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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635339
• 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.

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。