摘要
在保密通信中,最关键的是要有一个可靠的密码产生器。研究发现该密码产生器可由一个混沌系统来实现,混沌系统在混沌通信中起着“加密”的作用,构造一个好的混沌系统对保密通信是很有意义的。混沌同步在混沌保密通信中起着“解密”的作用,混沌同步是混沌保密通讯中的关键技术之一。
目前对非时滞混沌系统的大量研究成果表明,正的Lyapunov指数个数较少的非时滞低维混沌系统在保密性方面并不很强。为使系统具有多个正的Lyapunov指数,系统的维数将急剧增加,同步系统的结构随之变得庞大而复杂,这无形中就增加了同步的难度。时滞混沌系统为无穷维系统,它是可以创生具有多个正Lyapunov指数、结构相对简单却能产生具有极高随机性和不可预测性的时间序列的超混沌系统。因此在同步的过程中,它既可克服高维非时滞超混沌系统在结构上庞大而复杂的缺点,又能提高保密性。因此,对时滞混沌系统的构造与同步的研究都是具有重要实际意义的,它有望成为高新技术的一个新的生长点。另一方面,时滞混沌系统可以看成抗破译能力极强的时空混沌系统的特殊情况,对时滞混沌同步的深入研究定会有助于一般时空混沌同步的研究。所以,关于时滞混沌同步理论的研究,也是一件很有理论意义的工作。
论文围绕混沌保密通信的两个核心问题,即混沌系统的构造(“制造密钥”)和混沌系统同步(“解密”)的判定进行了理论探讨和数值仿真,在以下几个方面取得了一定的成果。
对于时滞混沌系统提出两种构造方法:其一是扰动,对已有的混沌系统作时滞线性扰动或者对线性系统作时滞非线性扰动;其二是变性,对非时滞混沌系统的非线性项作时滞变形。通过对稳定的线性系统和已有的非时滞混沌系统的改造,分别得到两个自由度为1的新时滞混沌系统,六个自由度为2的新时滞混沌系统,五个自由度为3的新时滞混沌系统,其中包括一个自由度为3的时滞反馈Chua电路系统(它易于用电子线路实现,且可用于模拟混沌保密通信)。
论文给出了时滞系统Lyapunov指数比较确切的定义,进而提出了判定时滞系统是否混沌的简单实用方法:首先考察波形图和吸引子投影图,初步判定系统是否混沌;再作出解对初值的敏感性示意图,进一步对系统是否混沌进行确认;最后通过估算最大Lyapunov指数作出系统是否混沌的确切结论。此方法用多个原有的著名的时滞混沌系统进行了检验,发现它是可靠的。用此方法还对文章中构造出的所有新时滞系统(以下简称新时滞系统)的混沌特性作了考察,结果表明它们在取一定的参数时都是混沌的。
In security communication, the cryptogram generator is a key device. It is showed that in chaotic security communication, this cryptogram generator can be a chaotic system. In chaotic security communication, the utility of chaotic system is "encryption", thus it is valuable to construct a proper chaotic system for chaotic security communication. While the duty of chaotic synchronization is "decryption", it is a key technology for chaotic security communication.It is showed by large amount of uptodate investigations of nondelay chaotic system that those low dimensional nondelay chaotic systems with few positive Lyapunov exponents were not safe enough. To increase the number of positive Lyapunov exponents, the dimension of a system should be increased drastically, and the structure of the synchronization system would be huge and complex, so it would be more difficult for synchronization. Time-delayed chaotic system is an infinite dimension system; it can be a hyperchaotic system with large numbers of positive Lyapunov exponents, and can generate highly stochastic and unpredictable time series. Thus in the process of synchronization, it overcomes the shortage of nondelay chaotic system with huge dimension and complex structure, and it can also increase safety for secure communication. Therefore the study of chaotic synchronization of a time-delayed chaotic system is of high practical importance for increasing the safety of secure communication. It may become a new increase point of new-higher technology. In addition, time-delayed chaotic system can be considered as a special case of spatio-temporal chaotic systems with extremely high safety for security communication. The study of synchronization of time-delayed chaotic system would be helpful for the study of synchronization for general spatio-temporal chaotic systems. Thus the study of synchronization of a time-delayed chaotic system is of theoretical significance.This thesis investigated two key problems for chaotic security communication, which is the construction of chaotic system ("cryptogram generator") and their synchronization ("decrypt"). By theoretical investigation as well as numerical simulation, some results with different aspect were established.Two methods were presented for constructing a time-delayed chaotic system: one is perturbation, time-delayed linear perturbation for chaotic system or time-delayed nonlinear perturbation for linear system, and the other is a time-delayed modification,
adjusting the nonlinear term of a chaotic system to a delayed one. By amendment of stable linear system and nondelay chaotic system, two novel time-delayed chaotic systems with freedom degree 1 , six novel time-delayed chaotic systems with freedom degree 2 , five novel time-delayed chaotic systems with freedom degree 3 including a time-delayed feedback Chua circuit with freedom of degree 3 were constructed.An original definition of Lyapunov exponent for time-delayed system was presented, upon which a simple practical method for determining whether a time-delayed system was chaotic or not was presented: one first exams the wave form and the projection plot of attractor to find out whether a system is chaotic in the first round, then he draws the 'sensitive' wave form to strengthen the affirmed result of the first step, and at last he estimates the largest Lyapunov exponent to make an affirmative conclusion. This method was tested by several well known time-delay chaotic system and it was found to be reliable. The chaotic activities of new time-delayed system constructed were demonstrated by this method and it was found that all the new time-delayed systems were chaotic with certain parameters.For the investigation of synchronization of time-delayed chaotic systems, several asymptotic stability theorems with matrix inequalities as sufficient conditions for linear time-delayed system were proved by using stability theory of functional differential equations. Upon the analysis of the structure of the new time-delayed chaotic system constructed, two classes of abstract time-delayed chaotic systems were presented, one is the time-delayed nonlinear perturbation of a nondelay linear system and the other is nondelay or time-delayed nonlinear perturbation of a time-delayed linear system. These two classes of systems cover all the new time-delayed chaotic system constructed. The asymptotic stability theorems were used to investigate three kinds of synchronization problems (complete synchronization, lag synchronization and anticipating synchronization) of these two classes of time-delayed systems, and several sufficient conditions were established. And then the synchronization parameters were tested by these sufficient conditions for all new time-delayed chaotic system constructed, and the numerical simulation with Matlab 6.5 showed satisfaction.Adaptive synchronization and control problems for time-delayed system with parameters not fully known were investigated. Firstly, the adaptive chaotic synchronization problem of the time-delayed linear perturbed parameters fully unknown Chen system was studied by using Lyapunov-Krassovskii functional method, a parameters update law was established and the complete synchronization with the
time-delayed linear perturbed Chen system as drive system was reached. Then the adaptive chaotic synchronization problem of the time-delayed feedback Chua system with two parameters unknown was studied by using Lyapunov function method, a parameters update law was established and the complete synchronization with time-delayed feedback Chua system as drive system were reached. Finally, the Lyapunov-Krassovskii functional method was used to study the adaptive synchronization and control problems for a general class of time-delayed system, two general theorem were proved and a new time-delayed chaotic system constructed were used to simulat it.
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