- Email: [email protected]
GM(1,1) grey prediction of Lorenz chaotic system
Chaos, Solitons and Fractals 42 (2009) 1003–1009
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
GM(1,1) grey prediction of Lorenz chaotic system Yagang Zhang a,b,*, Yan Xu a, Zengping Wang a a
Key Laboratory of Power System Protection and Dynamic Security Monitoring and Control under Ministry of Education, North China Electric Power University, Box 205, Baoding, Hebei 071003, PR China Center for Nonlinear Complex Systems, School of Physical Science and Technology, Yunnan University, Kunming, Yunnan 650091, PR China
b
a r t i c l e
i n f o
Article history: Accepted 23 February 2009
Communicated by Prof. Ji-Huan He
a b s t r a c t The grey prediction of Lorenz chaotic system will be discussed carefully in this paper. We are mainly using GM(1,1) model to predict data sequences, and the usual prediction precision has exceeded 90%. In the symbolic prediction of Lorenz chaotic dynamical system, the precision of grey prediction certainly will decrease as the length of symbolic sequence is increasing. But in this place we have found a generating rule that may realize chaotic synchronization at least in a short and medium term, and we can analysis and predict in this way. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Based on the characteristic data of related systems behavior, grey system theory [1,2] is mainly used to resolve mathematical relationships considering various factors. It focuses on models of uncertainty with insufficient information, predicts and makes decisions by conditional analyzing [3]. A system is defined as a white system if the information in it is known. Otherwise, a system will be a black box if nothing in it is clear. The grey system is just between the white system and the black-box system. Each stochastic variable that varies within a given range is called a grey quantity. We are not relying on statistical methods to process the grey quantity, but directly using them to process the original data and research the intrinsic regularity. The analysis mechanism of grey prediction (GP) in grey system can explore the large amount of unknown information with the existing small amount of information. The GP adopts the essential part of the grey system theory and can give an estimation of the grey system in detail. It has been successfully applied in predicting problems with excellent results in many fields [4–8], such as ecosystems, industrial systems, social systems and economic systems, etc. GP can be divided into series forecast, abnormal-value forecast, seasonal disaster forecast, system forecast and topology forecast in the aspects of functions and features [9]. There are two basic operations in GP: accumulated generating operator (AGO) and inverse accumulated generating operator (IAGO). Based on the small amount incomplete information that is already known, a grey model (GM) can be established. The limited original data sequence that is known can be used to produce new data sequence with exponential pattern via AGO, by which the uncertainty of the original data sequence is decreased. We can establish a differential equation to perform the fitting work and predict the unknown part, as well as to inversely acquire the predicted value of the original data sequence via IAGO. Among grey models, the first order grey model with one variable GM(1,1) has been applied widely. When the original data sequences imply exponential laws, it is very advisable to use GM(1,1) model for forecasting [10]. In this paper, we are mainly using GM(1,1) model to predict data sequences generated by Lorenz chaotic system, and the usual prediction precision has exceeded 90%. * Corresponding author. Address: Department of Mathematics and Physics, North China Electric Power University, Box 205, Baoding, Hebei 071003, PR China. Tel.: +86 312 3187992. E-mail address: [email protected] (Y. Zhang). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.02.031
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Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
This paper is organized as follows. In Section 2, the GM(1,1) prediction model is introduced. In Section 3, Lorenz equation, the symbolic dynamics and dual star products of Lorenz map are discussed carefully. In Section 4, grey prediction of Lorenz chaotic system based on GM(1,1) model is clarified in detail. 2. GM(1,1) prediction model Suppose there is an original time sequence with n samples (time point) xð0Þ and xð0Þ can be expressed as,
xð0Þ ¼ ½xð0Þ ð1Þ; xð0Þ ð2Þ; ; xð0Þ ðnÞ where, xð0Þ ðiÞ ði ¼ 1; 2; 3; Þ is the time series data at time i, n should be equal to or larger than 4. In order to reveal the objective law of systems, the grey system theory adopts a unique data preprocessing method before model is to be established. It uses AGO to accumulate the sequence xð0Þ and obtain xð1Þ , that is,
xð1Þ ¼ ½xð1Þ ð1Þ; xð1Þ ð2Þ; ; xð1Þ ðnÞ where;
xð1Þ ðkÞ ¼
k X
xð0Þ ðiÞ:
i¼1
The sequence xð1Þ ðkÞ has exponential increasing rules, and the solution of one-order differential equation is just exponential increasing form. In fact, the sequence xð1Þ satisfies one-order linear differential equation: ð1Þ
dx þ axð1Þ ¼ u: dt
ð1Þ
Using discrete one-order linear difference Eq. (1), one can obtain a matrix as follows,
0
xð0Þ ð2Þ
1
0
B xð0Þ ð3Þ C B C B B C B B B .. C ¼ B @ . A B @ xð0Þ ðnÞ
12 ½xð1Þ ð1Þ þ xð1Þ ð2Þ
1
1
C 1 C a C : .. C . C A u 12 ½xð1Þ ðn 1Þ þ xð1Þ ðnÞ 1 12 ½xð1Þ ð2Þ þ xð1Þ ð3Þ .. .
ð2Þ
Let
1
0
B xð0Þ ð3Þ C C B C Yn ¼ B B .. C; @ . A
B B B B¼B B @
0
xð0Þ ð2Þ
a A¼ ; u
xð0Þ ðnÞ
12 ½xð1Þ ð1Þ þ xð1Þ ð2Þ
1
1
C 1C C C; ... C A 12 ½xð1Þ ðn 1Þ þ xð1Þ ðnÞ 1 12 ½xð1Þ ð2Þ þ xð1Þ ð3Þ ...
the matrix can be expressed as,
Y n ¼ B A:
ð3Þ
In the Eq. (3), Y n and B can be obtained directly by the original data, but A needs more calculation. One usually uses leastsquare method to get the least-square approximation, and the Eq. (3) can be expressed as,
b þ eH ; Yn ¼ B A wherein, eH is an error term. Using matrix derivation formula, one gets,
b ¼ ðBT BÞ1 BT Y n ¼ A
^ a : ^ u
^ are obtained, furthermore, ^ and a So, u
^ ^ u u ea^t þ : xð1Þ ðtÞ ¼ xð1Þ ð1Þ ^ ^ a a Let xð1Þ ð0Þ ¼ xð0Þ ð1Þ, the time response function of GM(1,1) model is also obtained,
^ ^ u u xð0Þ ð1Þ ea^k þ ^ ^ a a
ðk ¼ 1; 2; 3; Þ:
ð4Þ
Then, one does inverse accumulated revivification with the time response function (4), and gets grey prediction model,
^ u ^xð0Þ ðk þ 1Þ ¼ ^xð1Þ ðk þ 1Þ ^xð1Þ ðkÞ ¼ xð0Þ ð1Þ ð1 ea^ Þea^k ^ a
ðk ¼ 1; 2; 3; Þ
ð5Þ
Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
1005
^. Then, the concrete pre^ and a After substituting the correlative data into expression (5), one can resolve the coefficients u diction formula can be confirmed. 3. Lorenz map 3.1. Lorenz equation The general form of Lorenz equation [11,12] is:
8 > < x_ ¼ rðy xÞ; y_ ¼ ðr zÞx y; > :_ z ¼ xy bz: In 1976, Guckenheimer and Williams introduced geometrical Lorenz model for the first time. This model is a Poincar map on the Poincar section which has Lorenz class flow system and usually used to describe the variation regularity of atmosphere movement. This geometrical structure of Lorenz flow may be reduced to a self-map on one-dimensional interval f : ½l; m ! ½l; m,
( f ðxÞ ¼
fL ðxÞ ¼ m ajxjk þ h:o:t;
x 6 0;
fR ðxÞ ¼ l þ bxk þ h:o:t; x > 0;
ðl; m > 0; k > 1Þ
where k is a constant greater than 1, h:o:t represents high-level term. Both of the branches fL and fR are monotone increasing. In order to get iterative sequences in the part of chaos, the Lorenz map should be:
( f ðxÞ ¼
fL ðxÞ ¼ 1 2jxj2 ; x 6 0; fR ðxÞ ¼ 1 þ 2x2 ; x > 0:
ð6Þ
The concrete symbolic definition rule is shown in Fig. 1. 3.2. The symbolic dynamics and dual star products of Lorenz map Seen from symbolic dynamical system [13], Lorenz map belongs to a more complex dynamical category because it has a discontinuity point. The Lorenz map presents more abundant dynamical actions [14,15] compared with Unimodal map. The symbolic dynamics of Lorenz map can consult paper [16]. Fig. 2 is the bifurcation diagram of Lorenz map. There are two kinds of dual star products in the Lorenz map, namely, the up-star product ðÞ and the down-star product ðÞ. Suppose there are two kneading pairs W ¼ ðCV 1 ; DU 1 Þ and Z ¼ ðCV 2 ; DU 2 Þ, thereinto,
U 1 ¼u11 u12 u1i u1k ;
V 1 ¼ v 11 v 12 v 1j v 1l ;
u1i ; v 1j 2 fL; Rg
U 2 ¼u21 u22 u2i u2n ;
V 2 ¼ v 21 v 22 v 2j v 2m ;
u2i ; v 2j 2 fL; Rg
then, the dual star products are just W Z ¼ ðCV 1 ; DU 1 Þ ðCV 2 ; DU 2 Þ; 2 f; g, and each symbol of Z can be replaced one by one according to the following rules,
Fig. 1. Symbolic definition rule of Lorenz map.
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Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
Fig. 2. The bifurcation diagram of Lorenz map.
8 C ! CV 1 RU 1 ; > > > < D ! DU LV ; 1 1 The up-star product ðÞ : > L ! LV 1 RU 1 ; > > : R ! RU 1 LV 1 : 8 C ! ðDU 2 LV 2 ÞT ; > > > > < D ! ðCV RU ÞT ; 2 2 The down-star product ðÞ : > L ! ðRU LV ÞT ; > 2 2 > > : R ! ðLV 2 RU 2 ÞT ;
ð7Þ
ð8Þ
thereinto, T is a keeping parity operator, which makes the following symbolic counterchange: L $ R; C $ D. Obviously, the length of compound sequences obtained by the dual star products rules (7) and (8) is
jWj jZj ¼ ðk þ l þ 2Þ ðn þ m þ 2Þ: 4. Grey prediction of Lorenz chaotic system based on GM(1,1) 4.1. Iterative process of Lorenz chaotic system The general dynamical iterative form of Lorenz map is expression (6). Giving an initial value x1 ¼ 0:65, the number of iterations is n ¼ 16, one can obtain Tables 1 and 2. Fig. 3 is the concrete iterative process. Let us transform these iterative values. Firstly, each f ðxn Þ is made absolute value transformation. Secondly, one carries out P cumulative sum Zðxk Þ ¼ ki¼1 j f ðxi Þ j, and gets Tables 3 and 4. 4.2. Random generating sequence’s grey prediction of Lorenz chaotic system In the GM(1,1) model, the expression of AGO is xð1Þ ðkÞ ¼ The formula of grey estimate value is,
Pk
m¼1 x
ð0Þ
ðmÞ, so one gets Table 5.
h ui Estim: ¼ xð1Þ ð0Þ eat ð1 ea Þ; a thereinto, xð1Þ ð0Þ ¼ 0:1550; a ¼ 0:10625; u ¼ 2:15158; t ¼ 1; 2; 3; , namely,
Estim: ¼ 20:4052 e0:10625t ð1 e0:10625 Þ:
Table 1 The iterative values of Lorenz chaotic system (n ¼ 1; 2; ; 8). n
1
2
3
4
5
6
7
8
xn f ðxn Þ
0:65 0:1550
0:1550 0:9520
0:9520 0:8124
0:8124 0:3200
0:3200 0:7951
0:7951 0:2645
0:2645 0:8601
0:8601 0:4795
1007
Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009 Table 2 The iterative values of Lorenz chaotic system (n ¼ 9; 10; ; 16). n
9
10
11
12
13
14
15
16
xn f ðxn Þ
0:4795 0:5402
0:5402 0:4163
0:4163 0:6534
0:6534 0:1461
0:1461 0:9573
0:9573 0:8329
0:8329 0:3874
0:3874 0:6999
Fig. 3. Sixteen iterations of Lorenz chaotic system (x1 ¼ 0:65).
Table 3 The cumulative sum results of Zðxk Þ (k ¼ 1; 2; ; 8). k
1
2
3
4
5
6
7
8
Zðxk Þ
0:1550
1:1069
1:9194
2:2394
3:0346
3:2991
4:1591
4:6386
Table 4 The cumulative sum results of Zðxk Þ (k ¼ 9; 10; ; 16). k
9
10
11
12
13
14
15
16
Zðxk Þ
5:1788
5:5951
6:2485
6:3946
7:3519
8:1848
8:5722
9:2721
Table 5 Main variable values of GP process. k
Zðxk Þ
AGO
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.1550 1.1069 1.9194 2.2394 3.0346 3.2991 4.1591 4.6386 5.1788 5.5951 6.2485 6.3946 7.3519 8.1848 8.5722 9.2721
0.1550 1.2619 3.1813 5.4207 8.4553 11.7544 15.9135 20.5521 25.7309 31.3260 37.5745 43.9691 51.3210 59.5058 68.0780 77.3501
AGO mean * 0.7085 2.2216 4.3010 6.9380 10.1049 13.8340 18.2328 23.1415 28.5285 34.4503 40.7718 47.6451 55.4134 63.7919 72.7141
Mean square
Mean AGO
Mean square AGO
Mean AGO square
*
*
*
*
0.5019 4.9355 18.4986 48.1358 102.1080 191.3782 332.4350 535.5290 813.8725 1186.820 1662.340 2270.051 3070.645 4069.407 5287.333
0.7085 2.9301 7.2311 14.1691 24.2739 38.1079 56.3407 79.4822 108.0106 142.4609 183.2327 230.8777 286.2911 350.0830 422.7971
0.5019 5.4374 23.9360 72.0719 174.1798 365.5580 697.9930 1233.5220 2047.3945 3234.2142 4896.5539 7166.6047 10237.2496 14306.6561 19593.9892
0.5019 8.5852 52.2881 200.7620 589.2222 1452.208 3174.269 6317.412 11666.29 20295.09 33574.20 53304.51 81962.59 122558.1 178757.3
So, the prediction values and corresponding errors of 16 iterations can be calculated (see Table 6). Fig. 4 shows the actual values and the prediction values of 16 iterations. Here are grey prediction results of the following 3 steps:
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Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
Table 6 Prediction values and corresponding errors of 16 iterations. k
Zðxk Þ actual value
Zðxk Þ estimate value
Error
Relative error (%)
Precision (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.1550 1.11 1.92 2.24 3.03 3.30 4.16 4.64 5.18 5.60 6.25 6.39 7.35 8.18 8.57 9.27
0.1550 2.29 2.54 2.83 3.15 3.50 3.89 4.33 4.81 5.35 5.95 6.62 7.36 8.19 9.10 10.12
0 1:18 0:62 0:59 0:12 0:2 0.27 0.31 0.37 0.25 0.3 0:23 0:01 0:01 0:53 0:85
0 106:31 32:29 26:34 3:96 6:06 6.49 6.68 7.14 4.46 4.80 3:60 0:14 0:12 6:18 9:17
100 6:31 67.71 73:66 96:04 93:94 93.51 93.32 92.86 95.54 95.2 96.4 99.86 99.88 93.82 90.83
Fig. 4. Sixteen iterations’ actual values and prediction values of Lorenz chaotic system.
k ¼ 17 : Estim: value ¼ 11:26; k ¼ 18 : Estim: value ¼ 12:52; k ¼ 19 : Estim: value ¼ 13:92: It will be intuitionistic and natural that returning these grey prediction results to actual symbolic space. Combined the exact position of peak and trough, the value of positive or negative can be confirmed. 5. Conclusion The GP of Lorenz chaotic system has been clarified carefully in this paper. We are mainly using GM(1,1) model to predict data sequences, and the usual prediction precision has exceeded 90%. Because of the intrinsic property of chaos, it is impossible to give an exact prediction in long-term. In the symbolic prediction of Lorenz chaotic dynamical system, the precision of GP certainly will decrease as the length of symbolic sequence is increasing. But in this place we have found a generating rule that may realize chaotic synchronization at least in a short and medium term, and we can analysis and predict in this way. Maybe on next idyllic weekend trip, we need not carry a superfluous umbrella. Acknowledgements This research was supported partly by the National Natural Science Foundation of China (50777016) and the Youthful Teacher Research Foundation of NCEPU.
Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
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References [1] Deng J. Introduction to grey system theory. J Grey Syst 1989;1(1):1–24. [2] Deng J. Control problems of grey system. Syst Control Lett 1982;5:288–94. [3] Lin CT, Yang SY. Forecast of the output value of Taiwan’s optoelectronics industry using the grey forecasting model. Technol Forecast Soc Change 2003;70:177–86. [4] Akay D, Atak M. Grey prediction with rolling mechanism for electricity demand forecasting of Turkey. Energy 2007;32:1670–5. [5] Mao M, Chirwa EC. Application of grey model GM(1, 1) to vehicle fatality risk estimation. Technol Forecast Soc Change 2006;73:588–605. [6] Li PL. Grey forecasting of earthquake (M greater than or equal to 6, 0) frequency in China mainland. J Grey Syst 1990;2(2):133–51. [7] Tan GJ. Applied study of grey system theory on steel products experiment engineering. ICAMT’99. New York: Science; 1999. [8] Tseng FM, Yu HC, Tzeng GH. Applied hybrid grey model to forecast seasonal time series. Technol Forecast Soc Change 2003;70:563–74. [9] Liu S, Lin Y. Grey information: theory and practical applications. Berlin: Springer; 2006. [10] Chiang JY, Chen CK. Application of grey prediction to inverse nonlinear heat conduction problem. Int J Heat Mass Transfer 2008;51:576–85. [11] Richter H. Controlling the Lorenz system: combining global and local schemes. Chaos, Solitons & Fractals 2001;12(13):2375–80. [12] Zhou TS, Liao HH, Zheng ZH, Tang Y. The complicated trajectory behaviors in the Lorenz equation. Chaos, Solitons & Fractals 2004;19(4):863–73. [13] Hao BL, Zheng WM. Symbolic dynamics and chaos. Directions in chaos, vol. 7. Singapore: World Scientific; 1998. [14] Zhang YG. Stochastic properties in Lorenz maps. Far East J Dyn Syst 2006;8(2):175–84. [15] Zhang YG, Wang ZP. Knot theory based on the minimal braid in Lorenz system. Int J Theoret Phys 2008;47:873–80. [16] Peng SL, Du LM. Dual star products and symbolic dynamics of Lorenz maps with the same entropy. Phys Lett A 1999;261:63–73.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
GM(1,1) grey prediction of Lorenz chaotic system Yagang Zhang a,b,*, Yan Xu a, Zengping Wang a a
Key Laboratory of Power System Protection and Dynamic Security Monitoring and Control under Ministry of Education, North China Electric Power University, Box 205, Baoding, Hebei 071003, PR China Center for Nonlinear Complex Systems, School of Physical Science and Technology, Yunnan University, Kunming, Yunnan 650091, PR China
b
a r t i c l e
i n f o
Article history: Accepted 23 February 2009
Communicated by Prof. Ji-Huan He
a b s t r a c t The grey prediction of Lorenz chaotic system will be discussed carefully in this paper. We are mainly using GM(1,1) model to predict data sequences, and the usual prediction precision has exceeded 90%. In the symbolic prediction of Lorenz chaotic dynamical system, the precision of grey prediction certainly will decrease as the length of symbolic sequence is increasing. But in this place we have found a generating rule that may realize chaotic synchronization at least in a short and medium term, and we can analysis and predict in this way. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Based on the characteristic data of related systems behavior, grey system theory [1,2] is mainly used to resolve mathematical relationships considering various factors. It focuses on models of uncertainty with insufficient information, predicts and makes decisions by conditional analyzing [3]. A system is defined as a white system if the information in it is known. Otherwise, a system will be a black box if nothing in it is clear. The grey system is just between the white system and the black-box system. Each stochastic variable that varies within a given range is called a grey quantity. We are not relying on statistical methods to process the grey quantity, but directly using them to process the original data and research the intrinsic regularity. The analysis mechanism of grey prediction (GP) in grey system can explore the large amount of unknown information with the existing small amount of information. The GP adopts the essential part of the grey system theory and can give an estimation of the grey system in detail. It has been successfully applied in predicting problems with excellent results in many fields [4–8], such as ecosystems, industrial systems, social systems and economic systems, etc. GP can be divided into series forecast, abnormal-value forecast, seasonal disaster forecast, system forecast and topology forecast in the aspects of functions and features [9]. There are two basic operations in GP: accumulated generating operator (AGO) and inverse accumulated generating operator (IAGO). Based on the small amount incomplete information that is already known, a grey model (GM) can be established. The limited original data sequence that is known can be used to produce new data sequence with exponential pattern via AGO, by which the uncertainty of the original data sequence is decreased. We can establish a differential equation to perform the fitting work and predict the unknown part, as well as to inversely acquire the predicted value of the original data sequence via IAGO. Among grey models, the first order grey model with one variable GM(1,1) has been applied widely. When the original data sequences imply exponential laws, it is very advisable to use GM(1,1) model for forecasting [10]. In this paper, we are mainly using GM(1,1) model to predict data sequences generated by Lorenz chaotic system, and the usual prediction precision has exceeded 90%. * Corresponding author. Address: Department of Mathematics and Physics, North China Electric Power University, Box 205, Baoding, Hebei 071003, PR China. Tel.: +86 312 3187992. E-mail address: [email protected] (Y. Zhang). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.02.031
1004
Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
This paper is organized as follows. In Section 2, the GM(1,1) prediction model is introduced. In Section 3, Lorenz equation, the symbolic dynamics and dual star products of Lorenz map are discussed carefully. In Section 4, grey prediction of Lorenz chaotic system based on GM(1,1) model is clarified in detail. 2. GM(1,1) prediction model Suppose there is an original time sequence with n samples (time point) xð0Þ and xð0Þ can be expressed as,
xð0Þ ¼ ½xð0Þ ð1Þ; xð0Þ ð2Þ; ; xð0Þ ðnÞ where, xð0Þ ðiÞ ði ¼ 1; 2; 3; Þ is the time series data at time i, n should be equal to or larger than 4. In order to reveal the objective law of systems, the grey system theory adopts a unique data preprocessing method before model is to be established. It uses AGO to accumulate the sequence xð0Þ and obtain xð1Þ , that is,
xð1Þ ¼ ½xð1Þ ð1Þ; xð1Þ ð2Þ; ; xð1Þ ðnÞ where;
xð1Þ ðkÞ ¼
k X
xð0Þ ðiÞ:
i¼1
The sequence xð1Þ ðkÞ has exponential increasing rules, and the solution of one-order differential equation is just exponential increasing form. In fact, the sequence xð1Þ satisfies one-order linear differential equation: ð1Þ
dx þ axð1Þ ¼ u: dt
ð1Þ
Using discrete one-order linear difference Eq. (1), one can obtain a matrix as follows,
0
xð0Þ ð2Þ
1
0
B xð0Þ ð3Þ C B C B B C B B B .. C ¼ B @ . A B @ xð0Þ ðnÞ
12 ½xð1Þ ð1Þ þ xð1Þ ð2Þ
1
1
C 1 C a C : .. C . C A u 12 ½xð1Þ ðn 1Þ þ xð1Þ ðnÞ 1 12 ½xð1Þ ð2Þ þ xð1Þ ð3Þ .. .
ð2Þ
Let
1
0
B xð0Þ ð3Þ C C B C Yn ¼ B B .. C; @ . A
B B B B¼B B @
0
xð0Þ ð2Þ
a A¼ ; u
xð0Þ ðnÞ
12 ½xð1Þ ð1Þ þ xð1Þ ð2Þ
1
1
C 1C C C; ... C A 12 ½xð1Þ ðn 1Þ þ xð1Þ ðnÞ 1 12 ½xð1Þ ð2Þ þ xð1Þ ð3Þ ...
the matrix can be expressed as,
Y n ¼ B A:
ð3Þ
In the Eq. (3), Y n and B can be obtained directly by the original data, but A needs more calculation. One usually uses leastsquare method to get the least-square approximation, and the Eq. (3) can be expressed as,
b þ eH ; Yn ¼ B A wherein, eH is an error term. Using matrix derivation formula, one gets,
b ¼ ðBT BÞ1 BT Y n ¼ A
^ a : ^ u
^ are obtained, furthermore, ^ and a So, u
^ ^ u u ea^t þ : xð1Þ ðtÞ ¼ xð1Þ ð1Þ ^ ^ a a Let xð1Þ ð0Þ ¼ xð0Þ ð1Þ, the time response function of GM(1,1) model is also obtained,
^ ^ u u xð0Þ ð1Þ ea^k þ ^ ^ a a
ðk ¼ 1; 2; 3; Þ:
ð4Þ
Then, one does inverse accumulated revivification with the time response function (4), and gets grey prediction model,
^ u ^xð0Þ ðk þ 1Þ ¼ ^xð1Þ ðk þ 1Þ ^xð1Þ ðkÞ ¼ xð0Þ ð1Þ ð1 ea^ Þea^k ^ a
ðk ¼ 1; 2; 3; Þ
ð5Þ
Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
1005
^. Then, the concrete pre^ and a After substituting the correlative data into expression (5), one can resolve the coefficients u diction formula can be confirmed. 3. Lorenz map 3.1. Lorenz equation The general form of Lorenz equation [11,12] is:
8 > < x_ ¼ rðy xÞ; y_ ¼ ðr zÞx y; > :_ z ¼ xy bz: In 1976, Guckenheimer and Williams introduced geometrical Lorenz model for the first time. This model is a Poincar map on the Poincar section which has Lorenz class flow system and usually used to describe the variation regularity of atmosphere movement. This geometrical structure of Lorenz flow may be reduced to a self-map on one-dimensional interval f : ½l; m ! ½l; m,
( f ðxÞ ¼
fL ðxÞ ¼ m ajxjk þ h:o:t;
x 6 0;
fR ðxÞ ¼ l þ bxk þ h:o:t; x > 0;
ðl; m > 0; k > 1Þ
where k is a constant greater than 1, h:o:t represents high-level term. Both of the branches fL and fR are monotone increasing. In order to get iterative sequences in the part of chaos, the Lorenz map should be:
( f ðxÞ ¼
fL ðxÞ ¼ 1 2jxj2 ; x 6 0; fR ðxÞ ¼ 1 þ 2x2 ; x > 0:
ð6Þ
The concrete symbolic definition rule is shown in Fig. 1. 3.2. The symbolic dynamics and dual star products of Lorenz map Seen from symbolic dynamical system [13], Lorenz map belongs to a more complex dynamical category because it has a discontinuity point. The Lorenz map presents more abundant dynamical actions [14,15] compared with Unimodal map. The symbolic dynamics of Lorenz map can consult paper [16]. Fig. 2 is the bifurcation diagram of Lorenz map. There are two kinds of dual star products in the Lorenz map, namely, the up-star product ðÞ and the down-star product ðÞ. Suppose there are two kneading pairs W ¼ ðCV 1 ; DU 1 Þ and Z ¼ ðCV 2 ; DU 2 Þ, thereinto,
U 1 ¼u11 u12 u1i u1k ;
V 1 ¼ v 11 v 12 v 1j v 1l ;
u1i ; v 1j 2 fL; Rg
U 2 ¼u21 u22 u2i u2n ;
V 2 ¼ v 21 v 22 v 2j v 2m ;
u2i ; v 2j 2 fL; Rg
then, the dual star products are just W Z ¼ ðCV 1 ; DU 1 Þ ðCV 2 ; DU 2 Þ; 2 f; g, and each symbol of Z can be replaced one by one according to the following rules,
Fig. 1. Symbolic definition rule of Lorenz map.
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Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
Fig. 2. The bifurcation diagram of Lorenz map.
8 C ! CV 1 RU 1 ; > > > < D ! DU LV ; 1 1 The up-star product ðÞ : > L ! LV 1 RU 1 ; > > : R ! RU 1 LV 1 : 8 C ! ðDU 2 LV 2 ÞT ; > > > > < D ! ðCV RU ÞT ; 2 2 The down-star product ðÞ : > L ! ðRU LV ÞT ; > 2 2 > > : R ! ðLV 2 RU 2 ÞT ;
ð7Þ
ð8Þ
thereinto, T is a keeping parity operator, which makes the following symbolic counterchange: L $ R; C $ D. Obviously, the length of compound sequences obtained by the dual star products rules (7) and (8) is
jWj jZj ¼ ðk þ l þ 2Þ ðn þ m þ 2Þ: 4. Grey prediction of Lorenz chaotic system based on GM(1,1) 4.1. Iterative process of Lorenz chaotic system The general dynamical iterative form of Lorenz map is expression (6). Giving an initial value x1 ¼ 0:65, the number of iterations is n ¼ 16, one can obtain Tables 1 and 2. Fig. 3 is the concrete iterative process. Let us transform these iterative values. Firstly, each f ðxn Þ is made absolute value transformation. Secondly, one carries out P cumulative sum Zðxk Þ ¼ ki¼1 j f ðxi Þ j, and gets Tables 3 and 4. 4.2. Random generating sequence’s grey prediction of Lorenz chaotic system In the GM(1,1) model, the expression of AGO is xð1Þ ðkÞ ¼ The formula of grey estimate value is,
Pk
m¼1 x
ð0Þ
ðmÞ, so one gets Table 5.
h ui Estim: ¼ xð1Þ ð0Þ eat ð1 ea Þ; a thereinto, xð1Þ ð0Þ ¼ 0:1550; a ¼ 0:10625; u ¼ 2:15158; t ¼ 1; 2; 3; , namely,
Estim: ¼ 20:4052 e0:10625t ð1 e0:10625 Þ:
Table 1 The iterative values of Lorenz chaotic system (n ¼ 1; 2; ; 8). n
1
2
3
4
5
6
7
8
xn f ðxn Þ
0:65 0:1550
0:1550 0:9520
0:9520 0:8124
0:8124 0:3200
0:3200 0:7951
0:7951 0:2645
0:2645 0:8601
0:8601 0:4795
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Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009 Table 2 The iterative values of Lorenz chaotic system (n ¼ 9; 10; ; 16). n
9
10
11
12
13
14
15
16
xn f ðxn Þ
0:4795 0:5402
0:5402 0:4163
0:4163 0:6534
0:6534 0:1461
0:1461 0:9573
0:9573 0:8329
0:8329 0:3874
0:3874 0:6999
Fig. 3. Sixteen iterations of Lorenz chaotic system (x1 ¼ 0:65).
Table 3 The cumulative sum results of Zðxk Þ (k ¼ 1; 2; ; 8). k
1
2
3
4
5
6
7
8
Zðxk Þ
0:1550
1:1069
1:9194
2:2394
3:0346
3:2991
4:1591
4:6386
Table 4 The cumulative sum results of Zðxk Þ (k ¼ 9; 10; ; 16). k
9
10
11
12
13
14
15
16
Zðxk Þ
5:1788
5:5951
6:2485
6:3946
7:3519
8:1848
8:5722
9:2721
Table 5 Main variable values of GP process. k
Zðxk Þ
AGO
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.1550 1.1069 1.9194 2.2394 3.0346 3.2991 4.1591 4.6386 5.1788 5.5951 6.2485 6.3946 7.3519 8.1848 8.5722 9.2721
0.1550 1.2619 3.1813 5.4207 8.4553 11.7544 15.9135 20.5521 25.7309 31.3260 37.5745 43.9691 51.3210 59.5058 68.0780 77.3501
AGO mean * 0.7085 2.2216 4.3010 6.9380 10.1049 13.8340 18.2328 23.1415 28.5285 34.4503 40.7718 47.6451 55.4134 63.7919 72.7141
Mean square
Mean AGO
Mean square AGO
Mean AGO square
*
*
*
*
0.5019 4.9355 18.4986 48.1358 102.1080 191.3782 332.4350 535.5290 813.8725 1186.820 1662.340 2270.051 3070.645 4069.407 5287.333
0.7085 2.9301 7.2311 14.1691 24.2739 38.1079 56.3407 79.4822 108.0106 142.4609 183.2327 230.8777 286.2911 350.0830 422.7971
0.5019 5.4374 23.9360 72.0719 174.1798 365.5580 697.9930 1233.5220 2047.3945 3234.2142 4896.5539 7166.6047 10237.2496 14306.6561 19593.9892
0.5019 8.5852 52.2881 200.7620 589.2222 1452.208 3174.269 6317.412 11666.29 20295.09 33574.20 53304.51 81962.59 122558.1 178757.3
So, the prediction values and corresponding errors of 16 iterations can be calculated (see Table 6). Fig. 4 shows the actual values and the prediction values of 16 iterations. Here are grey prediction results of the following 3 steps:
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Y. Zhang et al. / Chaos, Solitons and Fractals 42 (2009) 1003–1009
Table 6 Prediction values and corresponding errors of 16 iterations. k
Zðxk Þ actual value
Zðxk Þ estimate value
Error
Relative error (%)
Precision (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.1550 1.11 1.92 2.24 3.03 3.30 4.16 4.64 5.18 5.60 6.25 6.39 7.35 8.18 8.57 9.27
0.1550 2.29 2.54 2.83 3.15 3.50 3.89 4.33 4.81 5.35 5.95 6.62 7.36 8.19 9.10 10.12
0 1:18 0:62 0:59 0:12 0:2 0.27 0.31 0.37 0.25 0.3 0:23 0:01 0:01 0:53 0:85
0 106:31 32:29 26:34 3:96 6:06 6.49 6.68 7.14 4.46 4.80 3:60 0:14 0:12 6:18 9:17
100 6:31 67.71 73:66 96:04 93:94 93.51 93.32 92.86 95.54 95.2 96.4 99.86 99.88 93.82 90.83
Fig. 4. Sixteen iterations’ actual values and prediction values of Lorenz chaotic system.
k ¼ 17 : Estim: value ¼ 11:26; k ¼ 18 : Estim: value ¼ 12:52; k ¼ 19 : Estim: value ¼ 13:92: It will be intuitionistic and natural that returning these grey prediction results to actual symbolic space. Combined the exact position of peak and trough, the value of positive or negative can be confirmed. 5. Conclusion The GP of Lorenz chaotic system has been clarified carefully in this paper. We are mainly using GM(1,1) model to predict data sequences, and the usual prediction precision has exceeded 90%. Because of the intrinsic property of chaos, it is impossible to give an exact prediction in long-term. In the symbolic prediction of Lorenz chaotic dynamical system, the precision of GP certainly will decrease as the length of symbolic sequence is increasing. But in this place we have found a generating rule that may realize chaotic synchronization at least in a short and medium term, and we can analysis and predict in this way. Maybe on next idyllic weekend trip, we need not carry a superfluous umbrella. Acknowledgements This research was supported partly by the National Natural Science Foundation of China (50777016) and the Youthful Teacher Research Foundation of NCEPU.
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