随机布尔网络吸引子的鲁棒性与相对稳定性

生物网络中时常会出现多个表型或多条动力学路径共存的现象,但其表型和动力学路径在随机涨落影响下的稳定性问题迄今尚未得到完整的认识.本文通过分析随机布尔网络模型来尝试回答这个问题.表型和动力学路径的稳定性对应于随机布尔网络吸引子的鲁棒性和相对稳定性,而后者可以用指数扰动马氏链理论来加以刻画和分析.已有的指数扰动马氏链理论已经告诉我们,从某个吸引子的吸引域溢出的时间的对数正比于该吸引域的"非平衡态活化能势垒",于是本文首先推广了这个理论,证明了指数扰动马氏链模型中吸引子之间的最佳转移路径都是等概率的,因此在吸引子间"非平衡态活化能势垒"相等的情形下,吸引子的相对稳定性是由最佳转移路径的个数来决定的.该理论还预示着,当模型中的随机涨落很小时,将会出现如下的相变现象:在参数空间的某一区域内,这些表型和动力学路径将以某一比例共存在;而在另一些区域内,某个表型或某条动力学路径将占据主导地位,从而成为全局吸引子.最后,在人造模型以及蛋白质p53动力学模型中应用和验证了该理论,并且通过计算吸引子稳定性对于具体动力学参数的敏感性,提供了一种辨别网络中重要节点和节点间重要相互作用的新方法.     

动力系统中的Lipschitz稳定性

本文讨论动力系统的Ｌｉｐｓｃｈｉｔｚ稳定性与吸引性之间的关系．给出动力系统中弱吸引性、吸引性和强吸引性这三个概念相互等价的条件，在一定的条件下证明了（弱、强）吸引子与全局（弱、强）吸引子是一致的．本文还讨论了度量Ｌｙａｐｕｎｏｖ稳定性和拓扑Ｌｙａｐｕｎｏｖ稳定性及它们与Ｌｉｐｓｃｈｉｔｚ稳定性之间的关系．     

带有热记忆的非均匀柔性结构的长时间动力行为

本文研究了带有热效应的非均匀柔性结构方程，并且该热效应符合Coleman-Gurtin定律.利用半群方法，建立了系统的整体适定性.主要结论是该系统的长时间动力行为.本文证明了系统的拟稳定性，整体吸引子的存在性以及整体吸引子具有有限的分形维数.此外，还证明了指数吸引子的存在性.     

输入采用标准模糊分划的模糊控制系统性质及稳定性分析

具体定义模糊控制系统输入变量的标准模糊分划，研究和证明输入采用标准模糊分划的模糊控制系统的有关性质。在此基础上，采用Lyapunov直接法研究该类模糊控制系统的稳定性，提出一个判定模糊控制系统稳定性的充分条件。该条件将以往方法要在所有的子系统中寻找一个公共的正定矩阵满足Lyapunov不等式，放宽为在各最大交叠规则组内分别寻找公共的正定矩阵，从而减小稳定性判定的保守性和难度。     

Sine—Gordon方程的全局吸引子的维数估计

本文得到了阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的狄氏问题的全局吸引子的Ｈａｕｓｄｏｒｆｆ维数以偶数上界的参数条件，特别地，当阻尼与Ｌａｐｌａｅ算子的第一个特征值适当大时，全局吸引子是零维的，零维吸引子恰是系统的唯一平衡解并且指数吸引相空间的有界集。     

一类带有周期输入的人工神经网络的渐近性质

本文利用常微分方程定性理论研究一类带有周期输入的人工神经网络，得到了该网络周期吸引子的存在性。     

模糊仿射集与模糊向量子空间的再定义

利用模糊直线定义模糊仿射集，研究其有关性质。在此基础上，定义模糊向量子空间，证明一个模糊集为模糊仿射集的充要条件是该模糊集是一个模糊子空间的平移。     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数，得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时，吸引子的维数减小．特别地，得到了零维吸引子存在的参数条件．     

非线性Sobolev-Galpern方程的有限维整体吸引子

本文研究非线性Sobolev-Galpern方程解的渐近性态．首先证明了该方程在H^2(Ω)∩H0^1(Ω)中整体弱吸引子的存在性，然后利用一个能量方程证明了整体弱吸引子实际上是整体强吸引子，建立了整体吸引子的有限维性．     

Hasegawa-Mima方程的整体吸引子

考虑了带有耗散项的Hasegawa-Mima方程解的长时间性态, 研究了具有初值周期边值条件的Hasegawa-Mima方程的整体吸引子问题．运用关于时间的一致先验估计,证明了该问题整体吸引子的存在性,并获得了整体吸引子的维数估计．     

Structure of attractors for skew product semiflows

In this work we study the continuity and structural stability of the uniform attractor associated with non-autonomous perturbations of differential equations. By a careful study of the different definitions of attractor in the non-autonomous framework, we introduce the notion of lifted-invariance on the uniform attractor, which becomes compatible with the dynamics in the global attractor of the associated skew product semiflow, and allows us to describe the internal dynamics and the characterization of the uniform attractors. The associated pullback attractors and their structural stability under perturbations will play a crucial role.     

Analysis of dissipative structures in the Gierer-Mainhardt model

To analyze a manifold that approximates an attractor of a dissipative system we propose a Gaussian variational principle. For a Gierer-Mainhardt system we obtain closed-form expressions for the approximating stationary solution and study its stability. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 48–52.     

Random attractor for the stochastic Cahn–Hilliard–Navier–Stokes system with small additive noise

The objective of this paper is to study the asymptotic behavior of solutions, in terms of the upper semi-continuous property of random attractor, of the Cahn–Hilliard–Navier–Stokes system with small additive noise. We prove the existence of a random attractor for the Cahn–Hilliard–Navier–Stokes system with small additive noise. Furthermore, we consider the stability of global attractor and prove the random attractor of the Cahn–Hilliard–Navier–Stokes system with small additive noise will convergent to the global attractor of the unperturbed Cahn–Hilliard–Navier–Stokes system when the parameter of the perturbation ε tends to zero.     

Dynamical bifurcation of the damped Kuramoto–Sivashinsky equation

In this paper, we study the primary instability of the damped Kuramoto–Sivashinsky equation under a periodic boundary condition. We prove that it bifurcates from the trivial solution to an attractor which determines the long time dynamics of the system. Using the attractor bifurcation theorem and the center manifold theory, we describe the bifurcated attractor in detail.     

Horseshoe in a model of artificial neural network

In this paper, we study a three-dimensional general model of artificial neural network (ANN). To confirm the chaotic behavior in this neural network demonstrated in numerical studies, we consider a cross-section properly chosen for the attractor obtained and study the dynamics of the corresponding Poincaré map, and rigorously verify the existence of horseshoe by computer-assisted verification arguments.     

Attractivity,multistability, and bifurcation in delayed Hopfieldʼs model with non-monotonic feedback

For a system of delayed neural networks of Hopfield type, we deal with the study of global attractivity, multistability, and bifurcations. In general, we do not assume monotonicity conditions in the activation functions. For some architectures of the network and for some families of activation functions, we get optimal results on global attractivity. Our approach relies on a link between a system of functional differential equations and a finite-dimensional discrete dynamical system. For it, we introduce the notion of strong attractor for a discrete dynamical system, which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one. Our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations. Our abstract setting is not limited to applications in systems of neural networks; we illustrate its use in an equation with distributed delay motivated by biological models. We also obtain some results for neural systems with variable coefficients.     

Relation between Different Types of Global Attractors of Set-Valued Nonautonomous Dynamical Systems

The article is devoted to the study of the relation between forward and pullback attractors of set-valued nonautonomous dynamical systems (cocycles). Here it is proved that every compact global forward attractor is also a pullback attractor of the set-valued nonautonomous dynamical system. The inverse statement, generally speaking, is not true, but we prove that every global pullback attractor of an α-condensing set-valued cocycle is always a local forward attractor. The obtained general results are applied while studying periodic and homogeneous systems. We give also a new criterion of the absolute asymptotic stability of nonstationary discrete linear inclusions. Dedicated to our friend Professor Enrico Primo Tomasini on the occasion of his 55th birthdayMathematics Subject Classifications (2000) Primary: 34C35, 34D20, 34D40, 34D45, 58F10,58F12, 58F39; secondary: 35B35, 35B40.     

Research on the Evolution of Thinking Dynamics Attractor

ＲｅｓｅａｒｃｈｏｎｔｈｅＥｖｏｌｕｔｉｏｎｏｆＴｈｉｎｋｉｎｇＤｙｎａｍｉｃｓＡｔｔｒａｃｔｏｒＹａｎｇＱｉａｎ（杨谦）；ＺｈａｎｇＹｏｎｇｊｕｎ（张拥军）（ＳｏｕｔｈｗｅｓｔＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ，６１００３１）Ｂａ...