Mathematical Sciences: Large-scale Oscillatory Behavior in Asymmetric Systems with Loading

本项目将侧重于对非线性微分方程的数学研究，其动机是研究非对称系统上的大负载导致解的高度振荡行为。通过不对称系统，人们理解这样的系统，其中存在一个恢复力，该恢复力沿着从平衡到无穷大的射线渐近线性，但在无穷远处不是线性的。在所有类型的偏微分方程组以及常微分方程组和哈密顿系统的周期解中都观察到了这种振荡行为。最近的工作使人们对悬索桥、船舶和电梯中的非线性振荡有了新的见解。本工作的重点之一将是发展常微分方程组周期解的一般理论。最感兴趣的问题涉及不同解的数目、解的稳定性和在阻尼作用下解的持久性。哈密顿系统的计算方法以及描述扩散系统的具有非线性的抛物型问题的工作预计也将继续进行。除了电子学之外，在非线性电容器的行为方面，预计还会出现应用。来自非线性泛函分析的方法将与计算工具和奇点理论相结合，在非对称系统的背景下寻求广泛的一般原理。

This project will emphasize mathematical research on nonlinear differential equations motivated by studies of large loading on asymmetric systems which result in highly oscillatory behavior of solutions. By asymmetric systems one understands those systems in which there is a restoring force which is asymptotically linear along rays issuing from equilibrium to infinity, but not linear at infinity. The oscillatory behavior has been observed in partial differential equations of all types as well as with periodic solutions of ordinary differential equations and Hamiltonian systems. Recent work has led to new insights into nonlinear oscillations in suspension bridges, ships and elevators. One focus of the present work will be to develop a general theory of periodic solutions of systems of ordinary differential equations. Questions of primary interest concern the number of distinct solutions, the stability of solutions and the persistence of solutions under damping. Work is also expected to continue on computational approaches to Hamiltonian systems as well as on parabolic problems with nonlinearities which describe diffusion systems. Applications are expected to arise in areas already mentioned in addition to electronics, in the behavior of nonlinear capacitors. Methods from nonlinear functional analysis will be combined with computational tools and singularity theory in seeking broad general principles in the context of asymmetric systems.