Complex Oscillations and Invariant Manifolds

Cornell University, Ithaca NY

Investigators

Abstract

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.