【作者】 刘宇；

【导师】 刘铭；

【作者基本信息】 东北林业大学 ， 数学， 2022， 硕士

【摘要】 近年来,随机微分方程理论得到了深入的研究,其被广泛的应用于工程控制、生物数学、金融学等方面。一方面,神经网络与复杂网络被广泛地应用在保密通信、信号处理等多个领域,因此利用微分方程理论构建网络模型,研究其动力学性质就显得尤为重要。另一方面,利用微分方程理论建立生物种群模型,研究种群内生物的数量变化、持续存在和灭绝等问题可以使得我们更加了解生物种群的发展规律。因此带有时滞和随机扰动的网络模型和生物捕食系统的动力学性质是值得关注和研究的。本文研究的主要内容为以下三个方面:(1)在模糊细胞神经网络的框架下,引入了离散时滞、分布时滞以及随机扰动,结合Lyapunov方法与随机微分方程的相关理论,通过设计合适的反馈控制与自适应控制,实现了系统的固定时间同步,推导了系统实现固定时间同步的条件,计算出了稳定时间,最后,利用Matlab分别在两类控制下进行两次数值模拟,说明了理论结果的有效性。(2)在复杂网络的框架下,引入了Markov跳、随机扰动、离散时滞与分布时滞,第一次通过设计自适应非颤振控制研究了系统的固定时间同步,推导了系统实现固定时间同步的条件,计算出了稳定时间。除此之外,其得到的稳定时间只与控制参数有关,与系统的精确初始条件无关。最后,通过Matlab进行数值模拟,绘制了网络模型的轨迹,说明了所得的理论结果是有效的。(3)在修正的Leslie-Gower捕食者-食饵模型的框架下,引入了时滞、MichaelisMenten型收获、比率依赖型功能反应函数和随机扰动,使构建的模型与实际系统更加一致。首先证明该系统具有全局唯一的正解。其次,获得了系统的时间平均意义下持续存在和灭绝性的条件。此外,验证了系统在一定条件下是随机持久的。除此之外,当系统的参数满足相应条件时,证明了系统具有平稳分布,且是遍历的。最后,为了验证本节所得理论结果的有效性,利用Matlab进行了数值模拟。更多还原

【Abstract】 In recent years,the theory of stochastic differential equations has been intensively studied,and it is widely used in engineering control,biomathematics,finance,etc.On the one hand,neural networks and complex networks are widely used in many fields such as confidential communication and signal processing,so it is especially important to use differential equation theory to construct network models and study their dynamical properties.On the other hand,the use of differential equation theory to model biological populations and to study the change in number,persistence and extinction of organisms within a population can lead to a better understanding of the developmental patterns of biological populations.Therefore network models with time delay and stochastic perturbation and the dynamical properties of biological predation systems are worthy of attention and study.Based on the above discussion,the main work of this paper is as follows.(1)Under the framework of fuzzy cellular neural network,discrete time delay,distributed time delay and stochastic perturbation are introduced,combined with the Lyapunov method and the related theory of stochastic differential equation,the fixed-time synchronization of the system is achieved by designing suitable feedback control and adaptive control,the conditions for the system to achieve fixed time synchronization are derived and the stability time is calculated,and finally,using Matlab two numerical simulations were conducted under two types of control respectively to illustrate the validity of the theoretical results.(2)In the framework of complex networks,Markov jump,stochastic perturbation,discrete time delay and distributed time delay are introduced,and the fixed-time synchronization of the system is studied for the first time by designing adaptive non-chattering control,and the conditions for the system to achieve the fixed time synchronization are derived and the stabilization time is calculated.Besides,the stabilization time obtained is only related to the control parameters and not to the exact initial conditions of the system.Finally,the trajectory of the network model is plotted by numerical simulation in Matlab,which illustrates the validity of the obtained theoretical results.(3)In the framework of the modified Leslie-Gower predator-prey model,time delay,Michaelis-Menten type harvesting,ratio-dependent functional response function and stochastic perturbation are introduced to make the constructed model more consistent with the actual system.Firstly,it is shown that the system has a globally unique positive solution.Secondly,the conditions of persistence and extinction in mean of the system are obtained.In addition,it is verified that the system is stochastically persistent under certain conditions.In addition to this,it is proved that the system has a stationary distribution and is ergodic when the parameters of the system satisfy the corresponding conditions.Finally,in order to verify the validity of the theoretical results obtained in this section,numerical simulations were carried out using Matlab.更多还原