130–41. [15] Ro¨ssler OE. An equation for continuous chaos. Phys Lett 1976;57A(5):397–8. [16] El Naschie MS. Non-linear dynamics and infinite dimensional topology in high energy particle physics. Chaos, Solitons & Fractals 2003;17:591–9. [17] El Naschie MS. A review of E infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36. [18] El Naschie MS. Quantum gravity from descriptive set theory. Chaos, Solitons & Fractals 2004;19:1339–44. [19] El Naschie MS. The Higgs-physical and number theoretical arguments for the necessity of a triple elementary particle in super symmetric space–time. Chaos, Solitons & Fractals, in press. [20] Tanaka Y. Space–time symmetry, Chaos and E infinity theory. Chaos, Solitons & Fractals 2005;23:335–49.
Chaotic dynamics in the Friedmann equation Yosuke Tanaka b
a,*
, Yuzi Mizuno b, Tatsuhiko Kado
c
a Department of Physics, Faculty of Engineering, Kyushu Kyoritsu University, Kita-kyushu 807, Japan Department of Electrical, Electronic and Information Engineering, Faculty of Engineering, Kyushu Kyoritsu University, Kita-kyushu 807, Japan c Department of Mechanical Engineering, Faculty of Engineering, Kyushu Kyoritsu University, Kita-kyushu 807, Japan
Accepted 14 September 2004
Abstract We have studied relativistic equations and chaotic behaviors of the gravitational field on the basis of general relativity and chaotic dynamics. The Friedmann equation [the space component] shows chaotic behaviors in case of the _ inflation ðG=G > 0Þ and open (f = 1) universe. There occurs non-chaotic behaviors in other cases _ ðG=G 5 0; f ¼ 0; f ¼ þ1Þ. We have shown the following properties of the Friedmann chaos; (1) the sensitive dependence of solutions on parameters, (2) the self-similarity of solutions in the x–_x plane and the x–q plane. Numerical calculations were carried out with the use of the microsoft EXCEL. We have also discussed the self-similarity and the hierarchy structure of the universe on the basis of E infinity theory. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Recently, several authors have studied the metric tensors and the non-linear properties of the gravitational field on the basis of general relativity and non-linear theory [1–4]. Especially, El Naschie has studied the solitonic solution of the gravitational field equation with the use of the Eguchi–Hanson metric [1]. Rugh has analyzed the extended Robertson– Walker metric and discussed the chaos in the Einstein equation [3]. We have studied the compact Robertson–Walker metric and shown the Friedmann equation demonstrates the chaotic behaviors in the case the inflation universe _ ðG=G > 0Þ [4]. In this paper, we discuss the chaotic behaviors of the Friedmann equation in case of the open universe (f = 1). According to the general theory of relativity, the gravitational field equation is given as follows [5–7]: Gik þ jT ik þ kgik ¼ 0; j¼
8pK ¼ 1:86 1027 ½cm=g; c2
*
Corresponding author. Tel.: +81 93 693 3202; fax: +81 93 603 8186. E-mail address: [email protected] (Y. Tanaka).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.034
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
where the symbols Gik and Tik denote the Einstein tensor and the energy-momentum tensor, respectively. The third term is the cosmological term, and the symbols k and gik show the universal constant and the metric tensor of the fourdimensional continuum, respectively [8,9]. The above equation is called as ‘‘the Einstein equation’’ in this paper. From chaos theoretical point of view, the non-linear property of relativistic field equation is interesting and may introduce chaotic behaviors under certain conditions. In Section 2, following the text book of Einstein [7], we study the Friedmann model and show the chaotic behaviors of the gravitational field. In Section 3, we give numerical results and discuss the chaotic properties in the Friedmann equation. In Section 4, we give some discussions on the related problems. Summary is given in Section 5.
2. The Friedmann model Several authors discussed the Friedmann equation [3,4,7]. Here, we review briefly the Friedmann model of the universe [7]. 2.1. The Friedmann equation Friedmann found the solution of the Einstein equation in 1922. The metric is given as follows: ds2 ¼ ðdx4 Þ2 G2 A2 ½ðdx1 Þ2 þ ðdx2 Þ2 þ ðdx3 Þ2 ; 1 ; ðf ¼ þ1; 0; 1Þ: A¼ 1 þ 4f r2 The function G = G(t) satisfies the following equations: 1 € þ ðGÞ _ 2 g þ jp k ¼ 0; ff þ 2GG G2 3 _ 2 g þ jq þ k ¼ 0: 2 ff þ ðGÞ G Here, the symbols p and q show the pressure and the density, respectively. These equations are the space component and the time component of the Einstein equation, respectively. By eliminating the constant k from these equations, we have € G_ 2 f j G 2 þ ðp þ qÞ ¼ 0: 2 G G G Also, by eliminating the parameter f, we obtain € j 1 G þ ð3p þ qÞ k ¼ 0: 3 G 6 Among these four equations, two of them are independent, and we discuss the properties of three equations except for the second equation [the time component of the Einstein equation]. For convenience, we use the variable x instead of G; x¼
1 : G
With the use of this relation, the above three equations are transformed into the following form [4]: €x þ a
x_ 2 þ bx þ cx3 ¼ d F ðp; q; xÞ; x
where the function F(p,q,x) is defined as follows. 8 9 ½Case 1 > > < px; = F ðp; q; xÞ ¼ ðp þ qÞx; ½Case 2 : > > : ; ð3p þ qÞx; ½Case 3
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
409
Table 1 Values of coefficients a, b, c and d in the Friedmann equationa
Case 1 Case 2 Case 3 a
a
b
c
d
Comment
52
k 2
2f
1 2
0
f 0
k 2 k 2 k 6
F(p, q, x) = px F(p, q, x) = (p + q)x F(p, q, x) = (3p + q)x
k 3
The Friedmann equation is given as follows:
€x þ a
ð_xÞ2 þ bx þ cx3 ¼ d F ðp; q; xÞ: x
See text for details.
The coefficients a, b, c and d are listed in Table 1. At this stage, we have two independent field equations [the space component and the time component of the Einstein equation] and three unknowns (p, q and x = 1/G). To obtain the solutions of these equations, we must have one more equation. Here, we use the conservation law q_ ¼ 3
G_ ðp þ qÞ; G
which is obtained by the covariant derivative T ik;i ¼ 0;
ði; k ¼ 1; 2; 3; 4Þ:
Then, we have the standard form of equations. 8 > < x_ ¼ y; 2 y_ ¼ a yx bx cx3 þ d F ðp; q; xÞ; > : q_ ¼ 3 yx ðp þ qÞ: _ the standard form of ‘‘the Friedmann equation’’ in this paper. We summarize some _ qÞ We call these equations ð_x; y; chaotic equations and expansion rate of phase space volume in Table 2 [10–15]. As is well known, Lorenz introduced the two-hole type attractor [14], and Ro¨ssler derived the one-hole type attractor [15]. The expansion rate of phase space volume is given as follows:
Table 2 Nonlinear equation and expansion rate of phase space volume Nonlinear equation (1) Lorenz equationa 8 < x_ ¼ rðx yÞ y_ ¼ y xz þ rx : z_ ¼ xy bz (2) Ro¨ssler equationb 8 < x_ ¼ ðy zÞ y_ ¼ x þ ay : z_ ¼ b þ zðx lÞ (3) Friedmann equation 8 < x_ ¼ y 2 y_ ¼ a yx bx cx3 þ dF ðp; q; xÞ : q_ ¼ 3 yx ðp þ qÞ a b
Taken from Ref. [14]. Taken from Ref. [15].
Expansion rate
V_ V
¼ r 1 b
V_ V
¼aþxl
V_ V
¼ ð3 2aÞ xx_
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
V_ y ¼ ð3 2aÞ ; V x G_ ¼ ð3 2aÞ : G _ Therefore, there may occur chaos in case of the inflation universe ðG=G > 0Þ, and there may occur no chaos in case of _ the flat and the contraction universe ðG=G 5 0Þ [4]. 2.2. Properties of the Friedmann equation Here, we discuss the approximate solutions of the Friedmann equation and give comments on the properties of these solutions. 2.2.1. The approximate solution in the x–_x plane We consider the Friedmann equation in Case 1 (a = 5/2, b = k/2, c = f/2, d = j/2). The equation is written as follows: €x þ a
x_ 2 þ cx3 ¼ d p x; x
where we set k = 0 in the Friedmann model. By integrating the above equation, we have
Table 3 The numerical solutions of the Friedmann equation (Case 1) t
x
y
q
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.5 0.508 0.51628 0.524853 0.533735 0.542939 0.552483 0.562383 0.572658 0.583327 0.594412 0.605935 0.617919 0.630391 0.643377 0.656908 0.671014 0.685729 0.70109 0.717135 0.733907 0.751451 0.769815 0.789052 0.809217 0.830373 0.852585 0.875924 0.900466 0.926295 0.9535
0.2 0.207 0.214337 0.222032 0.230108 0.238588 0.247501 0.256873 0.266736 0.277122 0.288067 0.29961 1 0.311793 0.324661 0.338263 0.352651 0.367884 0.384023 0.401138 0.419301 0.438592 0.459098 0.480915 0.504144 0.528896 0.555294 0.583468 0.613561 0.645726 0.68013 0.716951
0.2 0.224 0.249622 0.277004 0.306295 0.337662 0.371287 0.407374 0.446146 0.487851 0.532766 0.581195 0.633481 0.690004 0.751188 0.817509 0.889499 0.967756 1.052953 1.145846 1.24729 1.358251 1.479824 1.61325 1.759941 1.921503 2.099773 2.296848 2.515131 2.757379 3.026763
For convenience, only the first 30 lines are shown.
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
x_ 2 þ 2a
Z
411
x_ 3 c dx þ x4 ¼ d p x2 þ const; x 2
where R 3 we simply assume the pressure p is constant. The variable x and x_ satisfy the following condition because of the ð_x =xÞdx term: If x ! 0; then x_ ! 0: c =0.5
3.5 3 2.5 2
x
1.5 1 0.5 0
0
500
1000
1500
2000
2500
3000
t
Fig. 1. The numerical solutions of the Friedmann equation [Case 1]. The figure is shown in the t–x plane for c = +0.5. The other parameters are fixed and given as follows: a = 2.5, b = 0, d = 0.25, p = 0.3 and x0 = 0.5. Only in the case of c = +0.5, there occurs chaos in the Friedmann equation [Case 1].
c=0.5 3 2 1
.
x 0 -1 -2 -3 -4 0
500
1000
1500
2000
2500
3000
t
Fig. 2. The numerical solutions of the Friedmann equation [Case 1] The figure is shown in the t–_x plane for c = +0.5. The other parameters are fixed and given as follows: a = 2.5, b = 0, d = 0.25, p = 0.3 and x0 = 0.2. Only in case c = +0.5, there occurs chaos in the Friedmann equation [Case 1].
c=0.5 120 100 80
ρ
60 40 20 0 -20 0
500
1000
1500
2000
250 0
3000
t
Fig. 3. The numerical solutions of the Friedmann equation [Case 1]. The figure is shown in the t–q plane for c = +0.5. The other parameters are fixed and given as follows: a = 2.5, b = 0, p = 0.3 and q0 = 0.2. Only in case c = +0.5, there occurs chaos in the Friedmann equation [Case 1].
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
In the next section, this property is shown to be approximately satisfied by the numerical solution of the Friedmann equation. (i) The vibrational mode (f = 1; c = +0.5). 3 For the positive value R of x, the factor x_ takes both positive and negative values in case of the vibrational mode. Therefore, the integral ð_x3 =xÞdx is very small and almost zero for the one cycle, and we have c x_ 2 d p x2 þ x4 ’ constðtÞ: 2 From this equation, we obtain 2 c 2 d p x ’ constðtÞ: x_ 2 þ 2 c This solution has the approximate symmetry under the following replacements: x ! ðxÞ and=or x_ ! ð_xÞ: In the next section, this property is also shown to be approximately satisfied by the numerical solution of the Friedmann equation. (ii) The exponential mode (f = 0, +1; c = 0, 0.5).
. x
c=-0.5
5 4 3 2 1 0 -1
-1 0 -2
x 1
2
3
4
2
3
4
3
4
-3 -4 -5
. x
c=0
5 4 3 2 1 0 -1
-1 0 -2
x 1
-3 -4 -5
c=0.5
5 4 3
-1
2 1 0 -1 0
x 1
2
-2 -3 -4 -5
Fig. 4. The parameter c dependence of solutions of the Friedmann equation [Case 1]. The figures are shown in the x–_x plane for c = +0.5, 0 and 0.5, respectively. The other parameters are fixed and given as follows: a = 2.5, b = 0, d = 0.25, p = 0.3, x0 = 0.5 and x_ 0 ¼ 0:2. Only in case c = +0.5 (f =1), there occurs chaos in the Friedmann equation [Case 1].
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
413
The solution is written as follows: c x_ 2 ¼ const: þ d p x2 x4 2a 2
Z
x_ 3 dx; x
R In case of the exponential mode, the factor x_ 3 takes only the positive value, and the integral ð_x3 =xÞdx increases as the variable x_ increases. Therefore, from above equation, the variable x_ increases as the variable x increases because of negative values of coefficients a and c (in Table 1). The exponential mode of solutions is shown in the next section. 2.2.2. The approximate solution in the x–q plane The density q satisfies the following equation: q_ x_ ’3 ; q x in case the pressure p is small and negligible. From this equation, we have the approximate solution q ’ cx3 ;
ρ
c=-0.5
14 12 10 8 6 4 2 0 -1
x 0
-2
1
ρ
2
3
4
2
3
4
3
4
c=0
14 12 10 8 6 4 2 0 -1
-2
x 0
1
ρ
c=0.5
14 12 10 8 6 4 2 0 -1
-2
x 0
1
2
Fig. 5. The parameter c dependence of solutions of the Friedmann equation [Case 1]. The figures are shown in the x–q plane for c = +0.5, 0 and 0.5, respectively. The other parameters are fixed as follows: a = 2.5, b = 0, d = 0.25, p = 0.3, x0 = 0.5 and q0 = 0.2. Only in case c = +0.5 (f =1), there occurs chaos in the Friedmann equation [Case 1].
414
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
where the factor c is the numerical constant. As is shown in the next section, the trajectory in the x–q plane is described approximately by this equation. (i) The vibrational mode (f = 1; c = +0.5). In the next section, we show that the numerical results satisfy approximately the above relation. The violation of the symmetry may be attributed to the inhomogeneous term (p 5 0) and the non-linear terms ð_x2 and x3 Þ. (ii) The exponential mode (f = 0, +1; c = 0, 0.5). In case of the large value of x, the numerical results satisfy approximately the above relation (q / x3), which are shown in the next section.
3. Numerical results We have analyzed the Friedmann equation with the use of the microsoft EXCEL and obtained the chaotic behaviors for the open universe (f = 1) in Case 1. Following the name of ‘‘the Lorenz Chaos’’ [14] and that of ‘‘the Ro¨ssler Chaos’’ [15], we may call these chaotic behaviors ‘‘the Friedmann chaos’’. 3.1. Results of Case 1 The numerical results of Case 1 are given in Table 3, where only the first 30 lines are shown for convenience. The full length of numerical results is 5994 lines in our calculation. The unit of time (t), space (x) and density (q) is arbitrary. We show the numerical solutions of the Friedmann equation [Case 1] in Figs. 1–3, where the figures are demonstrated in the t–x plane, t–_x plane and t–q plane, respectively, for c = +0.5(f = 1). The other parameters are fixed and given in each figures. These figures show the typical non-linear vibrational modes. Only in case c = +0.5(f = 1), there occurs chaos in the Friedmann equation [Case 1], as far as we use the parameters listed in Table 1. We show the parameter c dependence of solutions in the Friedmann equation [Case 1] in Figs. 4 and 5, where the figures are shown in the x–_x plane and x–q plane for c = +0.5, 0 and 0.5, respectively. The other parameters are fixed and given in each figures. Only in case c = +0.5(f = 1), there occurs chaos in the Friedmann equation [Case 1], as stated before.
. x
-1
5 4 3 2 1 0 -1 0 -2 -3 -4 -5
x 1
. x
-1
. x
p=0.28
2
3
-1
4
5 4 3 2 1 0 -1 0 -2 -3 -4 -5
. x
p=0.32
5
5
4 3
4 3
2 1 0
2 1 0
-1 0 -2
x 1
2
3
4
-1
-1 0 -2
-3 -4
-3 -4
-5
-5
p=0.30
x 1
2
3
4
2
3
4
p=0.34
x 1
Fig. 6. Chaotic behaviors of the Friedmann equation [Case 1]. Figures are shown in the x–_x plane. The parameter p is changed from 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0, c = +0.5, d = 0.25, x0 = 0.5 and x_ 0 ¼ 0:2.
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
ρ
ρ
p=0.28
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0 -1
-2
x 0
1
ρ
2
3
4
-2
12
12
10
10
8
8
6
6
4
4
2
2
-2
3
4
2
3
4
3
4
0
x 2
1
p=0.34
14
1
x 0
ρ
14
0
p=0.30
0 -1
p=0.32
0 -1
415
-1
-2
x 0
1
2
Fig. 7. Chaotic behaviors of the Friedmann equation [Case 1]. Figures are shown in the x–q plane. The parameter p is changed from 0.28 to 0.34. The other parameters are fixed and given as follows: a = 2.5, b = 0, c = +0.5, d = 0.25, x0 = 0.5 and q0 = 0.2.
Also, we show the chaotic behaviors in the Friedmann equation [Case 1] in Fig. 6 and 7, where the figures are demonstrated in the x–_x plane and the x–q plane for p = 0.28,0.30,0.32 and 0.34, respectively. The other parameters are fixed and given in each figures. These figures show the self-similarity, which is one of the typical properties of chaotic behaviors. 3.2. Results of Case 2 The numerical results of Case 2 are given in Table 4, where only the first 30 lines are shown for convenience. The full length of numerical results is 5994 lines in our calculation. The unit of time (t), space (x) and density (q) is arbitrary. We show the numerical solutions of the Friedmann equation [Case 2] in Figs. 8–10, where the figures are demonstrated in the t–x plane, the t–_x plane and the t–q plane, respectively, for c = 1.0(f = 1.0). The other parameters are fixed and given in each figures. These figures show the critical vibrational modes, and there occurs no chaos in Case 2, as far as we use the parameters listed in Table 1. We show the parameter c dependence of solutions in the Friedmann equation [Case 2] in Figs. 11 and 12, where the figures are shown in the x–_x plane and the x–q plane for c = +1.0, 0 and 1.0, respectively. The other parameters are fixed and given in each figures. From these figures, we see that there occurs no chaos in Case 2, even if in case c = +1.0(f = 1.0), as far as we use the parameters listed in Table 1. Also, we show the numerical solutions in the Friedmann equation [Case 2] in Figs. 13 and 14, where the figures are shown in the x–_x plane and the x–q plane for p = 0.28, 0.30, 0.32 and 0.34, respectively. The other parameters are fixed and given in each figures. From these figures, we see also that there occurs no chaos in Case 2, even if in case c = +1.0(f = 1.0), as stated before.
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Table 4 The numerical solutions of the Friedmann equation (Case 2) t
x
y
P
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.5 0.508 0.516028 0.52408 0.53215 0.540235 0.54833 0.55643 0.564529 0.572623 0.580708 0.588777 0.596825 0.604848 0.61284 0.620796 0.62871 0.636579 0.644395 0.652155 0.659854 0.667485 0.675046 0.682529 0.689932 0.69725 0.704477 0.71161 0.718645 0.725578 0.732404
0.2 0.2007 0.20129 0.201766 0.202127 0.202368 0.202489 0.202487 0.202359 0.202105 0.201723 0.201211 0.20057 0.199799 0.198898 0.197867 0.196706 0.195417 0.194001 0.192459 0.190792 0.189004 0.187097 0.185072 0.182934 0.180686 0.17833 0.175871 0.173312 0.170658 0.167913
0.2 0.224 0.248843 0.274533 0.301076 0.328473 0.356724 0.385826 0.415775 0.446563 0.478183 0.510621 0.543864 0.577895 0.612695 0.648241 0.684509 0.721472 0.759101 0.797363 0.836224 0.875648 0.915595 0.956025 0.996895 1.038159 1.079772 1.121685 1.163848 1.206212 1.248724
For convenience, only the first 30 lines are shown.
c=1.0
1 0.9 0.8 0.7 0.6
x 0.5 0.4 0.3 0.2 0.1 0 0
500
1000
1500
2000
2500
3000
t
Fig. 8. The numerical solutions of the Friedmann equation [Case 2]. The figure is shown in the t–x plane for c = +1. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3 and x0 = 0.5. There occurs no chaos even if c = 1.0 in the Friedmann equation [Case 2].
3.3. Results of Case 3 As discussed in the previous section (Section 2), the results of Case 3 show the exponential mode because of b = 0 and c = 0. The results are trivial and not shown. Needless to say, there occurs no chaos in Case 3.
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
417
c=1.0 0.25 0. 2 0.15
. x
0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0
500
1000
1500
2000
2500
3000
t
Fig. 9. The numerical solutions of the Friedmann equation [Case 2]. The figure is shown in the t–_x plane for c = +1. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3 and x_ 0 ¼ 0:2. There occurs no chaos even if c = 1.0 in the Friedmann equation [Case 2].
c=1.0
3 2.5 2
ρ 1.5 1 0.5 0 - 0.5 0
500
1000
1500
2000
2500
3000
t
Fig. 10. The numerical solutions of the Friedmann equation [Case 2]. The figure is shown in thet–q plane for c = +1. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3 and q0 = 0.2. There occurs no chaos even if c = 1.0 in the Friedmann equation [Case 2].
3.4. Properties of the Friedmann chaos Here, we discuss properties of the Friedmann chaos. (1) The sensitive dependence of solutions on the parameters. From Fig. 4, we see there occurs chaos in case c = +0.5 (f = 1.0). In case c = 0 (f = 0) and c = 0.5 (f = +1.0), there occurs no chaos, as far as we use the parameters listed in Table 1. These facts show that the trajectories in the x–_x plane demonstrate the sensitive dependence on the parameter c. Also, the trajectories in the x–q plane show the sensitive dependence on the parameter c, as in Fig. 5. As is well known, chaotic equations have the sensitive dependence on parameters. (2) The self-similarity of solutions. As demonstrated in Fig. 6, the trajectories in the x–_x plane show the self-similarity, which is the general property of chaotic motions. Also, the trajectories in the x–q plane show the self-similarity, as in Fig. 7. These properties (1) and (2) are general characters of chaotic behaviors and discussed by authors [10–13].
4. Discussions Here, we give some discussions on the related problems.
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x
- 0.5
1 0.8 0.6 0.4 0.2 0 - 0.2 0 - 0.4 - 0.6 - 0.8 -1
. x
- 0.5
1 0.8 0.6 0.4 0.2 0 - 0.2 0 - 0.4 - 0.6 - 0.8 -1
. x
- 0.5
1 0.8 0.6 0.4 0.2 0 - 0.2 0 - 0.4 - 0.6 - 0.8 -1
c=-1
x 0.5
1
1.5
1
1.5
c=0
x 0.5
c=1
x 0.5
1
1.5
Fig. 11. The parameter c dependence of solutions of the Friedmann equation [Case 2]. The figures are shown in the x–_x plane for c = +1, 0 and 1.0, respectively. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3, x0 = 0.5 and x_ 0 ¼ 0:2.
4.1. Computer simulations In Case 2, we have chaotic diagrams, if we change the values of parameters in the Friedmann equation. For example, using the parameter a = 0.3 instead of a = 1.0, we have chaotic behaviors in the x–_x plane and the x–q plane. However, these are not realistic figures but computational simulations, and the figures are not shown here. 4.2. A new theory of space–time structure Recently, El Naschie has studied a new theory of space–time structure on the basis of E infinity theory [16–19]. According to this theory, the Haussdorff dimension of the universe is given as follows: hDimðe1 ÞiH ¼
1 X
n/n ;
n¼0 3
¼/ ; ¼ 4 þ /3 ; where the symbol / denotes the golden ratio. The dimension of the universe turns out to be fractal (4 + /3 = 4.236), and the symbol / satisfies the relations
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
ρ
419
c=-1.0
1 0.8 0.6 0.4 0.2 0 - 0.2 0
- 0.5
x 0.5
1
1.5
1
1.5
1
1.5
- 0.4 - 0.6 - 0.8 -1
ρ
c=0
1 0.8 0.6 0.4 0.2 0 - 0.2 0
- 0.5
x 0.5
- 0.4 - 0.6 - 0.8 -1
ρ
c=1.0
1
0.8 0.6 0.4 0.2 0 - 0.2 0
- 0.5
x 0.5
- 0.4 - 0.6 - 0.8 -1
Fig. 12. The parameter c dependence of solutions of the Friedmann equation [Case 2]. The figures are shown in the x–q plane for c = +1, 0 and 1.0, respectively. The other parameters are fixed and given as follows: a = 1.0, b = 0, d = 0.25, p0 = 0.3, x0 = 0.5 and q0 = 0.2.
/¼ ¼
1 ; 1þ/ 1 : 1 þ 1þ1 1 1þ
In general, the symbol / satisfies the following relations: /n ¼ ¼
1 fn þ ð1Þnþ1 /n
;
1 fn þ ð1Þnþ1 f
;
1 n þð1Þ
nþ1
1 fn þ
where {fn} denotes the Fibonacci sequence with f1 = 1 and f2 = 3. For example, in case n = 3, we have
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x
- 0.5
x
p=0.28
1
1
0.8 0.6
0.8
0.4 0.2
0.4
0.6
0.2
0 - 0.2 0
0.5
1
- 0.5
1.5
x
0
x
0
- 0.2
- 0.4 - 0.6
- 0.4
- 0.8 -1
- 0.8
1
1.5
1
1.5
-1
p=0.32
x
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
p=0.34
0.2
x
0 - 0.2
0.5
- 0.6
x
- 0.5
p=0.30
0
0.5
1
1.5
x
0 - 0.5
- 0.2
- 0.4
- 0.4
- 0.6
- 0.6
- 0.8
- 0.8
-1
-1
0
0.5
Fig. 13. The numerical solutions of the Friedmann equation [Case 2]. Figures are shown in the x–_x plane. The parameter p is changed from 0.28 to 0.34. The other parameters are fixed and given as follows: a = 1.0, b = 0, c = +1.0, d = 0.25, x0 = 0.5 and x_ 0 ¼ 0:2.
ρ
p=0.28
1
ρ 1
0.8
0.8
0.6
0.6
0.4
0.4 0.2
0.2 0 - 0.5
-0.2 0
0 .5
1
- 0.5
1.5
-0.2 0 -0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
p=0.32
ρ
1
1
0.8 0.6
0.8
0.4 0.2
0.4
-0.2 0 -0.4 -0.6 -0.8 -1
x 0.5
1
1.5
p=0.34
0.6
0.2
0 -0.5
0
x
-0.4
ρ
p=0.30
x 0.5
1
1.5
0 - 0.5
-0.2 0
x 0.5
1
1.5
-0.4 -0.6 -0.8 -1
Fig. 14. The numerical solutions of the Friedmann equation [Case 2]. Figures are shown in the x–q plane. The parameter p is changed from 0.28 to 0.34. The other parameters are fixed and given as follows: a = 1.0, b = 0, c = +1.0, d = 0.25, x0 = 0.5 and q0 = 0.2.
Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
421
1 ; 4 þ /3 1 : ¼ 4 þ 4þ1 1
/3 ¼
4þ
These relations reflect the self-similarity of the universe. Indeed, as discussed in the previous section, the Friedmann chaos shows the self-similarity in the x–_x plane and the x–q plane (Fig. 6 and 7). E infinity theory is suitable for describing the self-similarity and the hierarchy properties of the universe, such as the mass spectrum of high energy particle physics [17,20]. The fractal dimension of chaotic attractors, such as the Lorenz attractor and the Friedmann attractor may be described well by E infinity theory. This problem will be discussed elsewhere.
5. Summary Finally, we summarize our work as follows. We have studied relativistic equations and chaotic motions of the universe on the basis of general relativity and chaotic dynamics. The Friedmann equation [the space component] shows _ the chaotic behaviors in case of the inflation ðG=G > 0Þ and open (f = 1) universe. In other cases _ ðG=G 5 0; f ¼ 0; f ¼ þ1Þ, there occurs no chaos in the Friedmann equation, as far as we use the parameters listed in Table 1. The Friedmann chaos has following properties: (1) The sensitive dependence of solutions on the parameters. (2) The self-similarity of solutions in the x–_x plane and the x–q plane. These characters are general properties of chaotic behaviors, such as the Lorenz chaos [14] and the Ro¨ssler chaos [15]. Especially, the self-similar characters of these chaos correspond to the mathematical properties of E infinity theory [17,18]. Acknowledgments We would like to thank the Information Processing Center of Kyushu Kyoritsu University for using the microsoft EXCEL, which is very powerful for analyzing linear and non-linear equations. Thanks are also to Ms. H. Shiroshita for typing the manuscript of this paper.
Appendix A Here, we show the difference equations for numerical calculations of the Friedmann equation. (1) The differential equations. The differential equations of the Friedmann model, which is mentioned in the text, are given as follows. 8 dX ¼ Y; > < dt 2 dY ¼ a YX bX cX 3 þ d F ðp; q; X Þ; dt > : dq ¼ 3 XY ðp þ qÞ; dt where the function F is defined in the following form. 8 > < pX ; F ðp; q; X Þ ¼ ðp þ qÞX ; > : ð3p þ qÞX ;
½Case 1; ½Case 2; ½Case 3:
(2) The difference equations. The difference equations, which correspond to the above differential equations, are written in the following form.
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Y. Tanaka et al. / Chaos, Solitons and Fractals 24 (2005) 407–422
8 X 2 ¼ X 1 þ DtðY 1 Þ; > > n o > < Y2 Y 2 ¼ Y 1 þ Dt a X 11 bX 1 cX 31 þ d F ðp; q1 ; X 1 Þ ; > n o > > : q ¼ q þ Dt 3 Y 1 ðp þ q Þ : 2 1 1 X1 where the function F is given as follows. 8 ½Case 1; > < pX 1 ; F ðp; q1 ; X 1 Þ ¼ ðp þ q1 ÞX 1 ; ½Case 2; > : ð3p þ q1 ÞX 1 ; ½Case 3: (3) Numerical calculations. With the use of the microsoft EXCEL (Office XP Professional), we have carried out the numerical calculations of the Friedmann equation. The EXCEL is powerful especially for analyzing non-linear equations.
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