Multi-frequency Oscillations: Regularity, Irregularity, and Complexity

该项目的重点是在生物，电气和机械振荡系统中产生的多频现象，有或没有阻尼。它由两个独立但密切相关的部分组成：1。准周期和弱准周期运动; 2.生物振荡和复杂性。 本项目的第一部分旨在研究适当退化的Hamilton系统、Poisson-Hamilton系统和Hamilton网络中规则和不规则多频振荡的存在性、稳定性和机制。该项目的第二部分致力于研究生物化学反应系统和生物网络中的生物振荡和复杂性，包括确定性和随机性设置，涉及诸如瞬时振荡，随机振荡，不变测度的稳定性以及简并性，鲁棒性，和系统的复杂性。该项目侧重于自然界中的多频现象，特别是那些在生物，电气和机械系统中出现的现象。大量的实验和计算机模拟表明，多频物理系统可以产生多相振荡的复杂动力学结果。但是，这种复杂性的机制还远未得到很好的理解，特别是当环境噪声存在时。在该项目中，PI计划开发一种数学理论，以从根本上理解这种复杂性。

The project focuses on multi-frequency phenomena arising in biological, electrical and mechanical oscillatory systems with or without damping. It consists of two separate but closely related parts: 1. Quasi-periodic and weakly quasi-periodic motions; 2. Biological oscillations and complexities. Part 1 of the project aims at the study of the existence, stability, and mechanism for both regular and irregular multifrequency oscillations in properly degenerate Hamiltonian systems, Poisson-Hamilton systems, and Hamiltonian networks. Part 2 of the project is devoted to the study of biological oscillations and complexity in bio-chemical reaction systems and bio-networks, in both deterministic and stochastic settings, with respect to issues such as stransient oscillations, stochastic oscillations, stability of invariant measures, and the connections among degeneracy, robustness, and system complexity.The project focuses on multi-frequency phenomena in nature especially those arising in biological, electrical, and mechanical systems. It is predicted by many experiments and computer simulations that physical systems involving multi-frequency can produce many complex dynamical outcomes oscillating in multiple phases. But the mechanism of such complexity is far from well-understood, especially when ambient noise is present. In the project, the PI plans to develop a mathematical theory towards a fundamental understanding of such complexity.