Almost Periodic Differential Equations and Lattice Dynamical Systems

沈提出了两个研究领域：概周期微分方程和格动力系统。最近，首席研究员和合作者系统地研究了各种类型的几乎周期微分方程的动力学。基本的动力学问题，如几乎周期解和几乎自同构解的存在性、有界解的渐近行为、极限集的结构等，已经得到了相当多的关注。首席研究员和她的合作者还研究了一些由生物学和物理学引起的几乎周期性微分方程的全局动力学，利用了首席研究员与Yi共同工作中建立的一般方法和理论。首席研究员计划继续她在周期微分方程方面的研究。特别是，她打算探索各种几乎周期性的种群模型的全球动态，如迁移选择模型和多物种竞争模型；研究几乎周期性强迫振子，如van der Pol振子和Josephson结；考虑空间概周期抛物方程；寻找概周期动力学存在的条件；研究一般微分方程中的几乎自同构动力学。在她最近的工作以及与合作者的合作中，她研究了格动力系统中驻波和行波的稳定性以及混沌动力学的存在性。在这些结果中，找到了耦合映射格中混沌动力学的存在准则；引入了一种与传统的偏微分方程运动坐标法不同但又类似的运动坐标法来处理行波；在离散的南云方程中存在空间混沌、传播失效、行波等多种动力学特性。以上工作为点阵动力系统的复杂行为提供了一些见解。继续她在晶格动力系统的研究，她想在不久的将来研究以下问题：其他可能的路线，如分岔到混沌动力学；行波的存在、稳定性及同步现象的出现。此外，她打算继续分析离散Nagumo方程中的动力学；并探索晶格种群模型中的动力学。几乎周期微分方程和格动力系统都被广泛地用作许多物理和生物问题的模型。例如，在单个物种的种群动态中，如果一个连续的环境有人居住，那么物种的动态是用一个几乎周期性的反应扩散方程来描述的，考虑了季节变化（注意季节变化可能不是完全周期性的，但几乎是周期性的），并且存在另一个固有的周期性变化。如果物种生活在一个斑驳的环境中，那么它的动态可以用晶格常微分方程来描述。此外，如果物种的每个个体以离散的时间方式从一个斑块迁移到另一个斑块，则动态特征为耦合映射晶格。在电路、气候动力学、图像处理和模式识别、材料科学等领域也发现了许多其他的例子。因此，研究概周期微分方程和格动力系统具有重要的意义。尽管已经做了大量的研究，但在这两个领域出现的许多情况还远远没有得到很好的理解。首席研究员建议在这两个领域进行研究。特别是，根据她之前的经验，她计划研究来自物理和生物学的各种模型，如竞争模型，几乎周期性的强迫振荡等。提出的研究旨在对几乎周期微分方程和晶格动力系统的定性和定量理论做出数学贡献，从而更好地理解这些系统的复杂行为。

9704245 Shen Research is proposed in two areas: almost periodic differential equations and lattice dynamical systems. Recently, the principal investigator and a collaborator systematically investigated the dynamics of various types of almost periodic differential equations. Fundamental dynamical issues such as the existence of almost periodic and almost automorphic solutions, asymptotic behavior of bounded solutions, the structure of omega limit sets, etc. have received a considerable amount of attention. The principal investigator and her collaborators also studied the global dynamics of some almost periodic differential equations arising from biology and physics by utilizing both the general approach and theory established in the principal investigator's joint works with Yi. The principal investigator plans to continue her research in almost periodic differential equations. In particular, she intends to explore the global dynamics of various almost periodic population models such as migration-selection models and multiple species competition models; to study almost periodically forced oscillators such as van der Pol oscillators and Josephson junctions; to consider spatially almost periodic parabolic equations; to find conditions on the existence of almost periodic dynamics; and to investigate almost automorphic dynamics in general differential equations. In her recent work and also in joint works with her collaborators, she studied the stability of standing and traveling waves and the existence of chaotic dynamics in lattice dynamical systems. Among the results, an existence criterion for chaotic dynamics in coupled map lattices is found; a moving coordinate approach, which is different from but analogous to the traditional moving coordinate approach in partial differential equations, is introduced to deal with traveling waves; and various dynamics such as spatial chaos, propagation failure, traveling waves, etc., are shown to exist in a discrete Nagumo equation. The above works provide some insight into the complex behavior of lattice dynamical systems. Continuing her commitments in lattice dynamical systemq, she would like to study the following problems in the near future: other possible routes such as bifurcations to chaotic dynamics; existence and stability of traveling waves and the appearance of synchronization. Moreover, she intends to continue the analysis of the dynamics in the discrete Nagumo equation; and to explore the dynamics in lattice population models. Both almost periodic differential equations and lattice dynamical systems are widely used as models for many physical and biological problems. For example, in the population dynamics of a single species, the dynamics of the species is described by an almost periodic reaction diffusion equation if a continuous environment is inhabited, seasonal variation is accounted for (note that seasonal variation may not be exactly periodic but rather almost periodic) and there is another inherent periodic variation. If the species inhabits a patchy environment, then its dynamics are described by a lattice ordinary differential equation. Furthermore, if each individual of the species migrates from patch to patch in a discrete time manner, then the dynamics are characterized by a coupled map lattice. Numerous other examples are found in electric circuits, climate dynamics, image processing and pattern recognition, material sciences, etc. It is therefore of great importance to study almost periodic differential equations and lattice dynamical systems. Though much research has been done, many scenarios appearing in both areas are far from being well understood. The principal investigator proposes to do research in these two areas. In particular, based on her previous experience, she plans to study various models arising from physics and biology such as competition models, almost periodically fo rced oscillators, etc. The proposed research is intended to be a mathematical contribution to the qualitative and quantitative theory of almost periodic differential equations and lattice dynamical systems, resulting in a better understanding of complex behavior in these systems.