Bifurcation in Dynamical Systems with Multiple Time Scales

Guckenheimer0101208研究了多时间尺度动力系统中轨道的慢-快分解和分叉。这将一般动力系统族中的分支理论推广到具有两个时间尺度的系统。重点放在松弛振荡上，即既有慢段又有快段的周期轨道。工作的初始阶段是对出现在具有弛豫振荡的单参数族的周期轨道上的简并分解进行分类。几何方法被用来确定与每个退化分解相关的分叉，实例的数值研究被用来激励工作并确保结果直接适用于生物模型。与合作者凯瑟琳·霍夫曼和沃伦·韦克瑟一起，这位研究人员正在重新研究一个经典的例子--催生了耗散动力系统混沌的强迫范德波尔系统，并期望给出这个系统中发生的分叉的完整描述。他还开发了用现有方法难以计算的结构的计算算法。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、腿部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。研究人员追求适用于多时间尺度动力系统的新的数学理论和计算方法。重点放在神经系统模型上，这是一个复杂动力学的存在和重要性显而易见的领域。这些研究借鉴了数十年来对在单一时间尺度的动力系统中观察到的一般现象进行的研究。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。这些扩展就是这个项目的目标。在更长的时间范围内，该项目为下一代细胞过程的生物模型(如基因表达和信号转导)奠定了基础。新的生物技术导致了更复杂的反应网络，这些网络被模拟为动力学系统。这个项目产生的结果有助于实现和解释这种复杂的、多时间尺度的动力系统的模拟。

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.