| An important application of Chaos theory in information science is information detection. Chaos system is sensitive to weak signals and immune to noise, which make chaos have good prospects in information detection technique. Chaos detection for weak signals belongs to this technique. Although it is a new subject, it has already made a great progress. There are two key problems in detection for weak signals based on chaos: one is to determine the critical value of the chaos system, the other is to judge the state of the system. Since people have always adopted experimental approach to determine the critical value and visual methods to judge the system's state, poor precision and low efficiency become the shortcomings of chaos detection for weak signals at present. So a new accurate approach to determine the critical value and a new quantitative criterion to judge the system's state are badly needed in chaos detection for weak signals in order to improve the precision and the efficiency. And Lyapunov Characteristic Exponent (LCE) can do these two jobs. LCE is one of the most important characteristic quantities of chaos, and also is an important criterion for the identification of chaos. The primary character of chaos is the sensitivity to initial condition, and LCE is just the measurement of this sensitivity. A positive LCE means the system is chaotic, while all the LCEs are negative, then the system is non-chaotic. So the sign of the LCE can be used to determine the state of the system, and the point at which the sign of the LCE changes from positive to negative can be used to achieve the critical value of the system. Nowadays, most algorithms to compute LCEs are more suited to higher dimensional systems. The computation processes of these methods are complicated, and the computational efficiency is low, which make them difficult to be used in practice. And there are few fast and effective algorithms suited to lower dimensional systems. The chaotic system for weak signals detection we discussed in this paper is only two dimensional. Although the existed algorithms can achieve the LCEs of the system, the efficiency is low and the precision is poor. So it's necessary to find a new fast and effective algorithm applied for lower dimensional systems. According to the analysis above, we determine the main work of this paper as follows. Firstly, find a fast and efficient algorithm applied to lower dimensional system to compute LCEs. Secondly, employ LCEs in the field of chaos detection for weak signals. Use LCEs to get the critical value of the system, and to judge the state of the weak signals detection system as a quantitative criterion. The following gives the main work of this paper in detail.Analysis and comparison of different methods for chaos identificationMake an analysis of the existed methods for chaos identification, and classify these methods into two categories: visual methods and quantitative methods. Visual methods include time evolvement of the system's variable , phase trajectory map, frequency flash sampling, Poincaré section and power spectrum. Quantitative methods include fractal dimension, Kolmogorov entropy and LCE. Then we used mathematical model to compare these methods, and got a conclusion: visual methods have low efficiency and poor precision, and even worse they can't accurately judge the system's state before and after the phase transition. While the quantitative methods judge the system's state by explicit numerical values, they are more accurate. Furthermore, LCE method is the most basic and also the most important method among them. With respect to the chaos identification in weak signals detection based on chaos, LCE method is the most desirable one. Research on the methods for computing the LCEs of deterministic systems with lower degrees of freedomBased on the standard QR factorization algorithm and the RHR (the first letter of the three authors' surnames) algorithm, which are two representative methods to compute the LCEs of deterministic systems...
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