The synchronization analysis of four symbolic complex dynamical systems will be discussed carefully in this paper. Grey system theory is mainly being used to study data sequences that are generated by 4letter chaotic dynamical system, and the usual prediction accuracy has exceeded 90%. In this place we have found a generating rule that may at least realize chaotic synchronization in short and medium terms. Considering the current study of DNA base sequences AGCT and the symbolic characteristic of four symbolic dynamical systems, which are formally in good corresponding relation. In this paper we have offered an effective research means to approach problems of this kind.
The problems on uncertainty exist commonly in nature: the ones in myriad sample can be resolved by probability and statistics ways, the ones in recognizing uncertainty can be dealt with by fuzzy mathematics. However, there also exists another kind of uncertainty in less data, little samples, incomplete information, and devoid of experience, which is just suitable to be dealt with by grey system theory [
Grey system theory mainly includes the theoretical system based on grey algebra system, grey equation, and grey matrix; the methodology system based on grey sequence generation; the analysis system based on grey correlation space; the greymodelcentered modeling system; and the technical system based on system analysis, evaluation, modeling, prediction and decision, and control and optimization. Since there is not any certain physical model in many social and economic issues, it is quite difficult to specify all the factors, not to mention establishing definite mapping relations, though some influence factors are known. For instance, those factors that affect prices, such as psychological expectation and government orientation, are immeasurable. While some data are measurable yet short of detailed information. If the available data alone are taken into consideration, the analysis outcome will surely be inaccurate. Grey system theory can generate new data sequences which indicate the variation trend of the original data and eliminate the fluctuation as well. Grey system theory can also solve some complex system issues with unknown parameters. The applications of grey system theory lie in many scientific fields, such as agriculture, industry, energy, traffic, petroleum, geology, meteorology, hydrology, ecology, environment, medicine, military affairs, economy, and society, which succeeds in solving good many practical problems in production and daily life.
As early as in 1947, Ulam and Neumann studied the densities distribution [
This paper is organized as follows. In Section
Grey system theory provides an approach to investigate the relationships of inputoutput process with unclear inner relationships, uncertain mechanisms, and insufficient information. It deals directly with the original data and searches the intrinsic regularity of the data rather than using statistical method. In the grey system theory, grey relational analysis and grey prediction model account for the essential parts [
The grey models have many different forms [
One of the most important properties of chaotic system is
One of the primary motivations of our research is designed to discuss the corresponding relationships between the exponential separate characteristic in chaotic system and the exponential increasing law of grey prediction. For example, let us choose logistic equation as a model, which has been studied extensively in nonlinear system. One of its equivalent forms can be expressed as
hereinto,
The bifurcation diagram of logistic map.
Let us adopt unimodal surjective map as
Given initial values
The exponential separate sequences of logistic map

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  31  32  33 
 








— 








— 
The exponential separate diagram of logistic map. (In this diagram, the solid line represents
According to Figure
The generic iterative form of 4letter surjective map is
The shape of 4letter surjective map is shown in Figure
The shape of 4letter surjective map
The general dynamical iterative form of 4letter surjective map is expressed in (
The iterative values of 4letter chaotic system (

1  2  3  4  5  6  7  8 



















The iterative values of 4letter chaotic system (

9  10  11  12  13  14  15  16 



















16 iterations of 4letter surjective map (
Let us transform these iterative values. First, each
The cumulative sum results of

1  2  3  4  5  6  7  8 










The cumulative sum results of

9  10  11  12  13  14  15  16 










In the grey system theory, the expression of accumulated generating operator (AGO) is
Main variable values of grey prediction process.


AGO  AGO mean  Mean square  Mean AGO  Mean square AGO  Mean AGO square 

1  0.9128  0.9128  *  *  *  *  * 
2  1.0248  1.9376  1.4252  2.0312  1.4252  2.0312  2.0312 
3  1.9256  3.8632  2.9004  8.4123  4.3256  10.4435  18.7108 
4  2.1498  6.0130  4.9381  24.3848  9.2637  34.8283  85.8161 
5  2.7680  8.7810  7.3970  54.7156  16.6607  89.5440  277.5789 
6  3.6568  12.4378  10.6094  112.5594  27.2701  202.1033  743.6584 
7  3.9840  16.4218  14.4298  208.2191  41.6999  410.3225  1738.882 
8  4.2193  20.6411  18.5315  343.4146  60.2314  753.7371  3627.816 
9  4.8007  25.4418  23.0415  530.9084  83.2728  1284.6455  6934.359 
10  5.5909  31.0327  28.2373  797.3423  111.5101  2081.9878  12434.49 
11  6.4671  37.4998  34.2663  1174.176  145.7763  3256.1637  21250.73 
12  6.8936  44.3934  40.9466  1676.624  186.7229  4932.7877  34865.44 
13  7.0843  51.4777  47.9356  2297.817  234.6585  7230.6047  55064.59 
14  7.8041  59.2818  55.3798  3066.917  290.0382  10297.5214  84122.16 
15  8.8015  68.0833  63.6826  4055.467  353.7208  14352.9886  125118.4 
16  9.7593  77.8426  72.9630  5323.592  426.6837  19676.5806  182059.0 
The formula of grey estimate value is
thereinto,
So, the prediction values and corresponding error of 16 iterations can be calculated (see Table
Prediction values and corresponding error of 16 iterations.



Error  Relative error  Precision 

1  0.9128  0.9128  0  0  100% 
2  1.0248  2.18 



3  1.9256  2.43 


73.80557% 
4  2.1498  2.72 


73.4766% 
5  2.7680  3.04 


90.17341% 
6  3.6568  3.40 


92.97747% 
7  3.9840  3.80 


95.38153% 
8  4.2193  4.25 


99.27239% 
9  4.8007  4.75 


98.9439% 
10  5.5909  5.31 


94.97576% 
11  6.4671  5.94 


91.84952% 
12  6.8936  6.64 


96.32123% 
13  7.0843  7.42 


95.26135% 
14  7.8041  8.30 


93.64565% 
15  8.8015  9.28 


94.56343% 
16  9.7593  10.38 


93.63991% 
Figure
16 iterations’ actual values and prediction values of 4letter chaotic system.
Here are grey prediction results of the following 3 steps:
It will be intuitionistic and natural to return these grey prediction results to actual symbolic space. Combining the exact position of peak and trough, the value of positive or negative can be confirmed.
Now let us present a research example.
According to a group of DNA based sequences “CCGGGGTCGCGCGGCA”, which double helix structure has been presented in Figure
DNA double helix structure.
The final four symbolic prediction results are “NNMMMMNNMNMNMMNL”, and the corresponding DNA based sequences are “CCGGGGCCGCGCGGCA”. Figure
DNA prediction structure.
It should be stressed that the biological system is highly complex; we cannot look to one method or two methods to solve all problems. But at least, we have provided an interesting thought.
The synchronization analysis of four symbolic complex dynamical systems has been clarified carefully in this paper. We are mainly using grey system theory to study data sequences that are generated by 4letter chaotic dynamical system, and the usual prediction accuracy has exceeded 90%. In this place we have found a generating rule that may at least realize chaotic synchronization in short and medium terms. And in this way we can analyze and predict different kinds of complex systems.
The world is essentially nonlinear and highly complex. A small stochastic force is usually able to have an unexpected impact on deterministic equation, and in certain conditions, it could determine the evolution of systems and even change the fate of macroscopic system. Nonlinear dynamics theory, the combination of Newtonian mechanics and statistical mechanics method, linear stochastic equation and nonlinear deterministic equation, and periodic solution and chaotic solution have achieved the high inherent unity of determinacy and randomicity, determinism, and indeterminism. The change of weather, the growth of species, the diffusion of molecular, the fluctuation of electrocardiogram and electroencephalogram signals, the periodic reform of society, the growth, the sudden plunge, and even the collapse of stock markets all have inherent nonlinear dynamics rules [
Considering the current study of DNA [
This research was supported partly by the NSFC (51277193), the Specialized Research Fund for the Doctoral Program of Higher Education (20110036110003), the Fundamental Research Funds for the Central Universities (11MG37, 12MS120), and the Natural Science Foundation of Hebei Province.