A geometric theory for non-standard relaxation oscillators

该项目旨在开发新的几何方法来分析生物节律的多尺度模型，并设计诊断工具来识别引起和控制这些信号的关键参数。节奏，如呼吸，神经和心脏的节奏和脉动激素分泌，是生命的核心。 由于这些模型中固有的非均匀时间尺度分裂，许多表现出松弛型行为的重要生化细胞信号无法用标准动力系统工具进行严格分析。该项目旨在开发一个统一的数学理论，将几何奇异摄动理论和代数几何的结果编织在一起，以解释生物，非标准，多尺度系统中复杂节奏和模式的起源，无论是在个人还是网络层面。

This project aims to develop new geometric methods for the analysis of multi-scale models of biological rhythms, and design diagnostic tools to identify key parameters that cause and control these signals. Rhythms, such as breathing, neural and cardiac rhythms and pulsatile hormone secretion, are central for life. Many important biochemical cell signals exhibiting relaxation-type behaviour cannot be rigorously analysed with standard dynamical systems tools due to an inherent non-uniform time-scale splitting in these models. This project aims to develop a unified mathematical theory that weaves together results from geometric singular perturbation theory and algebraic geometry to explain the genesis of complex rhythms and patterns in biological, non-standard, multi-scale systems, both at individual and network level.