Complex Oscillations and Invariant Manifolds

该项目研究多时间尺度动力系统的数学理论，并开发新的计算方法，将这一理论应用于生物现象的模型。本研究采用几何方法来研究这些问题。特别地，它研究了在组织复振荡中起关键作用的不变流形。计算这些流形的新方法是研究的热点之一。实际上，目前普遍缺乏用计算机研究流形的方法。创建一个全面的“光滑计算几何”是研究的长期目标。该项目还寻求开发将模型与数据拟合的方法。一个复杂的动态模型的所有参数都可以测量，或者用系统的方法从经验时间序列数据中估计这些参数是很少见的。对于多时间尺度系统，这是一个特别困难的优化问题，因为动态的突变不容易被平滑优化算法所基于的二次模型拟合。这个项目试图确定这些突变发生在哪里。该方法还可以进行精确的灵敏度分析，描述模型轨迹随参数变化的变化率。它们的目的是为设计具有周期性运行状态而不是稳定运行状态的工程系统提供可用的方法工具包。动力系统理论在将种群动力学、化学反应、激光等广泛不同的现象联系起来方面取得了惊人的成功。该项目遵循这一传统，试图解释在节奏过程中观察到的普遍动态行为，其中许多表现出复杂的振荡。呼吸、心跳、昼夜节律、月经周期和动物运动是该方法适用的生物节律的几个例子。高等动物的所有主要运动方式：行走、奔跑、滑行、游泳和飞行都是由身体的循环运动产生的。在神经系统中无处不在的爆发振荡是时间复杂性的例证：神经元的活跃放电与静止周期交替。在非平衡态化学反应器的混合模振荡中，大振幅和小振幅振荡的周期是交替的。在这些复杂的振荡中，多重时间尺度是固有的。因此，该项目开发了用于分析具有多个时间尺度的动力系统的新方法，其结果产生了对系统中快速变化如何由缓慢组分的变化引起的更深入的数学理解。几何模型将这种变化的机制简化为最简单的形式，并为从具有多个时间尺度的系统的数值模拟中获得的神秘结果提供数学解释。

This project studies the mathematical theory of dynamical systems with multiple time scales and develops new computational methods for bringing this theory to bear upon models of biological phenomena. The research employs geometric approaches to study these problems. In particular, it investigates invariant manifolds that play a key role in organizing complex oscillations. New computational methods for computing these manifolds are one focus of the research. Indeed, there is a general lack of methods for computer investigation of manifolds. Creation of a comprehensive ``smooth computational geometry'' is a long term goal of the research. The project also seeks to develop methods for fitting models to data. It is rare that all of the parameters of a complex dynamical model can be measured or that systematic methods are used to estimate these parameters from empirical time series data. With multiple time scale systems, this is a particularly difficult optimization problem because abrupt changes in the dynamics are not readily fit by the quadratic models upon which smooth optimization algorithms are based. This project seeks to identify where these abrupt changes occur. The methods also enable accurate sensitivity analysis that describes the rates of change of model trajectories as parameters are varied. They are designed to contribute to the toolkit of methods available for designing engineered systems with periodic operating states rather than ones which are steady.Dynamical systems theory is astonishingly successful in relating widely disparate phenomena observed in population dynamics, chemical reactions, lasers and much more. This project follows this tradition, seeking to explain universal dynamical behaviors observed in rhythmic processes, many of which display complex oscillations. Respiration, the heartbeat, circadian rhythms, menstrual cycles and animal locomotion are a few examples of biological rhythms to which the methods apply. All the primary modes of locomotion of higher animals: walking, running, slithering, swimming and flying result from cyclic motions of the body. Bursting oscillations that are ubiquitous in the nervous system exemplify temporal complexity: epochs of active firing of neurons alternate with quiescent periods. In mixed mode oscillations of non-equilibrium chemical reactors, epochs of large and small amplitude oscillations alternate. Multiple time scales are inherent in these complex oscillations. Thus, this project develops new methods for the analysis of dynamical systems with multiple time scales and the results yield a deeper mathematical understanding of how rapid changes in a system can result from variations of slow components. Geometric models reduce the mechanisms for such changes to their simplest forms and provide mathematical explanations for enigmatic results obtained from numerical simulations of systems with multiple time scales.