Bifurcation in Dynamical Systems with Multiple Time Scales

多时间尺度动力系统的分岔

• 批准号: 0101208
• 负责人: John Guckenheimer
• 金额: $ 37.73万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0101208&HistoricalAwards=false
• 关键词: Bifurcation; Dynamical; Systems; Multiple; Time

Guckenheimer0101208 The investigator studies slow-fast decompositions andbifurcations of trajectories in dynamical systems with multipletime scales. This extends the theory of bifurcation in genericfamilies of dynamical systems to those with two time scales.Emphasis is placed upon relaxation oscillations, periodic orbitsthat have both slow and fast segments. The initial stages of thework seek a classification of degenerate decompositions appearingin periodic orbits of one parameter families with relaxationoscillations. Geometric methods are used to determinebifurcations associated with each degenerate decomposition.Numerical investigation of examples is used to motivate the workand to ensure that the results are directly applicable tobiological models. Together with collaborators Kathleen Hoffmanand Warren Weckesser, the investigator is reexamining a classicalexample, the forced van der Pol system that gave birth to thediscovery of chaos for dissipative dynamical systems, and expectsto give a full description of the bifurcations that occur withinthis system. He also develops algorithms for the computation ofstructures that are difficult to compute with existing methods. Rhythmic phenomena are ubiquitous in biological systems.Examples include the heartbeat, the cell cycle, circadianrhythms, legged locomotion, and electrical signals in the nervoussystem. Most of these involve multiple time scales. Theinvestigator pursues new mathematical theory and computationalmethods that apply to dynamical systems with multiple timescales. Emphasis is given to models of neural systems, an areain which the presence and importance of complex dynamics aremanifest. The investigations draw upon decades of research incharacterizing generic phenomena observed in dynamical systemswith a single time scale. The resulting body of mathematics,sometimes called chaos theory, needs extension and modificationto fully explain the behavior of systems with multiple timescales. Those extensions are the goal of this project. On alonger time frame, the project lays foundations for cominggenerations of biological models for cellular processes such asgene expression and signal transduction. New biotechnology leadsto ever more complicated reaction networks that are simulated asdynamical systems. This project produces results that aid theimplementation and interpretation of such simulations of complex,multiple time scale dynamical systems.

Guckenheimer0101208 研究多时间尺度动力系统中轨迹的快慢分解和分岔。 这将动力系统的一般族中的分岔理论推广到具有两个时间尺度的动力系统，重点放在弛豫振荡，即具有慢段和快段的周期轨道上。 工作的初始阶段寻求一个退化分解的分类出现在周期轨道的一个参数的家庭与relaxationoscillations。 几何方法用于确定与每个退化分解相关的分支，数值例子的研究用于激励工作，并确保结果直接适用于生物模型。 与合作者Kathleen Hoffman和Warren Weckesser一起，研究人员正在重新研究一个经典的例子，强迫货车der Pol系统，该系统产生了耗散动力系统的混沌发现，并期望给出该系统内发生的分叉的完整描述。 他还开发了计算现有方法难以计算的结构的算法。 节律现象在生物系统中普遍存在，例如心跳、细胞周期、昼夜节律、腿运动和神经系统中的电信号。 其中大多数涉及多个时间尺度。 研究者追求新的数学理论和计算方法，适用于多个时间尺度的动力系统。 重点是神经系统的模型，这是一个复杂动力学的存在和重要性显而易见的领域。 这些调查利用了数十年来在单一时间尺度下对动力系统中观察到的一般现象进行表征的研究。 由此产生的数学体系，有时被称为混沌理论，需要扩展和修改，以充分解释具有多个时间尺度的系统的行为。 这些扩展是本项目的目标。 在更长的时间框架内，该项目为未来几代细胞过程（如基因表达和信号转导）的生物模型奠定了基础。 新的生物技术导致了越来越复杂的反应网络，这些网络被模拟为动力系统。 这个项目产生的结果，帮助实现和解释这样的模拟复杂的，多时间尺度的动力系统。