一类三维神经元模型的分支研究

考虑一类三维神经元模型的分支问题.利用常微分方程的定性与分支理论的知识,讨论了模型的平衡点个数及其稳定性,主要分析了平衡点的Hopf分支和Bogdanov-Takens分支,并得到了相应的鞍结点分支曲线,Hopf分支曲线与同宿分支曲线.     

高维系统极限环的分支

本文利用分支函数方法,讨论高维系统极限环的局部分支及一类全局分支,推广了 Hopf 分支定理.     

多时滞捕食-食饵系统正平衡点的稳定性及全局Hopf分支

本文首先用Cooke等人建立的关于超越函数的零点分布定理,研究了一类多时滞捕食-食饵系统正平衡点的稳定性及局部Hopf分支,在此基础上再结合吴建宏等人用等变拓扑度理论建立起的一般泛函微分方程的全局Hopf分支定理,进一步研究了该系统的全局Hopf分支.     

多时滞捕食-食饵系统正平衡点的稳定性及全局Hopf分支

本文首先用Cooke等人建立的关于超越函数的零点分布定理,研究了一类多时滞捕食-食饵系统正平衡点的稳定性及局部Hopf分支,在此基础上再结合吴建宏等人用等变拓扑度理论建立起的一般泛函微分方程的全局Hopf分支定理,进一步研究了该系统的全局Hopf分支.     

具有时滞的基因表达模型的稳定性与分支分析

应用频域法研究了一类具有三个时滞的基因表达模型的Hopf分支问题.基于Nyquist稳定性准则和Hopf分支定理,选取三个时滞的和τ作为分支参数,发现当τ超过某个临界值时,系统产生了Hopf分支.最后,对系统进行了数值仿真,数值仿真的结果验证了理论分析的正确性.     

一类具时滞和阶段结构的捕食模型的稳定性与Hopf分支

研究一类具有时滞和阶段结构的捕食模型的稳定性和Hopf分支的存在性问题.通过分析特征方程,得到了正平衡点局部稳定的条件.同时,应用中心流形定理和规范型理论,得到了确定Hopf分支方向和分支周期解的稳定性的计算公式.最后对所得理论结果进行了数值模拟.     

一类微生物絮凝模型平衡点的稳定性与Hopf分支

考虑了一类具有营养循环的微生物絮凝常微分方程动力学模型.该类动力学模型平衡点可呈现前向与后向分支.利用常微分方程相关理论,研究了其边界平衡点的全局稳定性,以及正平衡点的局部稳定性析与Hopf分支存在性.同时,计算机数值模拟结果显示了与理论结果的一致性,并进一步揭示了该类动力系统可能呈现的复杂的振荡性质.     

具有时滞的比率型三种群捕食模型的分支分析

本文研究了一类具有时滞的比率型三种群捕食模型.通过分析该模型的特征方程,证明了该模型在正平衡点的稳定性.通过选择时滞τ为分支参数,得到了当时滞τ通过一系列的临界值时,Hopf分支产生.应用中心流形和规范型理论,得到了关于确定Hopf分支特性的计算公式.最后进行数值模拟验证了我们所得结果的正确性.所得结果是对前人工作的补充.     

具阶段结构的多时滞SIR模型的稳定性分析

考虑了一类具有两个阶段结构的SIR模型,得到了解的正性和有界性,通过分析特征方程根的分布,以T1和T2为参数分析了平衡点的稳定性和局部 Hopf 分支存在性.进一步地,利用规范型和中心流型理论,给出了决定Hopf分支方向和分支周期解的稳定性的隐式算法.最后利用一些数值模拟来支持所得到的理论分析结果.     

双时滞HIV病毒的CD4$^+$T-细胞模型的稳定性和分支分析

本文研究双时滞HIV病毒的CD4~+ T-细胞模型.利用特征方程根的分别理论,分别研究具有单时滞和双时滞模型的线性稳定性.发现当模型中的时滞经过一系列临界值时,模型在平衡点经历了Hopf分支和Hopf-zero分支,并给出Hopf分支和Hopf-zero分支存在的充分条件.最后数值模拟验证了理论结果.     

一类变时滞神经网络的全局指数稳定性

本文研究一类变时滞神经网络平衡点的全局指数稳定性.在不要求激活函数全局Lipschitz条件下,利用Lyapunov函数方法,并结合Young不等式和Halanay时滞微分不等式,得到了系统全局指数稳定的充分条件.文末,一个数值例子用以说明本文结果的有效性.     

距离空间中插值神经网络的误差估计

研究距离空间中的神经网络插值与逼近问题.首先引进一类广义的激活函数,用比较简洁的方法讨论距离空间中插值神经网络的存在性,然后给出插值神经网络逼近连续函数的误差估计.     

Computing the Approximation Error for Neural Networks with Weights Varying on Fixed Directions

We obtain a sharp lower bound estimate for the approximation error of a continuous function by single hidden layer neural networks with a continuous activation function and weights varying on two fixed directions. We show that for a certain class of activation functions this lower bound estimate turns into equality. The obtained result provides us with a method for direct computation of the approximation error. As an application, we give a formula, which can be used to compute instantly the approximation error for a class of functions having second order partial derivatives.     

Global output convergence of recurrent neural networks with distributed delays

This paper discusses the global output convergence of a class of recurrent neural networks with distributed delays. The inputs of the neural networks are required to be time varying and the activation functions to be globally continuous and monotone nondecreasing. By using the definiteness of matrix and the properties of M-matrix, several sufficient conditions are established to guarantee the global output convergence of this class of neural networks. Symmetry in the connection weight matrices and the boundedness of the activation functions are not required in this paper. The convergence results are useful in solving some optimization problems and in the design of recurrent neural networks with distributed delays.     

Positive invariant sets and global exponential attractive sets of a class of neural networks with unbounded time-delays

In this paper, we study the positive invariant sets and global exponential attractive sets for a class of neural networks with unbounded time-delays. Based on the assumption for the activation function satisfying the global Lipschitz condition, several algebraic criterions for the aforementioned sets are obtained by constructing proper Lyapunov functions and employing Young inequality. Finally, examples are given and analyzed to demonstrate our results.     

Stability analysis for periodic solution of neural networks with discontinuous neuron activations

In this paper, we present a general class of neural networks with discontinuous neuron activations and varying coefficients, where the neuron activation function is a discontinuous monotone increasing and bounded function. By using the fixed point theorem in differential inclusion theory and constructing suitable Lyapunov functions, a condition is derived which ensures the existence and global exponential stability of a unique periodic solution for the neural network. Furthermore, under certain conditions global convergence in finite time of the state is investigated. The obtained results show that Forti’s conjecture for neural networks without delays is true. Finally, two numerical examples are given to demonstrate the effectiveness of the results obtained in this paper.     

Exponential convergence for high-order recurrent neural networks with a class of general activation functions

In this paper, we consider high-order recurrent neural networks with a class of general activation functions. By using some mathematical analysis techniques, we establish new results to ensure that all solutions of the networks converge exponentially to zero point.     

Global exponential convergence for delayed cellular neural networks with a class of general activation functions

In this paper, we consider delayed cellular neural networks with a class of general activation functions. By using some mathematical analysis techniques, we establish new results to ensure that all solutions of the networks converge exponentially to zero point.     

Existence and global exponential stability of anti‐periodic solutions for delayed quaternion‐valued cellular neural networks with impulsive effects

In this paper, we consider a class of delayed quaternion‐valued cellular neural networks (DQVCNNs) with impulsive effects. By using a novel continuation theorem of coincidence degree theory, the existence of anti‐periodic solutions for DQVCNNs is obtained with or without assuming that the activation functions are bounded. Furthermore, by constructing a suitable Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of anti‐periodic solutions for DQVCNNs. Our results are new and complementary to the known results even when DQVCNNs degenerate into real‐valued or complex‐valued neural networks. Finally, an example is given to illustrate the effectiveness of the obtained results.     

Vector Wirtinger-type inequality and the stability analysis of delayed neural network

This paper proposes some new stability criteria for a class of delayed neural networks with sector and slope restricted nonlinear neuron activation function. By using the convex express of the nonlinear neuron activation function, the original delayed neural network is transformed into a linear uncertain system. The proposed method employs an improved vector Wirtinger-type inequality for constructing a novel Lyapunov functional. Based on the Lyapunov stable theory, new delay-dependent and delay-independent stability criteria for the researched system are established in terms of linear matrix inequality technique, delay partitioning approach and characteristic root method. Three illustrative examples are presented to verify the effectiveness of the main results.