右端不连续时滞神经网络的多稳定性与分岔控制

In recent years, some valuable results have been obtained about multistability of neural networks. However, it should be noted that in most existing multistability results, there is a basic assumption: the neuron activation functions are continuous. In fact, neural networks with discontinuous activation functions are of importance and do frequently arise in practice. Based on the theory of differential inclusion and set-valued map, we study the issues of multistability analysis and bifurcation control for delayed neural networks with discontinuous activation functions. Firstly, a general class of delayed neural networks model is established, according to the peculiar features of discontinuous dynamical system. Secondly, the exact number of equilibrium points(periodic solutions) is given and the positively invariant sets and basins of attraction for these stationary equilibrium points(periodic solutions) are estimated. Thirdly, the dynamical behavior of equilibrium points(periodic solutions) in saturation regions is completely analyzed, some testable sufficient conditions are derived which ensure local convergence of equilibrium points(periodic solutions). Moreover, how the discontinuities and time delays lead to local convergence in finite time is revealed and the finite convergence time is accurately estimated. Finally, by choosing the time delay as a bifurcation parameter, the Hopf bifurcation behavior in unsaturation regions is discussed and efficient control strategy is presented in order to avoid the occurrence of harmful bifurcation. The study of these issues mentioned above not only promote the development of the theory of multistability and differential equations with discontinuous right-hand side, but also provide strong theoretical support for some practical applications of discontinuous dynamical system.

近年来，关于神经网络的多稳定性研究涌现出了许多有价值的成果，但这些成果基本上都是建立在网络右端的激活函数是连续的假设基础上。事实上，具有右端不连续激活函数的神经网络，称之为右端不连续神经网络，是非常重要的而且在实际中有着广泛的应用。本项目拟运用微分包含与集值映射理论研究右端不连续时滞神经网络的多稳定性和分岔控制问题。内容包括：基于不连续的动力学特征，建立更贴近实际的右端不连续时滞神经网络模型；确定新建模型微分包含平衡态的精确数目、正向不变集和吸引域；深入分析系统在饱和区域内的复杂动力学行为，建立可验证的局部稳定性准则，揭示有限时间收敛产生的机理，精确估计系统达到局部稳定的有限时间区间；探讨系统在不饱和区域内的分岔行为，并提出相应的控制策略。这些问题的解决不仅可以促进右端不连续微分方程和神经网络多稳定性理论的进一步发展与完善，而且为不连续系统在实际工程中的应用提供有力的理论保障。

神经网络的多稳定性主要研究网络多个平衡点的存在性及其复杂的动力学行为，包括平衡点的数目、位置、吸引域、局部稳定性和不稳定性等。神经网络的多稳定性可应用于诸多领域，如联想记忆、组合优化、模式识别、图像处理等。在目前文献中，绝大部分关于神经网络多稳定性的研究工作，都是建立在网络右端的激活函数是连续的假设基础上。事实上，具有右端不连续激活函数的神经网络，称之为右端不连续神经网络，是非常重要的而且在实际中有着广泛的应用。本项目运用微分包含与集值映射理论研究了右端不连续神经网络的多稳定性问题。内容包括：基于不连续的动力学特征，建立了几类贴近实际的右端不连续神经网络模型；确定了新建模型微分包含平衡点的数目、正向不变集和吸引域；深入分析了系统的局部稳定性和不稳定性，建立了可验证的局部稳定性准则，揭示了激活函数的不连续性对网络存储能力的影响。这些问题的解决促进了右端不连续微分方程和神经网络多稳定性理论的进一步发展与完善，同时为不连续系统在实际工程中的应用提供了有力的理论保障。在国家自然科学基金(项目批准号：61203300)的资助下，项目组成员已在国际重要学术刊物发表相关论文14篇，其中包括SCI论文8篇，EI论文12篇。