Mathematical Sciences: Oscillatory Behaviour in Asymmetric Systems with Loading

这个项目将强调数学研究， 非线性微分方程的研究 非对称系统上的负载导致高度振荡 解决方案的行为。 通过非对称系统， 这些系统中有一个恢复力， 渐近线性沿着射线发出的平衡， 无穷大，但在无穷大不是线性的。 振荡行为 在所有类型的偏微分方程中都可以观察到 以及常微分方程的周期解 方程和哈密顿系统。 最近的工作导致了新的 深入了解悬索桥、船舶的非线性振动 还有电梯 目前工作的一个重点将是制定一个通用的 常微分方程组周期解理论 方程 主要关心的问题是， 不同的解决方案，解决方案的稳定性和 阻尼下溶液的持久性。 预计工作还将 继续对哈密顿系统的计算方法， 以及具有非线性的抛物问题， 扩散系统 预计将在已经存在的领域提出申请 除了电子产品之外， 非线性电容器 非线性泛函分析方法 将结合计算工具和奇点理论 在不对称的背景下寻求广泛的一般原则， 系统.

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.