混沌和异步布尔网络中若干问题的研究
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摘要
复杂性科学作为系统学发展的新阶段,是当代科技发展的前沿领域之一。混沌系统和基因调控网络作为具有代表性的复杂系统是复杂性科学重要的研究领域和热点。本文利用理论分析和计算机仿真模拟相结合的方法对混沌系统的动力学特性、同步及在数字保密通信中的应用进行了讨论；并且,在逻辑系统线性化的基础上,对基因调控网络的量化模型——布尔网络的动力学特性以及布尔控制网络的可控性进行了研究。全文主要工作包括：
(1)将分数阶混沌系统从实数空间扩展到复数空间,提出了一类新的混沌模型——分数阶复混沌系统。利用相图、最大Lyapunov指数及分岔图等研究工具对分数阶复Lorenz系统和分数阶复Chen系统的动力学特性分别进行了详细讨论。通过改变系统参数和系统阶次,在参数空间选取几条代表性的路径,对新系统“通向混沌的道路”进行了研究,发现模型可通过Pomeau-Mannevilla途径、倍周期分岔等不同方式通向混沌,并观察到周期窗口和各类分岔现象等丰富的动力学行为。在对分数阶复混沌系统动力学研究基础上,将该模型应用于混沌保密通信领域,利用其混沌性强和信息携带量大的特点,分别提出了基于同步和异步两类数字保密通信方案。在同步方案中,使用耦合分数阶复Chen系统的交叉同步,实现了混沌数字键控保密通信,在不损失安全性的同时,有效简化系统设计的复杂性。在异步方案中,实现了可自纠错的信息保密传输,通过仿真实验验证,当传输过程中出现数据错误时,利用该方案可高概率的实时侦测及纠正。
(2)修正函数投影同步在混沌同步领域中具有高度的广义性,在本文中,分别对分数阶混沌系统、复混沌系统和不同维混沌系统间的修正函数投影同步进行了讨论,利用Lyapunov稳定性理论设计了自适应控制器,为增加方案的普适性,在同步过程中考虑了时滞、参数辨识以及参数干扰等因素。并且,本文提出了在复混沌系统的模空间和相空间同时实现两类不同方式的同步行为,即在模空间实现修正投影同步,在相空间实现反同步。该方案适用于只能采集到信号模和相数值的应用领域。
(3)利用矩阵半张量积,实现了广义异步布尔／多值网络的线性化表示,提出了系统网络状态转移矩阵的统一公式。在逻辑系统线性化基础上,对广义异步布尔／多值网络的动力学特性进行详细讨论,给出了判定任意长度吸引子的充要条件,并提出了在状态空间搜索全部吸引子和对应吸引域的具体算法。在线性化的基础上,对异步布尔控制网络的可控性进行研究。首先,对异步控制网络中状态可确定性控制的充要条件进行讨论；其次,基于三种不同控制方法,分别给出了系统概率可达状态集合的表达形式以及概率最大目标状态的控制路径等,从而对异步布尔控制网络的可控性进行了较全面的讨论。
In recent years, complexity science has been one of the front research fileds. Chaotic systems and gene regulatory newtorks as typical complex systems are the important and hot research fields of complexity science. In this dissertation, by using theoretical analysis and numerical simuliations, some problems in chaotic systems, such as the dynamics, synchronization and applications in secure commnucation have been discussed. Furthermore, based on linear representations of logical systems, the dynamics of asynchronous Boolean networks and controllability of asynchronous Boolean control networks have been studied. The main work of this paper is as follows:
(1) The fractional-order complex chaotic systems are first proposed, which extend the chaotic systems from real space into complex space. By utilizing phase portraits, the largest Lyapunov exponents and the bifurcation diagrams, dynamical behaviors of fractional-order complex Lorenz system and Chen system are studied, respectively. Chaotic regions and periodic windows are explored as well as different types of motions such as tangent bifurcations, flip bifurcations and chaotic crisis shown along the routes to chaos. By means of the fractional-order complex chaotic systems, two kinds of digital secure communication schemes have been achevied based on synchronous and asynchronous methods. In the synchronous scheme, an improved switch-modulated digital secure communication system is designed, which is based on a hybrid synchronization in coupled fractional-order complex Chen systems. And in the asynchronous scheme, an error-correcting secure communication is discussed, which has the ability of error checking and correcting in real time and high probability against the errors existing in transmitted signals. Numerical simuliations show the feasibility of the proposed schemes.
(2) Modified function projective synchronization (MFPS) is a kind of generalized synchronization scheme, and some other chaotic synchronizations, such as complete synchronization, anti-synchronization, projective synchronization and function projective synchronization, can be seen as the special cases of MFPS. Here, MFPS in frational-order chaotic systems, complex chaotic systems and different dimensions chaotic systems have been discussed, respectively. Based on Lyapunov stability theory, the adaptive controllers have been achieved. Theoretical proof and numerical simulations demonstrate the effectiveness of the proposed schemes. A hybrid modulus-phase synchronization in coupled complex systems is first proposed, which means different synchronization schemes can be achieved simultaneously in modulus and phase spaces of complex systems. A general controller is designed to implement modified projective synchronization in modulus space and anti-synchronization in phase space. And the theoretical proof is given based on the Lyapunov stability theory. This proposed scheme is suitable for some real cases that only modulus and phase of complex state variables are measurable.
(3) By using the semi-tensor product, the linear representations of generalized asynchronous Boolean networks (GABNs) and generalized asynchronous mutilple-valued networks (GAMVNs) are achevied. The dynamical behaviors of GABNs and GAMVNs are investigated analytically. Some results about the fixed points and loose attractors are provided and algorithms to find all of attractors and basins of attrations are given. Based on the algebraic representation, controllability of asynchronous Boolean control networks (ABCNs) is discussed. Firstly, deterministic controllability in ABCNs is discussed. Secondly, via three different kinds of controls, controllability of ABCNs is analytically discussed by revealing the reachable sets, respectively. Examples are shown to demonstrate the feasibility of the related discussions.
(1)将分数阶混沌系统从实数空间扩展到复数空间,提出了一类新的混沌模型——分数阶复混沌系统。利用相图、最大Lyapunov指数及分岔图等研究工具对分数阶复Lorenz系统和分数阶复Chen系统的动力学特性分别进行了详细讨论。通过改变系统参数和系统阶次,在参数空间选取几条代表性的路径,对新系统“通向混沌的道路”进行了研究,发现模型可通过Pomeau-Mannevilla途径、倍周期分岔等不同方式通向混沌,并观察到周期窗口和各类分岔现象等丰富的动力学行为。在对分数阶复混沌系统动力学研究基础上,将该模型应用于混沌保密通信领域,利用其混沌性强和信息携带量大的特点,分别提出了基于同步和异步两类数字保密通信方案。在同步方案中,使用耦合分数阶复Chen系统的交叉同步,实现了混沌数字键控保密通信,在不损失安全性的同时,有效简化系统设计的复杂性。在异步方案中,实现了可自纠错的信息保密传输,通过仿真实验验证,当传输过程中出现数据错误时,利用该方案可高概率的实时侦测及纠正。
(2)修正函数投影同步在混沌同步领域中具有高度的广义性,在本文中,分别对分数阶混沌系统、复混沌系统和不同维混沌系统间的修正函数投影同步进行了讨论,利用Lyapunov稳定性理论设计了自适应控制器,为增加方案的普适性,在同步过程中考虑了时滞、参数辨识以及参数干扰等因素。并且,本文提出了在复混沌系统的模空间和相空间同时实现两类不同方式的同步行为,即在模空间实现修正投影同步,在相空间实现反同步。该方案适用于只能采集到信号模和相数值的应用领域。
(3)利用矩阵半张量积,实现了广义异步布尔／多值网络的线性化表示,提出了系统网络状态转移矩阵的统一公式。在逻辑系统线性化基础上,对广义异步布尔／多值网络的动力学特性进行详细讨论,给出了判定任意长度吸引子的充要条件,并提出了在状态空间搜索全部吸引子和对应吸引域的具体算法。在线性化的基础上,对异步布尔控制网络的可控性进行研究。首先,对异步控制网络中状态可确定性控制的充要条件进行讨论；其次,基于三种不同控制方法,分别给出了系统概率可达状态集合的表达形式以及概率最大目标状态的控制路径等,从而对异步布尔控制网络的可控性进行了较全面的讨论。
In recent years, complexity science has been one of the front research fileds. Chaotic systems and gene regulatory newtorks as typical complex systems are the important and hot research fields of complexity science. In this dissertation, by using theoretical analysis and numerical simuliations, some problems in chaotic systems, such as the dynamics, synchronization and applications in secure commnucation have been discussed. Furthermore, based on linear representations of logical systems, the dynamics of asynchronous Boolean networks and controllability of asynchronous Boolean control networks have been studied. The main work of this paper is as follows:
(1) The fractional-order complex chaotic systems are first proposed, which extend the chaotic systems from real space into complex space. By utilizing phase portraits, the largest Lyapunov exponents and the bifurcation diagrams, dynamical behaviors of fractional-order complex Lorenz system and Chen system are studied, respectively. Chaotic regions and periodic windows are explored as well as different types of motions such as tangent bifurcations, flip bifurcations and chaotic crisis shown along the routes to chaos. By means of the fractional-order complex chaotic systems, two kinds of digital secure communication schemes have been achevied based on synchronous and asynchronous methods. In the synchronous scheme, an improved switch-modulated digital secure communication system is designed, which is based on a hybrid synchronization in coupled fractional-order complex Chen systems. And in the asynchronous scheme, an error-correcting secure communication is discussed, which has the ability of error checking and correcting in real time and high probability against the errors existing in transmitted signals. Numerical simuliations show the feasibility of the proposed schemes.
(2) Modified function projective synchronization (MFPS) is a kind of generalized synchronization scheme, and some other chaotic synchronizations, such as complete synchronization, anti-synchronization, projective synchronization and function projective synchronization, can be seen as the special cases of MFPS. Here, MFPS in frational-order chaotic systems, complex chaotic systems and different dimensions chaotic systems have been discussed, respectively. Based on Lyapunov stability theory, the adaptive controllers have been achieved. Theoretical proof and numerical simulations demonstrate the effectiveness of the proposed schemes. A hybrid modulus-phase synchronization in coupled complex systems is first proposed, which means different synchronization schemes can be achieved simultaneously in modulus and phase spaces of complex systems. A general controller is designed to implement modified projective synchronization in modulus space and anti-synchronization in phase space. And the theoretical proof is given based on the Lyapunov stability theory. This proposed scheme is suitable for some real cases that only modulus and phase of complex state variables are measurable.
(3) By using the semi-tensor product, the linear representations of generalized asynchronous Boolean networks (GABNs) and generalized asynchronous mutilple-valued networks (GAMVNs) are achevied. The dynamical behaviors of GABNs and GAMVNs are investigated analytically. Some results about the fixed points and loose attractors are provided and algorithms to find all of attractors and basins of attrations are given. Based on the algebraic representation, controllability of asynchronous Boolean control networks (ABCNs) is discussed. Firstly, deterministic controllability in ABCNs is discussed. Secondly, via three different kinds of controls, controllability of ABCNs is analytically discussed by revealing the reachable sets, respectively. Examples are shown to demonstrate the feasibility of the related discussions.
引文
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