一类新超混沌系统的产生、同步及其应用
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摘要
混沌作为一种复杂的非线性运动行为,在生物学、物理学和信息学等领域得到了广泛的研究。由于混沌具有内在的随机性、连续宽带谱和对初始值的高度敏感性等特点,使其特别适用于保密通信、信号处理、图像处理等方面。
1979年,R?ssler首次提出了超混沌的概念,与一般的混沌系统相比较,超混沌系统具有两个或两个以上的正Lyapunov指数,其相轨在更多方向上分离,动力学行为更加复杂,用于诸如保密通信中具有更大的潜在应用价值。因此有关超混沌的产生是目前混沌研究领域中的一个热点,尤其是有目的地、利用简单的手段控制原有的混沌系统进入到超混沌状态。
本学位论文研究了一类新四维超混沌系统的产生、广义投影同步、电路设计与实现(包括模拟电路设计与实现以及可编程逻辑器件中的设计与实现)及其在USB KEY中的应用,具体研究内容如下:
(1)用状态反馈控制法获得了一类新四维自治超混沌系统。在一个具有一个鞍点、二个稳定结焦点的新三维自治系统基础上,提出用线性状态反馈控制法和非线性状态反馈控制法产生超混沌;在另一个新三维自治混沌系统的基础上,提出用非线性状态反馈控制法产生超混沌。线性状态反馈控制法的核心是设计一个简单的线性状态反馈控制器,用它和原三维系统构成一个满足产生超混沌必要条件的四维自治系统,并使得四维系统可以产生超混沌。类似原理,非线性状态反馈控制法的核心是设计一个非线性状态反馈控制器,并把非线性状态控制器反馈到原三维系统中,使四维系统可以产生超混沌。通过对这类四维自治系统的基本动力学行为进行研究,包括平衡点性质、Lyapunov指数谱、分岔图及MATLAB仿真等,从理论上证实了这类四维自治系统可以产生超混沌吸引子。
(2)研究了有关分数阶四维超混沌系统。对提出的这类四维自治超混沌系统,以其中一个系统为例,用时频域转换分析方法和预估-校正时域方法,对这个分数阶四维系统进行了分析,结果表明q = 0.1时,系统仍然可以产生超混沌。在分数阶积分算子电路的基础上,设计了有关分数阶超混沌模拟电路,并进行了硬件电路实验。
(3)实现了这类超混沌系统间的广义投影同步。用主动控制方法的思想,分别设计了合适的非线性反馈控制器,实现了这类四维自治超混沌系统的有关广义投影同步,包括同结构投影同步、异结构投影同步以及错位投影同步,并用MATLAB进行了相关数值仿真,结果表明同步方案的可行性,同步速度快且稳定。
(4)根据这类四维超混沌系统状态方程的特点,基于电路理论,设计了这类四维自治超混沌系统的模拟电路,整个电路由反相求和电路、积分电路、反相电路和乘法器四部分组成。相同原理,设计了同结构投影同步电路、异结构投影同步电路、错位投影同步电路。对所设计的超混沌电路及投影同步电路进行了电路实验,并给出了相关实验结果。实验结果与数值仿真结果基本一致,在模拟电路上证实了超混沌吸引子的存在及投影同步方案的可行性。
(5)可编程门阵列技术的超混沌系统实现。用模拟电路实现的超混沌系统及投影同步方案易受实际器件精度、外界干扰等影响,不利于工程上的应用,而采用现代数字信号处理技术则可以克服这些问题。为此,提出采用二阶Runge-Kutta法来离散这类四维自治超混沌系统,对离散化后的数字化超混沌系统用FPGA技术来实现。离散化后的数字化超混沌系统可采用DSP Builder工具箱来搭建,也可采用硬件描述语言中的状态机来描述。结合实例,详细阐述了这两种设计方法的具体实现过程,最后通过数字模拟转换器将数字序列转换为模拟信号,在示波器上能观察到超混沌吸引子。FPGA实验结果与数值仿真结果完全一致,证实了这两种设计方法的可行性。
(6)在国民技术Z32安全芯片上,开发了相应的芯片操作系统(COS)、驱动程序等,并用可编程逻辑器件扩展了一种新的硬件加密算法,即超混沌加密算法。扩展的超混沌加密算法可以与芯片内部集成的DES等算法进行级联构成级联加密算法。将这种级联加密算法应用于Outlook 2007中,实现了电子邮件内容的加解密。
As we know, chaotic system can generate very complex nonlinear dynamic behavior, and has been intensively investigated in many fields such as biology, physics and information. Because of chaotic system possesses the following features: internal randomicity, wide band spectrum, high sensitivity to initial conditions, thus the application of chaos can be especially found in secure communication, signal processing and image processing.
Historically, hyperchaos was firstly reported by R?ssler in 1979. As we know, the normal chaotic system has one positive Lyapunov exponent, but hyperchaotic system has at least two positive Lyapunov exponents, implying that its dynamics are expended in several different directions simultaneously. It means that hyperchaotic system has more complex dynamic behavior, which can be used to improve the security of chaotic communication. So about the generation of hyperchaos, especially purposefully designing a hyperchaotic system from a chaotic system with simple ways, becomes a focus of chaotic study. So far, there is no systematic theoretical method about the generation of hyperchaos.
This thesis studies on a class of novel hyperchaotic systems including the generation, generalized projective synchronization, conventional analog circuit design and implementation, field programmable gate array based design and implementation, and its application. The main contributions are listed as follows:
(1) A class of four-dimensional autonomous hyperchaotic systems are obtained by incorporating state feedback controller. By incorporating a simple dynamical linear state feedback controller and a simple dynamical nonlinear state feedback controller based on a new three-dimensional autonomous chaotic system with one saddle and two stable node-foci, two four-dimensional systems which can generate hyperchaos are obtained. Similarly, by incorporating another simple dynamical nonlinear state feedback based on another new three-dimensional autonomous chaotic system, a four-dimensional system which can generate hyperchaos is obtained. These four-dimensional systems are analyzed by investigating the Lyapunov exponent spectrum and bifurcation diagram.
(2) Study on one four-dimensional hyperchaotic system with fractional order by time-frequency transformation approach and predictor-corrector approach, the results show that it still can generate hyperchaos when the fractional order q equals 0.1. The corresponding fractional order analog circuit is designed and implemented.
(3) Using active control method, we present generalized projective synchronization of this class of hyperchatic systems, including the same hyperchaotic system, different hyperchaotic system, and dislocated projective synchronization. Some numerical simulations are show that the schemes of synchronization have a good performance.
(4) According to the character of state equations for this class of hyperchaotic systems, a class of hyperchaotic analogue circuits are designed, which are composed of four parts: anti-adder, integrator, inverter and multiplier. Similarly, the generalized projective circuits are also designed. Some hardware platforms are built and some experiments are done. The observations are agreement with numerical simulations, which verify that hyperchaotic attractors exist and the schemes of synchronization are feasible.
(5) Implementation of this class of hyperchaotic systems based on FPGA. As we know, the conventional analogue circuits will be easily influenced by the device accuracy and outside interference, but such problems can be overcome by modern digital signal processing technology. Two-step Runge-Kutta algorithm is used, in order to convert the continuous hyperchaotic systems into discrete hyperchaotic systems, which is appropriate for FPGA processing. We use two methods to realized discrete hyperchaotic systems, one is based on DSP Builder tool, another is based on hardware description language and state machine. We give the detail developing process of the two methods with examples. Using high-speed digital-to-analog converter (DAC), continuous analog hyperchaotic signal can be observed. The experimental observations are agreement with numerical simulations, which verify the two methods are feasible.
(6) Based on Z32 secure chip, some applications are developed including chip operating system, drivers and so on. A hyperchaotic encryption and decryption algorithms is implemented by FPGA technology. It can be combined with DES algorithm, so a scheme of cascade can be obtained. Applying this scheme of cascade to Outlook 2007, we realize encryption and decryption in E-mail system.
1979年,R?ssler首次提出了超混沌的概念,与一般的混沌系统相比较,超混沌系统具有两个或两个以上的正Lyapunov指数,其相轨在更多方向上分离,动力学行为更加复杂,用于诸如保密通信中具有更大的潜在应用价值。因此有关超混沌的产生是目前混沌研究领域中的一个热点,尤其是有目的地、利用简单的手段控制原有的混沌系统进入到超混沌状态。
本学位论文研究了一类新四维超混沌系统的产生、广义投影同步、电路设计与实现(包括模拟电路设计与实现以及可编程逻辑器件中的设计与实现)及其在USB KEY中的应用,具体研究内容如下:
(1)用状态反馈控制法获得了一类新四维自治超混沌系统。在一个具有一个鞍点、二个稳定结焦点的新三维自治系统基础上,提出用线性状态反馈控制法和非线性状态反馈控制法产生超混沌;在另一个新三维自治混沌系统的基础上,提出用非线性状态反馈控制法产生超混沌。线性状态反馈控制法的核心是设计一个简单的线性状态反馈控制器,用它和原三维系统构成一个满足产生超混沌必要条件的四维自治系统,并使得四维系统可以产生超混沌。类似原理,非线性状态反馈控制法的核心是设计一个非线性状态反馈控制器,并把非线性状态控制器反馈到原三维系统中,使四维系统可以产生超混沌。通过对这类四维自治系统的基本动力学行为进行研究,包括平衡点性质、Lyapunov指数谱、分岔图及MATLAB仿真等,从理论上证实了这类四维自治系统可以产生超混沌吸引子。
(2)研究了有关分数阶四维超混沌系统。对提出的这类四维自治超混沌系统,以其中一个系统为例,用时频域转换分析方法和预估-校正时域方法,对这个分数阶四维系统进行了分析,结果表明q = 0.1时,系统仍然可以产生超混沌。在分数阶积分算子电路的基础上,设计了有关分数阶超混沌模拟电路,并进行了硬件电路实验。
(3)实现了这类超混沌系统间的广义投影同步。用主动控制方法的思想,分别设计了合适的非线性反馈控制器,实现了这类四维自治超混沌系统的有关广义投影同步,包括同结构投影同步、异结构投影同步以及错位投影同步,并用MATLAB进行了相关数值仿真,结果表明同步方案的可行性,同步速度快且稳定。
(4)根据这类四维超混沌系统状态方程的特点,基于电路理论,设计了这类四维自治超混沌系统的模拟电路,整个电路由反相求和电路、积分电路、反相电路和乘法器四部分组成。相同原理,设计了同结构投影同步电路、异结构投影同步电路、错位投影同步电路。对所设计的超混沌电路及投影同步电路进行了电路实验,并给出了相关实验结果。实验结果与数值仿真结果基本一致,在模拟电路上证实了超混沌吸引子的存在及投影同步方案的可行性。
(5)可编程门阵列技术的超混沌系统实现。用模拟电路实现的超混沌系统及投影同步方案易受实际器件精度、外界干扰等影响,不利于工程上的应用,而采用现代数字信号处理技术则可以克服这些问题。为此,提出采用二阶Runge-Kutta法来离散这类四维自治超混沌系统,对离散化后的数字化超混沌系统用FPGA技术来实现。离散化后的数字化超混沌系统可采用DSP Builder工具箱来搭建,也可采用硬件描述语言中的状态机来描述。结合实例,详细阐述了这两种设计方法的具体实现过程,最后通过数字模拟转换器将数字序列转换为模拟信号,在示波器上能观察到超混沌吸引子。FPGA实验结果与数值仿真结果完全一致,证实了这两种设计方法的可行性。
(6)在国民技术Z32安全芯片上,开发了相应的芯片操作系统(COS)、驱动程序等,并用可编程逻辑器件扩展了一种新的硬件加密算法,即超混沌加密算法。扩展的超混沌加密算法可以与芯片内部集成的DES等算法进行级联构成级联加密算法。将这种级联加密算法应用于Outlook 2007中,实现了电子邮件内容的加解密。
As we know, chaotic system can generate very complex nonlinear dynamic behavior, and has been intensively investigated in many fields such as biology, physics and information. Because of chaotic system possesses the following features: internal randomicity, wide band spectrum, high sensitivity to initial conditions, thus the application of chaos can be especially found in secure communication, signal processing and image processing.
Historically, hyperchaos was firstly reported by R?ssler in 1979. As we know, the normal chaotic system has one positive Lyapunov exponent, but hyperchaotic system has at least two positive Lyapunov exponents, implying that its dynamics are expended in several different directions simultaneously. It means that hyperchaotic system has more complex dynamic behavior, which can be used to improve the security of chaotic communication. So about the generation of hyperchaos, especially purposefully designing a hyperchaotic system from a chaotic system with simple ways, becomes a focus of chaotic study. So far, there is no systematic theoretical method about the generation of hyperchaos.
This thesis studies on a class of novel hyperchaotic systems including the generation, generalized projective synchronization, conventional analog circuit design and implementation, field programmable gate array based design and implementation, and its application. The main contributions are listed as follows:
(1) A class of four-dimensional autonomous hyperchaotic systems are obtained by incorporating state feedback controller. By incorporating a simple dynamical linear state feedback controller and a simple dynamical nonlinear state feedback controller based on a new three-dimensional autonomous chaotic system with one saddle and two stable node-foci, two four-dimensional systems which can generate hyperchaos are obtained. Similarly, by incorporating another simple dynamical nonlinear state feedback based on another new three-dimensional autonomous chaotic system, a four-dimensional system which can generate hyperchaos is obtained. These four-dimensional systems are analyzed by investigating the Lyapunov exponent spectrum and bifurcation diagram.
(2) Study on one four-dimensional hyperchaotic system with fractional order by time-frequency transformation approach and predictor-corrector approach, the results show that it still can generate hyperchaos when the fractional order q equals 0.1. The corresponding fractional order analog circuit is designed and implemented.
(3) Using active control method, we present generalized projective synchronization of this class of hyperchatic systems, including the same hyperchaotic system, different hyperchaotic system, and dislocated projective synchronization. Some numerical simulations are show that the schemes of synchronization have a good performance.
(4) According to the character of state equations for this class of hyperchaotic systems, a class of hyperchaotic analogue circuits are designed, which are composed of four parts: anti-adder, integrator, inverter and multiplier. Similarly, the generalized projective circuits are also designed. Some hardware platforms are built and some experiments are done. The observations are agreement with numerical simulations, which verify that hyperchaotic attractors exist and the schemes of synchronization are feasible.
(5) Implementation of this class of hyperchaotic systems based on FPGA. As we know, the conventional analogue circuits will be easily influenced by the device accuracy and outside interference, but such problems can be overcome by modern digital signal processing technology. Two-step Runge-Kutta algorithm is used, in order to convert the continuous hyperchaotic systems into discrete hyperchaotic systems, which is appropriate for FPGA processing. We use two methods to realized discrete hyperchaotic systems, one is based on DSP Builder tool, another is based on hardware description language and state machine. We give the detail developing process of the two methods with examples. Using high-speed digital-to-analog converter (DAC), continuous analog hyperchaotic signal can be observed. The experimental observations are agreement with numerical simulations, which verify the two methods are feasible.
(6) Based on Z32 secure chip, some applications are developed including chip operating system, drivers and so on. A hyperchaotic encryption and decryption algorithms is implemented by FPGA technology. It can be combined with DES algorithm, so a scheme of cascade can be obtained. Applying this scheme of cascade to Outlook 2007, we realize encryption and decryption in E-mail system.
引文
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