人工神经网络系统的动态复杂性研究
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摘要
神经网络的非线性动力学性质,主要采用动力学系统理论来分析神经网络的演化过程和吸引子的性质,探索神经网络的协同行为和集体计算功能,了解神经信息处理机制。混沌神经网络是近年来发展起来的一门新的科学,该领域的的研究和发展对于混沌加密和混沌通信具有十分重要的应用价值和实际意义,是当前非线性领域内发展最迅速、最活跃、最引人注目的热点方向之一
众所周知,混沌加密和混沌通信是混沌发展的两大热点,这需要设计各种非线性电路来模拟、研究混沌系统。我们发现利用混沌神经网络设计出的混沌信号发生器具有全局稳定的优异性质,这克服了Chua电路吸引性质不佳,遇到扰动不容易收敛的不足,有助于混沌电路在密码学和通信等领域得到更好的工程应用。因此发现更多具有不同混沌吸引子的神经网络模型,并对其进行动力学研究就有了重要的理论和实际意义。
在混沌神经网络的动力学问题之中,混沌性判定和混沌同步是两大核心问题,由于混沌系统研究的难度较大,以及混沌动力系统理论知识的不完善,使得这两项工作变的非常艰难。系统的混沌性判定问题,除了需要较深的动力系统理论以外,还需要大量的数值计算、计算机仿真和算法分析,使得研究非常的艰难。而混沌同步由于其在混沌加密和混沌通信等领域重要的应用价值,也是近年来学者们研究的焦点。目前,研究混沌同步还是以传统的控制理论为主,并没有能很好的利用混沌系统自身的特点。
本文重点研究了连续人工神经网络的动态复杂性问题,特别是利用动力系统中的拓扑马蹄理论与计算机辅助计算相结合,对多个著名的神经网络模型的混沌性进行了严格的判定,并对神经网络的拓扑结构与混沌行为的关联关系进行了细致的理论分析,还研究混沌神经网络的一类同步控制问题,取得了下列成果：
(1)利用动力系统中由Smale马蹄理论发展得到的拓扑马蹄理论,结合计算机的辅助计算,对连续人工神经网络中非常具有代表性的Hopfield神经网络模型、细胞神经网络模型和一类具有物理意义的人工神经网络模型的混沌性给出严格的判定。该方法具有很强的适用性、灵活性、可靠性和数学上的严密性,可以有效地对系统的混沌性进行验证。
(2)研究了低维连续神经网络的混沌性与系统拓扑结构的联系,以著名的Hopfield神经网络为例,根据其神经元连接的拓扑结构分类,讨论了混沌行为与系统拓扑类型之间的联系,我们发现拓扑结构中出现环路是神经网络系统有混沌性质的必要条件,以及连接数小于4的神经网络系统是不会出现混沌行为的,并给出了理论证明。
(3)以细胞神经网络为例,研究了神经网络系统的混沌完全同步问题。与以往的混沌同步问题不同,本文讨论的同步为两个不具有混沌性质的子系统经过同步后得到具有混沌性质的耦合系统,我们利用拓扑马蹄理论对这个新颖的同步系统的混沌性给出了验证,还运用全局伸展定理,证明了该细胞神经网络模型与Chua电路模型是拓扑共轭的。
(4)研究了在混沌的发展中起到重要作用的两个非神经网络的系统模型：平滑Chua系统和NSG系统。虽然这两类系统均包含了丰富的动力学行为,但是在此之前的研究都局限在数值仿真上,本文利用动力系统中拓扑马蹄理论严格的验证了这两个系统的混沌性。
As a new research field developed in these recent years, the chaotic neural network is one of the most rapid developed, the most popular and the most attractive subjects in the nonlinear science. The complex dynamics of the chaotic neural network is different from normal neural networks which have the character of gradient descent. The chaotic neural network has plenty of dynamical behavior and different kinds of attractors, so a great deal of attention has been paid to this area. The nonlinear dynamics of neural network mainly make use of the dynamical system theory to analyze the evolution of the system and the character of the attractor, explore the cooperative behaviors of the nerve cells, and study the mechanism of information processing.
As we know, chaos encryption and chaos communication is the most attractive field in the research about chaos. Meanwhile, nonlinear circuit is very helpful to simulate the real chaotic system for studying chaos encryption and chaos communication. We find that chaotic signal generators based on chaotic neural network have the great character of global stability, which overcome the shortcoming of the famous Chua's circuit. So it has very important academic and practical significance to find more chaotic neural network models which have different chaotic attractors.
Chaos verification and chaos synchronization has become the focus of the research of the chaotic systems. However, the theory of dynamical complexity is profound to understand, which makes these two issues become very difficult problems. Especially, chaos verification needs not only the theory of dynamical system but also a great deal of numerical calculation, computer simulation and algorithm analysis. As for chaos synchronization, people have paid a great deal of attention on it for its important application in chaotic encryption and chaotic communication. Nevertheless now the research about chaos synchronization still base on the traditional control theory.
This paper studies the dynamical complexity in continuous artificial neural network, by computer simulation, Lyapunov exponent calculation, computer-assisted verification and so on. Rich dynamical behaviors are found, such as chaos, limit cycle, equilibrium point. In this paper, chaos verifications of several chaotic systems are given by the combination of topological horseshoe theory and computer-assisted verification. Using Poincare section and Poincare map, this paper gives an effective verification for the existence the existence of attracting sets in chaotic system. The research work carried out includes the following:
(1) Prove chaotic character on the Hopfield neural network model, cellular neural network model and a simple artificial neural network model strictly. The method of computer-assisted verification makes use of topological horseshoe theory developed by the Smale horseshoe of dynamical system combined with computer assisted. In these cases, we have shown this method works well and make chaos verification easily and reliability.
(2) Take the famous Hopfield neural network for instance to study the dynamical complexity of the low-dimensional continuous neural network according to topology structure of the models. We are concerned with this interesting problem in dynamics of neural networks that what connection topology will prohibit chaotic behavior in continuous time neural network and to what extent a continuous time neural network described by continuous ordinary differential equations is simple enough while can still exhibit chaos. This paper shows that the existence of directed loop in connection topology is necessary for chaos to occur.
(3) Present that chaos synchronization can take place in the coupled non-chaotic neural network systems. This is demonstrated by coupling two non-chaotic cellular neural networks. The couplings give rise to a synchronous chaotic dynamics and in the meanwhile the synchronous dynamics is globally asymptotically stable, thus chaos synchronization takes place under the suitable couplings. And it is interesting to show that the cellular neural network is topologically conjugate to an unfolded Chua's circuit by virtue of "The Global unfolding Theorem".
(4) The two non-neural network system models which play the important roles in chaos development are reviewed:smooth Chua system and NSG system. We present the rigorous verification of chaos by giving the computer-assisted verification based on horseshoe theory of dynamical system, which has been numerically observed before.
众所周知,混沌加密和混沌通信是混沌发展的两大热点,这需要设计各种非线性电路来模拟、研究混沌系统。我们发现利用混沌神经网络设计出的混沌信号发生器具有全局稳定的优异性质,这克服了Chua电路吸引性质不佳,遇到扰动不容易收敛的不足,有助于混沌电路在密码学和通信等领域得到更好的工程应用。因此发现更多具有不同混沌吸引子的神经网络模型,并对其进行动力学研究就有了重要的理论和实际意义。
在混沌神经网络的动力学问题之中,混沌性判定和混沌同步是两大核心问题,由于混沌系统研究的难度较大,以及混沌动力系统理论知识的不完善,使得这两项工作变的非常艰难。系统的混沌性判定问题,除了需要较深的动力系统理论以外,还需要大量的数值计算、计算机仿真和算法分析,使得研究非常的艰难。而混沌同步由于其在混沌加密和混沌通信等领域重要的应用价值,也是近年来学者们研究的焦点。目前,研究混沌同步还是以传统的控制理论为主,并没有能很好的利用混沌系统自身的特点。
本文重点研究了连续人工神经网络的动态复杂性问题,特别是利用动力系统中的拓扑马蹄理论与计算机辅助计算相结合,对多个著名的神经网络模型的混沌性进行了严格的判定,并对神经网络的拓扑结构与混沌行为的关联关系进行了细致的理论分析,还研究混沌神经网络的一类同步控制问题,取得了下列成果：
(1)利用动力系统中由Smale马蹄理论发展得到的拓扑马蹄理论,结合计算机的辅助计算,对连续人工神经网络中非常具有代表性的Hopfield神经网络模型、细胞神经网络模型和一类具有物理意义的人工神经网络模型的混沌性给出严格的判定。该方法具有很强的适用性、灵活性、可靠性和数学上的严密性,可以有效地对系统的混沌性进行验证。
(2)研究了低维连续神经网络的混沌性与系统拓扑结构的联系,以著名的Hopfield神经网络为例,根据其神经元连接的拓扑结构分类,讨论了混沌行为与系统拓扑类型之间的联系,我们发现拓扑结构中出现环路是神经网络系统有混沌性质的必要条件,以及连接数小于4的神经网络系统是不会出现混沌行为的,并给出了理论证明。
(3)以细胞神经网络为例,研究了神经网络系统的混沌完全同步问题。与以往的混沌同步问题不同,本文讨论的同步为两个不具有混沌性质的子系统经过同步后得到具有混沌性质的耦合系统,我们利用拓扑马蹄理论对这个新颖的同步系统的混沌性给出了验证,还运用全局伸展定理,证明了该细胞神经网络模型与Chua电路模型是拓扑共轭的。
(4)研究了在混沌的发展中起到重要作用的两个非神经网络的系统模型：平滑Chua系统和NSG系统。虽然这两类系统均包含了丰富的动力学行为,但是在此之前的研究都局限在数值仿真上,本文利用动力系统中拓扑马蹄理论严格的验证了这两个系统的混沌性。
As a new research field developed in these recent years, the chaotic neural network is one of the most rapid developed, the most popular and the most attractive subjects in the nonlinear science. The complex dynamics of the chaotic neural network is different from normal neural networks which have the character of gradient descent. The chaotic neural network has plenty of dynamical behavior and different kinds of attractors, so a great deal of attention has been paid to this area. The nonlinear dynamics of neural network mainly make use of the dynamical system theory to analyze the evolution of the system and the character of the attractor, explore the cooperative behaviors of the nerve cells, and study the mechanism of information processing.
As we know, chaos encryption and chaos communication is the most attractive field in the research about chaos. Meanwhile, nonlinear circuit is very helpful to simulate the real chaotic system for studying chaos encryption and chaos communication. We find that chaotic signal generators based on chaotic neural network have the great character of global stability, which overcome the shortcoming of the famous Chua's circuit. So it has very important academic and practical significance to find more chaotic neural network models which have different chaotic attractors.
Chaos verification and chaos synchronization has become the focus of the research of the chaotic systems. However, the theory of dynamical complexity is profound to understand, which makes these two issues become very difficult problems. Especially, chaos verification needs not only the theory of dynamical system but also a great deal of numerical calculation, computer simulation and algorithm analysis. As for chaos synchronization, people have paid a great deal of attention on it for its important application in chaotic encryption and chaotic communication. Nevertheless now the research about chaos synchronization still base on the traditional control theory.
This paper studies the dynamical complexity in continuous artificial neural network, by computer simulation, Lyapunov exponent calculation, computer-assisted verification and so on. Rich dynamical behaviors are found, such as chaos, limit cycle, equilibrium point. In this paper, chaos verifications of several chaotic systems are given by the combination of topological horseshoe theory and computer-assisted verification. Using Poincare section and Poincare map, this paper gives an effective verification for the existence the existence of attracting sets in chaotic system. The research work carried out includes the following:
(1) Prove chaotic character on the Hopfield neural network model, cellular neural network model and a simple artificial neural network model strictly. The method of computer-assisted verification makes use of topological horseshoe theory developed by the Smale horseshoe of dynamical system combined with computer assisted. In these cases, we have shown this method works well and make chaos verification easily and reliability.
(2) Take the famous Hopfield neural network for instance to study the dynamical complexity of the low-dimensional continuous neural network according to topology structure of the models. We are concerned with this interesting problem in dynamics of neural networks that what connection topology will prohibit chaotic behavior in continuous time neural network and to what extent a continuous time neural network described by continuous ordinary differential equations is simple enough while can still exhibit chaos. This paper shows that the existence of directed loop in connection topology is necessary for chaos to occur.
(3) Present that chaos synchronization can take place in the coupled non-chaotic neural network systems. This is demonstrated by coupling two non-chaotic cellular neural networks. The couplings give rise to a synchronous chaotic dynamics and in the meanwhile the synchronous dynamics is globally asymptotically stable, thus chaos synchronization takes place under the suitable couplings. And it is interesting to show that the cellular neural network is topologically conjugate to an unfolded Chua's circuit by virtue of "The Global unfolding Theorem".
(4) The two non-neural network system models which play the important roles in chaos development are reviewed:smooth Chua system and NSG system. We present the rigorous verification of chaos by giving the computer-assisted verification based on horseshoe theory of dynamical system, which has been numerically observed before.
引文
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