| Chaos control is an important part of the chaos theory. Since the OGY method proved that by applying small perturbation, the chaotic phenomenon could be controlled, there have been lots of methods in the area of chaos control. Based on the aim of chaos control, it can be divided into two parts: restraining chaos and strengthening chaos. The basic idea of chaos control is that taking advantage of the extreme sensitivity to the small perturbation, causing the system's phase trajectory change greatly by means of applying the given small perturbation, and making the positive Lyapunov exponent of the chaos system to be negative. By utilizing this method, we can realize the control of the chaos system.An important application of Chaos theory in information science is signal detection. The main direction of the study is using the chaos oscillators to detect the weak signal. Considering the theory and the aim of chaos control, this method of detection by chaos oscillators also belongs to the chaos control. In chaos oscillators detecting, we use the weak signal as the small perturbation, and add it to the system, which will cause the phase trajectory bouncing from the chaotic state to the long periodic state.There are many methods of chaos control. The control variable and the phenomenon are different in different methods. We can take advantage of the relationship between the small perturbation we applied during the process of chaos controlling and some special variable of the system, to build a model, by which we can detect the weak signal using the chaos control perturbation. In other words, we could use the weak signal to be a special variable of the system, and put it into the chaos system, then, we can control the system by applying small perturbation, and make the state of the system to be periodic. At last, we can detect the weak signal by analyzing the relationship between the weak signal and the small perturbation we applied.According to the analysis above, we determine the main work of this paper as follows: based on the theory and characteristics of the chaos control, we can analyze the relationship between the small perturbation we applied in the chaos control and the special variable corresponded, and finally built a weak signal detecting model. So we could affiliate the theory of chaos control with the study of weak signal detection.The following gives the main work of this paper in detail.1. Analysis and study on the theory and main methods of chaos control. Firstly, we studied the theory, characteristics and aim of chaos control, and analyzed several main methods of chaos control, like the OGY control, the Continuous feedback control, the Linear feedback control, and the Adaptive control .etc. Then, by analysis, we found that, we could use the relationship between the small perturbation and some special variable of the chaos system in some chaos control methods to build the model of detecting weak signal, and realize the weak signal detection by chaos control.2. The model of weak signal frequency detection by small perturbation of chaotic orbitsAt first, we analyzed the theory of the chaos control method based on the small perturbation of chaotic orbits——the OGY method. Through the analysis, we found that, the small perturbation we applied in OGY method has a linear relationship with the"small disturbance"of the system. So we could see the signal to be detected as the"small disturbance", and add it to the system. Then, we used the OGY method to apply small perturbation to the system, and make it to be stable. Because of the linear relationship between the two, we could detect the weak signal's frequency by observing the small perturbation's trajectory. At last, we realized this detecting model on the Chua's Circuit, and detected a weak sine signal's frequency in an ideal situation。3. The model of weak signal frequency detection by small perturbation of resonant parameterFirstly, we studied the application and phenomenon of the small perturbation of resonant parameter control. And we analyzed the Homeles Duffing function with the small perturbation in x 3 concretely. By our analysis, we found that when the frequencies of the small perturbation we used and the periodic driving force of the system satisfy some condition, the chaos system will take on the resonant phenomenon. Then, making use of this discovery, we brought forward a frequency detecting method of weak signal. That is to detect a weak sine signal with fixed frequency by the small perturbation of resonant parameter.Through the mathematical experiment, we proved that this model is sensitive to the change of the frequency. We also proved that this model is"immune"to the noise in a way. When the parameter of the system is fixed, the frequency it can detect is fixed, too. The operation of the model is very simple, so it has some practicability.4. The frequency scanning by the sign changing of xIn the model of weak signal frequency detection by small perturbation of resonant parameter, we needed to find what frequency of the system signal can cause a resonant phenomenon. And we should realize this by frequency scanning. In nonlinear dynamics, the Lyapunov exponent is usually used to analyze the state of the system quantitatively. But if we use the Lyapunov exponent to judge whether the resonant phenomenon has taken place, the calculation is too complex, and will cause the fact that the efficient is low.In the mathematical experiment, by observing the phase trajectory both of the chaotic state and the resonant state, we can find that, when the system is in chaotic state, the sign of x will change, but when it is in the resonant state, the sign of x won't change. So utilizing this special phenomenon, we construct a method of frequency scanning by judging the sign changing of x. Using this method, the calculation has been reduced greatly. Compared with the method that using the Lyapunov exponent, the speed of the detection has risen more than one time, and we could get a high efficiency. |
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