Weak Signals Detection On The Property Of Sensitive Dependence On Initial Conditions Of Chaos System

Chaotic motion, which widely exists, can describe the very typical behavior of many nonlinear systems. An important application of chaos theory in information science is information detection. A chaotic system is sensitive to regular signal but immune to noise, which make chaos have good prospect in information detection technique. Chaos detection for weak signals belongs to this technique. Although it is a new subject, it has already make a great progress, At present, by acting on a chaotic detector, a small signal can cause a transition between the states of the system from the chaotic and periodic and thus be detected. Yet it is easy to bring about such questions as: firstly, the precision is low, the main reason for which is that the detection method has only exploited some qualitative aspects of the property of sensitive dependence on initial conditions, which are only associated with the state transition of the Duffing oscillator, to detection the existence of the signal with known frequency and its amplitude. For more complicated case such as the determination of the amplitude and phase of a weak signal (as considered in the present paper), we have to find the relation between the varieties of the system state and the parameters of the signal. In other words, we need to define first an appropriate index to quantitatively represent the varieties of the system state. Such an index should be sensitive to the signal parameters and insensitive to the noise. The Lyapunov exponents, who reflect the rate of expansion and contraction near a limited set, are most appropriate for use as an index. Secondly, the critical value is not accurate. In the above-mentioned method, it is necessary to make certain the bifurcation threshold value of the chaotic critical state, but from now, there isn't any analytical algorithm to present the value. In fact, the value is usually determined experimentally, from which, we only get the range of the value not accurate value. To solve this problem, we either choose a detection method that is independent of the critical value or find a new accurate approach to determine the critical value. The Lyapunov exponent is one of the most important characteristic quantities of chaos. The calculation of the value in usually state is so complicated and time-consuming (by definition, t →∞) that it is difficult to be used in a practical signal processing. In this brief, we make the chosen system always keep in a periodic motion ( γ> γc) and the calculation of the Lyapunov exponent in this state is easy, in this detection method the accurate critical value isn't need. In this brief, we choose Duffing equation as the detector, and presented the algorithm of the Lyapunov exponents of this system under periodic state. The Lyapunov exponents are employed to quantitatively represent the relation between the varieties of the system state and the parameters of the signals, and the detection of signals immerged in strong noise is developed. The main work of this paper is given as follows, 1. Analysis on the deficiency of the existed methods for the weak signal detection on chaos system and the Improvement scheme we will adopted in this paper. Firstly we summarized and analyzed the research actuality about the weak signal detection on chaos system, and indicate the deficiency of the existing methods that detects the weak signals by transition of the detector's state from chaotic to periodic state. Propose the solution: to define an index to represent the property of sensitive dependence on initial conditions quantitatively, and find the relation between the index and the parameters of the signal. In the paper, represent in detail two physics index that are relevant to the property of sensitive dependence on initial conditions: the length of the laminar phase and the Lyapunov exponents. 2. Analysis on the bifurcation theory of the chaos detection model. The present method was carried out when the detection system is periodic, so we have to choose the periodic orbit. The corresponding bifurcation process for Duffing oscillator is discussed in detail, and we analyzed the feasibility and rationality of the periodic orbit chosen in the present method, the corresponding parameter of detection system was presented. 3. Analysis on the application of the Lyapunov characteristic exponents under period state in chaotic for weak signal Firstly we presented the algorithm of the Lyapunov characteristic exponents of chaos detection system under period, and the basic thought of using the exponents to detect the weak signal was given. According to the relation curve of the...