分数阶时滞复值神经网络动力学行为研究

分数阶微积分最主要的优点是能够描述系统的记忆性和遗传性。研究发现，分数阶神经网络具有比整数阶神经网络更加有效的计算能力和信息处理能力。近年来，分数阶复值神经网络动力学行为的研究获得了许多成果，但这些成果几乎都是在激活函数分解为实部函数和虚部函数情形下建立的，且与分数阶次无关，因此没有更好地揭示网络的本质。本项目拟在激活函数不分解为实部函数和虚部函数情形下，建立分数阶时滞复值神经网络动力学行为的判定准则。针对不同种类的时滞，建立相应的分数阶时滞复值神经网络模型。通过构造网络模型的比较系统，建立用以分析网络动力学行为的比较原理。以分数阶微积分、Laplace变换、复分析、矩阵分析、泛函分析等数学理论为工具，通过构造酉空间上的Lyapunov泛函或者Lyapunov-Krasovskii泛函，建立分数阶时滞复值神经网络稳定性、同步性、耗散性和无源性等动力学行为且与分数阶次相关的判据。

The main advantage of the fractional-order calculus lies in that it can describe the memory and heredity of the system. It is found that the fractional-order neural network is more efficient than the integral-order neural network on the ability of calculation and information processing. In recent years, some results on dynamic behaviors have been obtained for fractional-order complex-valued neural networks. However, it is worth pointing out that almost all the results are derived under the situation that the activation functions can be expressed by separating their real and imaginary parts, and the provided criteria of dynamic behaviors are independent on the order of fractional derivative. Hence it does not reveal the essence of the network. In this project, the problems of the dynamic behaviors analysis for fractional-order complex-valued neural networks with time delays will be investigated and some criteria to guarantee the dynamical behaviors of the networks will also be established under the situation that the activation functions do not need to be decomposed into their real and imaginary parts. According to different types of time delays, the corresponding fractional-order complex-valued neural networks with time delays are established. By constructing the comparison system of network models, the comparison principle for analyzing the dynamic behaviors of neural networks will be established. By employing some mathematical theories, such as fractional-order calculus, Laplace transform, complex analysis, matrix analysis, functional analysis, and constructing Lyapunov functional or Lyapunov-Krasovskii functional in unitary space, some criteria of dynamic behaviors such as stability, synchronization, dissipation and passivity for the fractional-order complex-valued neural networks are to be established, and these criteria are dependent on the order of fractional derivative.