高秩锥单调动力系统与指数分离性

The proposed project will focus on the dynamics of monotone systems with respect to high-rank cones, the exponential separations associated with invariant high-rank cones, as well as the interplay between monotone systems and the exponential separations in the setting of high-rank cones. More precisely, in the general framework of infinite dimensional systems, we will study the structure of the limit-sets, the generic form of convergence and the structural stability of monotone systems with respect to high-rank cones. Equivalent characterization of the exponential separations will be thoroughly investigated in terms of invariant high-rank cones. We will also try to establish the theory of invariant manifolds and foliations in nonlinear systems associated with exponential separations, by which we make an attempt to study the dynamical behavior of smooth monotone systems with respect to high-rank cones. The so obtained theory will be also, in this project, applied to study the dynamics of several systems of differential equations including structural stability of the competitive systems, the generic convergence to slow oscillations in scalar delayed systems governed by monotone negative feedback, and the occurrence of almost automorphic circle flows generated by the scalar parabolic equations with periodic boundary conditions.

本项目研究高秩锥单调系统动力学，并围绕高秩不变锥研究指数分离性及其与高秩锥单调系统的相互作用。在无穷维系统框架下，研究高秩锥单调动力系统的极限集性态、通有收敛形式以及结构稳定性；围绕高秩不变锥，研究无穷维系统指数分离性的等价刻划；建立无穷维系统指数分离性下的不变流形存在光滑性及叶层理论，利用其考察高秩锥下光滑单调系统的极限集性态及其在扰动下的保持；通过建立高秩锥下光滑单调系统理论，用统一的观点讨论若干微分方程系统的不变集特性及其扰动稳定性，包括竞争系统不变集扰动稳定性、单调负反馈(时滞)系统的通有慢振荡性、以及一维非自治抛物系统周期边值问题的几乎自守圆周流属性。

本项目深入研究了高秩锥单调系统动力学及其应用。具体包括：1)围绕不变锥，通过研究指数分离性及其与高秩锥单调系统的相互作用，建立了高秩锥光滑单调系统的通有动力学理论。证明了系统通有轨道或者成为伪有序轨或者收敛至平衡点，进而证明了高维系统的“通有Poincare-Bendixson定理”。2)系统地发展了C^1光滑低正则性下的非自治单调系统理论：跨越了Pesin理论的约束困难,对C^1光滑的强单调离散动力系统，证明了“动力学两分法”与通有收敛性；对C^1光滑的强单调斜积半流系统，证明了任何线性稳定极小集具有几乎自守属性。3)完整刻画了几乎周期外频驱动下的一维抛物方程周期边值问题的动力学性态。我们的研究显示了多频驱动与单频驱动的周期系统具有本质不同：证明了，当中心流形维数不超过1时，极限集可嵌入几乎周期驱动圆周流；而当其中心流形维数大于1时，给出反例说明该嵌入属性不再成立。证明了系统的任何极小集均可以剩余地嵌入几乎自守驱动的圆周流。4)建立了圆周上自治/周期的抛物方程的非游荡点的结构特征。在无需任何系统双曲性假设下，证明了自治系统任何非游荡点或是平衡点或是旋波；对周期系统任何非游荡点或是周期点或是旋波。5)通过建立的动力系统不变锥理论，获得多类微分方程动力学不变集特性及扰动稳定性。