Dynamics of maps with memory, random maps, multi-valued maps and the geometric Markov Renewal processes

具有记忆的映射动力学、随机映射、多值映射和几何马尔可夫更新过程

• 批准号: RGPIN-2017-05321
• 负责人: Islam, MdShafiqul
• 金额: $ 1.02万
• 依托单位: University of Prince Edward Island
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711611
• 关键词: Dynamics; maps; memory; random; multi

Dynamical systems are mathematical models for many problems in science, engineering, economics, finance and other areas. Complicated chaotic behaviors occur for many of these dynamical system models. An absolutely continuous invariant measure (acim) is a powerful tool for the study of chaotic behavior of discrete dynamical system models. An acim measures asymptotic relative frequencies of points of chaotic orbits generated by a discrete dynamical system with any initial point. An orbit of a dynamical system can be very complicated in deterministic sense, however it may not be chaotic in probabilistic or statistical sense. An acim is a very useful mathematical tool for the study of long term behavior and their chaotic nature. How do we know that such an acim exists? If it exists, how can we find acims analytically and numerically? What are properties of these acims. These are interesting, important and challenging questions in Ergodic Theory and Dynamical Systems. My long term objective is to contribute largely for the development of theoretical and computational methods via acims and other dynamics. In the next 5 years, I plan to study a number of chaotic discrete dynamical systems in one and higher dimensions. Firstly, I will study the existence of infinite acims for a family of random maps (closed systems). Moreover, we will study absolutely continuous conditional invariant measures and escape rates of the corresponding open dynamical systems with holes. A random map is a discrete time dynamical system, where one of a number of maps on the state space is selected randomly according to fixed probabilities or position dependent probabilities and applied in each iteration of the process. Random maps have applications in many areas of science and engineering such as in the study of fractals, in modelling interference effects in quantum mechanics, in computing metric entropy, and in forecasting the financial markets. Secondly, I will study dynamics of multi-valued maps. Multi-valued maps play an important role in many area of Science and Engineering such as in chaos synchronization, economics, rigorous effects in quantum mechanics, numerics and differential inclusions. We are interested in existence, approximations and properties of absolutely continuous invariant measures. Thirdly, I will study dynamics of maps with memory which are two dimensional chaotic dynamical systems generated by one dimensional map via a process which uses current and past information of the one dimensional map. There are many practical situations (such as stock market) where these type of two dimensional dynamical systems are useful mathematical models for analyzing various quantities. We will study SRB measures, acims and other dynamics of maps with memory. Finally, I will study the stability and control of the Geometric Markov Renewal Processes (GMRP). A GMRP is a process for the study of option prices in finance.

动力系统是科学、工程、经济、金融和其他领域许多问题的数学模型。许多动力系统模型都会出现复杂的混沌行为。绝对连续不变测度（acim）是研究离散动力系统模型混沌行为的有力工具。 acim 测量由具有任意初始点的离散动力系统生成的混沌轨道点的渐近相对频率。动力系统的轨道在确定性意义上可能非常复杂，但在概率或统计意义上可能不是混沌的。 acim 是研究长期行为及其混沌本质的非常有用的数学工具。我们怎么知道这样的acim存在呢？如果存在，我们如何通过分析和数值方法找到acim？这些acims有什么特性？这些是遍历理论和动力系统中有趣、重要且具有挑战性的问题。我的长期目标是通过 acims 和其他动力学为理论和计算方法的发展做出巨大贡献。在接下来的5年里，我计划研究一些一维及更高维度的混沌离散动力系统。首先，我将研究一系列随机映射（封闭系统）的无限 acim 的存在性。此外，我们还将研究相应的带孔开放动力系统的绝对连续条件不变测度和逃逸率。随机映射是一种离散时间动态系统，其中状态空间上的多个映射之一是根据固定概率或位置相关概率随机选择的，并应用于过程的每次迭代。随机地图在科学和工程的许多领域都有应用，例如分形研究、量子力学中的干扰效应建模、计算度量熵以及预测金融市场。其次，我将研究多值映射的动力学。多值映射在科学和工程的许多领域发挥着重要作用，例如混沌同步、经济学、量子力学中的严格效应、数值和微分包含。我们对绝对连续不变测度的存在性、近似值和性质感兴趣。第三，我将研究具有记忆的地图的动力学，这是由一维地图通过使用一维地图的当前和过去信息的过程生成的二维混沌动力系统。在许多实际情况（例如股票市场）中，这些类型的二维动力系统是分析各种数量的有用数学模型。我们将研究带有记忆的 SRB 测量、acim 和其他地图动态。最后，我将研究几何马尔可夫更新过程（GMRP）的稳定性和控制。 GMRP 是研究金融期权价格的过程。