【作者】 曾以成；

【导师】 童勤业；

【作者基本信息】 浙江大学 ， 生物医学工程， 2002， 博士

【摘要】 因混沌系统具有初值敏感性和参数敏感性，混沌理论初步建立，人们就想到利用这种特性进行微弱信号检测或测量，但混沌系统是不稳定的，且对噪声存在相同的敏感性，所以，抑制噪声成为混沌测量的关键问题。在本工作中，通过对混沌测量系统实施耦合，探索如何抑制测量过程中的噪声；同时，采用简单的非线性电路，以降低混沌测量系统的复杂性：采用符号动力学方法，以降低混沌系统描述上的复杂性。 通过耦合来抑制噪声的基本思想源于混沌同步研究。在合适条件下，耦合的混沌系统能达同步，我们认为其走向同步的过程是被耦合的混沌轨道相互靠近到重合的过程，由相互靠近的两段轨道测量出的初始值就有可能被约束在一定范围内，而不至于发散得比实际的噪声扰动更远，这是有利于测量的。我们对所用混沌测量系统的可同步性进行了研究，通过建立两个及三个混沌测量系统的耦合方程，具体分析了两电路耦合的同步过程及同步稳定性；理论分析与数值实验表明，系统是可稳定同步的，即是可进行耦合测量的。 两个混沌电路耦合能否抑制噪声，进行了四个方面的数值实验：(1)只初值受噪声干扰，轨道其它地方不受噪声干扰；(2)轨道中间某处有噪声，而别处无噪声；(3)只初值受噪声干扰，耦合强弱变化的影响；(4)只初值受噪声干扰，电路参数有涨落的影响。在实验范围内，均观察到了噪声被抑制的效果。 三个电路耦合时，与两个电路的耦合进行了对比数值实验。结果表明测量差异比两个电路耦合时更小。即抑制噪声的效果，三个测量系统耦合时比两个测量系统耦合要好。 推广到N(N是大的整数)个的情形会如何，我们用锯齿映射的多个耦合实验了轨道靠近的趋势。接下来，对从耦合Logistic映射格子模型中恢复初值的统计特性进行了数值实验，给定的初始信号分布为高斯分布，在映象过程中不存在噪声的情况下，实验显示恢复信号的均值等于给定信号的均值，而恢复信号的方差小于给定信号的方差，并且还表明，若同样的耦合系数，全局耦合与局部耦合相比，前者恢复信号的方差比给定信号的方差小的更多。接着，针对不同的参数变化，对从耦合混沌测量系统映射格子模型中恢复初值的统计特性进行了类似的数值实验，得知在合适的参数范围内，亦有相似的实验结果。从统计意义上，证明N个测量系统的耦合能有效地抑制噪声。 本论文的结果是初步的，但是，是富有启发性的。用单个系统独立重复测量，相当于对一个随机过程在时间维上独立取样；而这里的耦合测量相当于在空间维(或系综)上同时取样，且取样是不独立的，分析处理是独立的，这是一种新的思路。 单个神经元或多个神经元的集合均表现出混沌特性，生物神经系统是一个复杂的耦合混沌系统，这里的耦合混沌信号测量模型可以为复杂的生物信息处理系统建模所借鉴。从复杂系统恢复初值的统计方法也可以为类似的耦合系统求逆问题提供参考。更多还原

【Abstract】 Chaotic system has initial-conditions-sensitivity and parameters-sensitivity,and can be used to detect or measure weak signals. But it is unstable and it is also sensitive to noise,reducing noise is the key step in measuring procedure. This problem is solved in this paper by coupled several measuring systems. Meanwhile,we use a simple nonlinear circuit for decreasing the complexity of chaotic measuring system and use symbolic dynamics for simplifying analysis on the system.The idea to reduce the noise with coupling the measuring systems was from the chaos synchronization. In suitable coupling intensity,coupled chaotic systems can be synchronized,so we think that synchronizing is the procedure of coupled trajectories attracting and coinciding each other at last. Two initial conditions,measured by coupled two trajectories,may be confined tightly and do not diverge over initial noise distortion,this point is useful for chaotic measuring. We studied the synchronizable property of our chaotic measuring system,established the equations of coupled two subsystems and three subsystems respectively,analyzed the synchronizing process and synchronization stability on the basis of symbolic dynamics. Theoretical analysis and numerical simulations show that the chaotic measuring systems can be synchronized stably.Whether two coupled chaotic measuring systems can reduce noise or not,we carried out the following simulations in four situations:(1) noise only in starting point,no noise elsewhere in trajectories,(2)noise only in somewhere on trajectories,elsewhere no noise,(3) coupling intensity change,noise only in initial point,(4) system parameters fluctuation,noise only in initial point. The noise-reduction effect was observed in all cases.Comparing the results from coupled three circuits with coupled two circuits,it is discovered that the error in the former is smaller than that in the latter. Namely,the former has the stronger noise-reduction function.Extended measuring system to coupled N units,firstly,we investigated the trajectory distance between unit one and unit two in coupled many units with units number increasing,secondly,we tested the statistical properties of the initial conditions recovery from the coupled Logistic map lattices. Gaussian white noise as the initial signals and no noise in mapping,simulations show that the mean of the recovered signals equals that of the given,and the variance of the former is smaller than that of the latter. In the same coupling coefficient,the global coupling model has the smaller variance than local coupling. Following this,we tested the statistical properties of the initial conditions recovery from the coupled chaotic measuring system lattices,some results are the similar to the former case. All tests indicate noise-reduction is effective in coupled TV systems.These achievements are preliminary,but they are heuristic. Use single system to measure many times is to sample the stochastic process in time dimension. Here we use N coupled units to measure is to sample the stochastic process in space dimension or ensemble,and this sampling is not independent,but data processing is independent. This is a new measurement model.A single neuron or a small ensemble of neurons is known to exhibit chaotic behaviors. Biological neural systems are complex coupled chaotic systems. Our model may be model these systems. In some situations,one must solve reverse problem,our model can be taken as reference.更多还原