混沌系统函数投影同步的理论与应用研究
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摘要
在过去的几十年中,由于混沌同步在物理系统、生态网络和安全通讯等领域的潜在应用,其已成为非线性科学领域中的一个热门课题,并得到海内外科学家和工程师的广泛关注。混沌同步是指两个或更多的混沌系统通过耦合或外力所具有的共同的动态特性。迄今为止,人们提出了很多种混沌同步的方式,如混沌系统的完全同步、相同步、广义同步、滞后同步和投影同步等。其中投影同步由于具有不可预料的比例因子可进一步增加保密通信的安全性及通信的传输速度,其在混沌保密通讯领域中具有潜在的应用前景,因而近期被广泛的研究。
目前大多数针对混沌系统投影同步的研究都是基于常数比例因子进行的,而对函数比例因子研究较少。由于不可预料的比例函数可进一步增加保密通信的安全性,因此对混沌系统的函数投影同步问题的研究,具有十分重要的理论意义和应用价值。本文主要针对混沌系统的函数投影同步和改进函数投影同步问题进行研究,给出了各种情况下达到这两种同步的控制器的设计方法,及它们在保密通信中的应用方法。本文取得的主要成果归纳如下:
首先在混沌系统函数投影同步的研究方面,详细阐述了函数投影同步的概念,并验证了适当的选取比例函数可增加同步混沌系统的复杂程度及混沌程度。基于Lyapunov稳定性理论研究了函数投影同步的通用方法,给出了任意两个相同混沌系统和不同混沌系统达到函数投影同步的控制器设计的通用方法。基于自适应控制的方法研究了一类带有未知参数的混沌系统的函数投影同步问题,给出了使该类混沌系统达到函数投影同步的控制器设计方法,及驱动系统中未知参数的估计方法。
其次提出了改进函数投影同步的定义。改进函数投影同步指驱动系统和响应系统的状态被同步到一个期望的比例函数矩阵,它是函数投影同步的一种更普遍的定义方式,其新特性是可以单独的设置不同状态的比例函数。很明显函数投影同步是改进函数投影同步的特例,当改进函数投影同步中的比例函数矩阵的每一个比例函数都取相同的函数时就是函数投影同步,因此改进函数投影同步的研究更具有普遍意义。很明显适当的选取比例函数矩阵中的比例函数,可进一步增加保密通信中信息的安全性。之后基于自适应控制的方法研究了带未知参数的统一混沌系统的改进函数投影同步问题,最后基于Lyapunov稳定性定理,分别研究了一般情况下两个相同和不同混沌系统发生改进函数投影同步的控制器设计方法。
研究了滞后函数投影同步和滞后改进函数投影同步问题。由于在远程通信中时滞广泛存在,因此研究具有时间延迟的混沌系统的函数投影同步和改进函数投影同步问题非常重要。基于Lyapunov稳定性定理,分别给出了达到这两种同步的控制器设计方法。
最后研究了函数投影同步和改进函数投影同步在保密通信中的应用。在理论上研究了利用这两种同步方式进行保密通信的方法。在所提通信方法中同步被实现,并且信息信号被准确的恢复。此外,由于预先指定的比例函数是任意的并且不被拦截方所知,使得这种通信方法的信号传送方式更加灵活且很难被拦截方所破解,因此可获得更安全的通信。
In the past decades, chaos synchronization has become a hot subject in the field of nonlinear science due to its great potential applications in physical systems, biological networks, and secure communications etc, which has attracted much attention from scientists and engineers. Chaos synchronization means two or more chaotic systems share a common dynamical behavior, which could be induced by coupling or by external forcing. Up to now, there exist many types of synchronization, such as, complete synchronization, phase synchronization, lag synchronization, generalized synchronization, and projective synchronization, etc. Amongst all kinds of chaos synchronization, projective synchronization has been extensively investigated in recent years because it can obtain faster and securer communication with its proportional feature.
At present, most of research efforts about projective synchronization have concentrated on studying the constant scaling factor, but it is seldom studying the function scaling factor. It is obvious that the unpredictability of the scaling function can additionally enhance the security of communications, which could be used to get more secure communications. So the studies of function projective synchronization are very important for theory and practical applications. This paper studies the function projective synchronization and the modified function projective synchronization problems, and gives variety control schemes in variety conditions and their applications in the field of secure communication. The main research results in this dissertation can be given as the following four parts.
Firstly, we illustrate the definition of function projective synchronization in detail and prove the synchronized chaotic system more complex and chaos. Based on Lyapunov stability theory, the general methods for function projective synchronization between two identical or different chaotic systems are investigated, respectively. The function projective synchronization of a general class of chaotic systems with unknown parameters is investigated by adaptive control scheme. The adaptive control law and the parameter update law are derived to make the states of two chaotic systems asymptotically synchronized up to a desired scaling function.
Secondly, we give the definition of modified function projective synchronization, which makes the responses of the synchronized dynamical states synchronizing up to a desired scaling function matrix. Modified function projective synchronization is the more general definition of function projective synchronization, which allows us to flex the scale functions of the different states independently. It is easy to see that function projective synchronization is the special cases of modified function projective synchronization, when all scaling functions of the scaling function matrix take the same scaling function. It is obvious that the unpredictability of the scaling function matrix in modified function projective synchronization can additionally enhance the security of communication. Then, based on Lyapunov stability theory, the general methods for modified function projective synchronization between two identical or different chaotic systems are investigated, respectively. Later, the modified function projective synchronization of unified chaotic systems with unknown parameters is investigated by adaptive control scheme.
Thirdly, the problems of lag function projective synchronization and the lag modified function projective synchronization are investigated. Due to that delay may exist extensively in the remote communication system, the studies of the function projective synchronization and modified function projective synchronization with lag time are very important. Based on Lyapunov stability theory, the general methods for lag function projective synchronization and lag modified function projective synchronization are investigated, respectively.
Finally, based on function projective synchronization and modified function projective synchronization, the schemes for secure communication are investigated. The secure communication methods using function projective synchronization and modified function projective synchronization are investigated in theory. In this case, synchronization can be completely realized and the information signal can be recovered accurately. Furthermore, these schemes offer much more flexibility in signal transmission and make the third party difficult to break the information signal because the scaling functions are arbitrary and unknown to the interceptor, which could be used to get more secure communications.
目前大多数针对混沌系统投影同步的研究都是基于常数比例因子进行的,而对函数比例因子研究较少。由于不可预料的比例函数可进一步增加保密通信的安全性,因此对混沌系统的函数投影同步问题的研究,具有十分重要的理论意义和应用价值。本文主要针对混沌系统的函数投影同步和改进函数投影同步问题进行研究,给出了各种情况下达到这两种同步的控制器的设计方法,及它们在保密通信中的应用方法。本文取得的主要成果归纳如下:
首先在混沌系统函数投影同步的研究方面,详细阐述了函数投影同步的概念,并验证了适当的选取比例函数可增加同步混沌系统的复杂程度及混沌程度。基于Lyapunov稳定性理论研究了函数投影同步的通用方法,给出了任意两个相同混沌系统和不同混沌系统达到函数投影同步的控制器设计的通用方法。基于自适应控制的方法研究了一类带有未知参数的混沌系统的函数投影同步问题,给出了使该类混沌系统达到函数投影同步的控制器设计方法,及驱动系统中未知参数的估计方法。
其次提出了改进函数投影同步的定义。改进函数投影同步指驱动系统和响应系统的状态被同步到一个期望的比例函数矩阵,它是函数投影同步的一种更普遍的定义方式,其新特性是可以单独的设置不同状态的比例函数。很明显函数投影同步是改进函数投影同步的特例,当改进函数投影同步中的比例函数矩阵的每一个比例函数都取相同的函数时就是函数投影同步,因此改进函数投影同步的研究更具有普遍意义。很明显适当的选取比例函数矩阵中的比例函数,可进一步增加保密通信中信息的安全性。之后基于自适应控制的方法研究了带未知参数的统一混沌系统的改进函数投影同步问题,最后基于Lyapunov稳定性定理,分别研究了一般情况下两个相同和不同混沌系统发生改进函数投影同步的控制器设计方法。
研究了滞后函数投影同步和滞后改进函数投影同步问题。由于在远程通信中时滞广泛存在,因此研究具有时间延迟的混沌系统的函数投影同步和改进函数投影同步问题非常重要。基于Lyapunov稳定性定理,分别给出了达到这两种同步的控制器设计方法。
最后研究了函数投影同步和改进函数投影同步在保密通信中的应用。在理论上研究了利用这两种同步方式进行保密通信的方法。在所提通信方法中同步被实现,并且信息信号被准确的恢复。此外,由于预先指定的比例函数是任意的并且不被拦截方所知,使得这种通信方法的信号传送方式更加灵活且很难被拦截方所破解,因此可获得更安全的通信。
In the past decades, chaos synchronization has become a hot subject in the field of nonlinear science due to its great potential applications in physical systems, biological networks, and secure communications etc, which has attracted much attention from scientists and engineers. Chaos synchronization means two or more chaotic systems share a common dynamical behavior, which could be induced by coupling or by external forcing. Up to now, there exist many types of synchronization, such as, complete synchronization, phase synchronization, lag synchronization, generalized synchronization, and projective synchronization, etc. Amongst all kinds of chaos synchronization, projective synchronization has been extensively investigated in recent years because it can obtain faster and securer communication with its proportional feature.
At present, most of research efforts about projective synchronization have concentrated on studying the constant scaling factor, but it is seldom studying the function scaling factor. It is obvious that the unpredictability of the scaling function can additionally enhance the security of communications, which could be used to get more secure communications. So the studies of function projective synchronization are very important for theory and practical applications. This paper studies the function projective synchronization and the modified function projective synchronization problems, and gives variety control schemes in variety conditions and their applications in the field of secure communication. The main research results in this dissertation can be given as the following four parts.
Firstly, we illustrate the definition of function projective synchronization in detail and prove the synchronized chaotic system more complex and chaos. Based on Lyapunov stability theory, the general methods for function projective synchronization between two identical or different chaotic systems are investigated, respectively. The function projective synchronization of a general class of chaotic systems with unknown parameters is investigated by adaptive control scheme. The adaptive control law and the parameter update law are derived to make the states of two chaotic systems asymptotically synchronized up to a desired scaling function.
Secondly, we give the definition of modified function projective synchronization, which makes the responses of the synchronized dynamical states synchronizing up to a desired scaling function matrix. Modified function projective synchronization is the more general definition of function projective synchronization, which allows us to flex the scale functions of the different states independently. It is easy to see that function projective synchronization is the special cases of modified function projective synchronization, when all scaling functions of the scaling function matrix take the same scaling function. It is obvious that the unpredictability of the scaling function matrix in modified function projective synchronization can additionally enhance the security of communication. Then, based on Lyapunov stability theory, the general methods for modified function projective synchronization between two identical or different chaotic systems are investigated, respectively. Later, the modified function projective synchronization of unified chaotic systems with unknown parameters is investigated by adaptive control scheme.
Thirdly, the problems of lag function projective synchronization and the lag modified function projective synchronization are investigated. Due to that delay may exist extensively in the remote communication system, the studies of the function projective synchronization and modified function projective synchronization with lag time are very important. Based on Lyapunov stability theory, the general methods for lag function projective synchronization and lag modified function projective synchronization are investigated, respectively.
Finally, based on function projective synchronization and modified function projective synchronization, the schemes for secure communication are investigated. The secure communication methods using function projective synchronization and modified function projective synchronization are investigated in theory. In this case, synchronization can be completely realized and the information signal can be recovered accurately. Furthermore, these schemes offer much more flexibility in signal transmission and make the third party difficult to break the information signal because the scaling functions are arbitrary and unknown to the interceptor, which could be used to get more secure communications.
引文
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