【作者】 杨芳；

【导师】 段金桥； 孙旭；

【作者基本信息】 华中科技大学 ， 统计学， 2021， 博士

【副题名】分析方法与计算

【摘要】 近年来随机动力系统引起了人们广泛的关注.气候系统和地球物理系统在动态演化的过程中往往会受到随机扰动的影响.即使随机扰动看起来非常小或者很弱,但是它们对系统演化的长期影响可能是显著的,甚至是深远的,这些在随机分岔、随机共振和由噪声诱导的迁移现象中都观测到了.高斯扰动具有连续的轨道,因而被广泛地用于描述系统在建立数学模型中的不确定性涨落.然而,许多复杂现象的涨落是非高斯的,具有间歇性跳跃、反常扩散和重尾分布等特性.L(?)vy过程被认为是描述非高斯噪声的一个合适的模型.本博士论文研究了气候系统中的一些随机迁移现象并提供了一种分析非平衡系统极大似然动力学的新方法.具体地说,第一,我们给出了一个由非高斯-平稳L(?)vy过程驱动的随机北极海冰简化模型,并且研究了从常年有冰覆盖状态(季节性无冰覆盖状态)迁移到季节性无冰覆盖状态(常年无冰覆盖状态)的最大可能转移轨道.第二,由于观测资料的缺乏限制了我们对气候系统的深入研究.因此,在给定初始时刻状态和最终时刻状态的情况下,我们提出了新的方法来研究随机动力系统的极大似然动力学行为.第三,北极海冰的消融能够引起海表淡水通量的改变,从而可能触发对其极为敏感的温盐环流在不同状态之间发生转移.我们利用极大似然轨道刻画了一个随机温盐环流简化模型在不同状态之间随时间的演化.本博士论文主要的研究内容和创新性的研究成果如下:(1)我们研究了一个由非高斯-平稳L(?)vy过程驱动的随机北极海冰简化模型.通过数值计算概率密度函数所满足的非局部Fokker-Planck方程,我们可以得到随机北极海冰模型在某些噪声强度的扰动下,从一个稳态迁移到另一个稳态的最大可能转移轨道.在这个简化系统中,我们用早期预警临界时间预测了北极可能出现无冰覆盖的时间,用灾难发生临界时间预测了北极可能出现全年无冰覆盖的时间.(2)我们提出了一种在桥条件下研究随机微分方程动力学行为的新方法,即:用极大似然轨道来描述非平衡系统的状态随时间的演化.一方面,我们给出了极大似然轨道的解析表示.另一方面,通过数值求解向前Fokker-Planck方程和向后Fokker-Planck方程,我们也给出了极大似然轨道的数值计算方法,并举例说明了该方法的有效性和适用性.(3)我们考虑了在给定初始时刻状态和最终时刻状态情况下,由高斯噪声激励的一个随机温盐环流简化模型.我们利用(2)中提出的新方法,给出了随机温盐环流简化模型在两个稳定状态之间随时间演化的极大似然轨道,同时也确定了极大似然状态发生突变的时间.这为我们研究随机温盐环流系统状态发生突变的机制,以及系统未来的演化趋势提供了一定的理论支撑.更多还原

【Abstract】 Stochastic dynamical systems have attracted a lot of attention recently.The dynamical evolution of climate and geophysical systems is often subject to random fluctuations.Even if random perturbation may appear to be very small or very weak,their long-term impact on the system evolution may be significant or even profound.This has been observed in stochastic bifurcation,stochastic resonance,and noise-induced transitions.Gaussian white noise,which has continuous trajectories,is widely used in describing fluctuations in mathematical modeling of complex systems under uncertainty.However,many fluctuations in complex phenomena are non-Gaussian.An important class of non-Gaussian processes,namely -stable Lévy processes,describe fluctuations with features such as heavy tails.L(?)vy processes describe the fluctuation with heavy tail distributions and band jumps,which are considered as suitable models to simulate non-Gaussian white noise.This thesis is about some stochastic transition phenomena in a reduced climate system and provides a new analysis method.Specifically,we firstly present the maximum likely trajectory for stochastic Arctic sea ice model driven by a non-Gaussian -stable L(?)vy process to transit from a perennially ice-covered state(seasonally ice-free state)to a seasonally ice-free state(perennially ice-free state).Secondly,as the lack of observational data limits our in-depth study of the climate system,we propose a new analytic method to study the maximum likelihood dynamics of stochastic dynamical systems with given initial and final time states.Thirdly,the melting of Arctic sea ice may cause the change of sea surface freshwater flux.We use the aforementioned maximum likelihood trajectory to describe the evolution orbits of a reduced stochastic thermohaline circulation model between different stable states.The main contents and innovation of this thesis are as follows:(1)We study a specific climate model,namely,the Arctic sea ice model driven by the non-Gaussian -stable L(?)vy process.By numerically solving the nonlocal Fokker-Planck equation satisfied by the probability density function,we obtain the maximum likely trajectory of the stochastic Arctic sea ice model from one stable state to another under some noise intensity.We use the early warning tipping time to predict the approximate time when the system changes from a perennially ice-covered state to seasonally ice-free one,and the disaster-happening tipping time to predict the approximate time from seasonally ice-free state to perennially ice-free state.(2)We propose a new method to study the dynamic behavior of stochastic differential equations under some bridge conditions,i.e.,the maximum likelihood trajectory is used to describe the evolution of non-equilibrium systems during transitions.At the same time,we present the analytical expression and a numerical method for the maximum likelihood trajectory,and illustrate the effectiveness and applicability of the method is illustrated.(3)We consider another climate model: A reduced stochastic thermohaline circulation model driven by Brownian motion under the condition that both the initial and final time states are given.Using the method proposed in(2),we explain the maximum likelihood trajectory of the stochastic thermohaline circulation model with time evolution between the two stable states,and determine the time when the maximum likelihood state changes abruptly.This provides a theoretical support to study the mechanism of abrupt change in the state of the thermohaline circulation system and the future evolution of the system.更多还原