【作者】 李萍；

【导师】 李骥；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2023， 博士

【摘要】 近年来,快慢系统被广泛地应用于金融、生态学、化学、医学等领域。随着有关确定性快慢系统的研究成果越来越丰富,随机快慢系统也受到了广泛的关注。现实世界中随机扰动是客观存在的,所以有关随机快慢系统的研究十分具有现实意义。本文主要研究了三个具有分岔点的随机快慢系统模型,第一个是具有折点和跨临界分岔点的Holling-II型捕食-食饵模型,第二个是典型的具有带分岔延迟的跨临界分岔点的随机快慢系统模型,第三个是典型的具有折点/同宿轨分岔的神经元模型。对于前两个模型,本文利用样本路径法,通过对整个过程的路径进行精确控制,估计扰动后解轨道保持在某个区域里的概率或者从某个区域逃逸的概率来探讨不同强度的加性噪声对分岔点附近的动力学行为的影响。对于第三个模型,本文从随机动力系统的角度出发,探讨了一致有界实噪声对系统的影响。本文分为六章。前两章是简介和预备知识。第三章,主要研究一个具有Holling-II型功能反应函数的捕食-食饵模型,该模型由具有一个折点和一个跨临界分岔点的二维快慢微分方程表示。在两个分岔点的邻域里,利用样本路径法研究随着慢变量变化的小加性噪声对快变量的动力学行为的影响。本文通过对单个路径的行为给出精确的概率估计,证明了充分小但非指数小的加性噪声会破坏跨临界分岔点附近的分岔延迟现象。同时,证明了在高概率的意义下,充分小的加性噪声不会破坏折点附近的临界跃迁现象,但是会改变轨道发生临界跃迁的位置。第四章,主要在第三章的研究基础上,继续探讨适当大强度的加性噪声对带分岔延迟的跨临界分岔点附近的动力学行为的影响。估计了在不同强度加性噪声扰动下随机解会以高概率发生临界跃迁的位置,总结了不同强度的加性噪声对带分岔延迟的跨临界分岔点附近的动力学行为影响的结果。除此之外,在分析过程中,利用样本路径法证明了,对于非自治快慢系统,小的加性噪声对快变量发生较大偏移的轨道所造成的影响是可以忽略不计的。第五章,利用随机动力系统不变流形理论,探讨一致有界实噪声对于因折点/同宿轨分岔而产生的放电和振荡行为的影响,解释一种由一致有界随机外力产生混沌振荡的机制,并给出数值模拟用于佐证。第六章,总结本文的主要工作和结果,并指出接下来可以继续研究的问题。更多还原

【Abstract】 Fast-slow systems have found broad applications in various domains such as finance,ecology,chemistry,and medicine in recent years.Stochastic fast-slow systems have garnered significant attention due to the growing abundance of research findings on deterministic fastslow systems.Stochastic perturbation exists objectively in the real world,so the research on stochastic fast-slow systems is of great practical significance.In this paper,three models of stochastic fast-slow systems with bifurcation points are studied.One is a predator-prey model with a fold point and a transcritical bifurcation point,one is a typical model of stochastic fast-slow systems with a transcritical bifurcation point accompanied by bifurcation delay,the other is a typical neuronal model with fold/homoclinic bifurcations.For the first two models,by using the sample paths approach,we estimate the probability that the paths of the perturbed solutions remain in some region or that the paths of the perturbed solutions escape from some region by precisely controlling the whole paths of the process,in order to investigate the effect of additive noise with different intensities on the dynamic behavior near the bifurcation points.For the third model,we explored the effect of uniformly bounded real noise on the system from the perspective of random dynamical systems.The paper is structured into six chapters.The first two chapters are introduction and preliminary knowledge.In Chapter 3,we study a predator-prey model with Holling-II type functional response function,which is represented by two-dimensional fast-slow differential equations with a fold point and a transcritical bifurcation point.In the neighborhoods of two bifurcation points,we investigate the effect of small additive noise varying with slow variable on the dynamics of fast variables by the sample-paths approach.In this paper,by giving accurate probability estimates for the behavior of individual paths of the stochastic solutions,we show that the sufficiently small but non-exponentially small additive noise destroys the bifurcation delay phenomenon near the transcritical bifurcation point.Meanwhile,we show that,with high probability,the critical transition phenomenon near the fold point remains unaffected as long as the additive noise is sufficiently small,while the locations where the critical transitions of the paths occur are altered.In Chapter 4,on the basis of the study in Chapter 3,we continue to investigate the effect of additive noise with reasonable large intensity on the dynamical behavior near the transcritical bifurcation point with bifurcation delay.We also provide estimations of the locations where the stochastic solutions perturbed by additive noise with different intensities experience critical transitions with high probability,and summarize the results of the effect of additive noise with different intensities on the dynamics near the transcritical bifurcation point with bifurcation delay.In addition,during the analysis,we demonstrate that for nonautonomous fast-slow systems,the effect of small additive noise on trajectories with large shifts in the fast variables is negligible by the sample-paths approach.In Chapter 5,we investigate the effect of uniformly bounded real noise on the discharge and oscillation behavior due to fold/homoclinic bifurcations,using the invariant manifold theory of random dynamical systems to explain a mechanism of chaotic oscillation generated by uniformly bounded random external forces and present numerical simulations to support it.In Chapter 6,we summarize the main work and results of this paper,and point out the problems that can be further studied.更多还原