Almost Periodic Differential Equations and Lattice Dynamical Systems

9704245沈从文的研究主要集中在两个方面：概周期微分方程组和格子动力系统。最近，主要研究人员和一位合作者系统地研究了各种类型的概周期微分方程的动力学。一些基本的动力学问题，如概周期解和几乎自同构解的存在性，有界解的渐近行为，omega极限集的结构等，都受到了相当大的关注。首席研究员和她的合作者还利用首席研究员与易的联合工作中建立的一般方法和理论，研究了一些源于生物学和物理学的概周期微分方程的全局动力学。首席研究员计划继续她对几乎周期微分方程式的研究。特别是，她打算探索各种概周期种群模型的全球动力学，如迁徙-选择模型和多物种竞争模型；研究几乎周期强迫振子，如van der Pol振子和Josephson结；考虑空间概周期抛物型方程；找到概周期动力学存在的条件；以及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她的合作者的合作中，她研究了格子动力系统中驻波和行波的稳定性以及混沌动力学的存在。在这些结果中，找到了耦合映射格子中混沌动力学的存在准则；引入了一种不同于但类似于偏微分方程组中传统移动坐标方法的移动坐标方法来处理行波；离散的Nagumo方程中存在空间混沌、传播失败、行波等各种动力学。上述工作为揭示格子动力系统的复杂行为提供了一定的理论基础。继续她对格子动力系统的承诺，她希望在不久的将来研究以下问题：其他可能的途径，如分叉到混沌动力学；行波的存在和稳定性以及同步的出现。此外，她打算继续分析离散Nagumo方程中的动力学，并探索格子种群模型中的动力学。概周期微分方程组和格子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力学中，如果居住着一个连续的环境，物种的动态可以用一个几乎周期的反应扩散方程来描述，季节变化可以考虑季节性变化(注意，季节变化可能不是完全周期的，而是几乎周期的)，并且存在另一个固有的周期变化。如果物种生活在斑驳的环境中，那么它的动态可以用格子常微分方程式来描述。此外，如果物种的每个个体以离散的方式从一个斑块迁移到另一个斑块，那么动力学特征是一个耦合的映象格子。在电路、气候动力学、图像处理与模式识别、材料科学等领域都有大量的例子。因此，研究概周期微分方程和格子动力系统具有重要意义。虽然已经做了很多研究，但这两个领域出现的许多情景还远未得到很好的理解。首席研究员建议在这两个领域进行研究。特别是，根据她以前的经验，她计划研究来自物理和生物学的各种模型，如竞争模型、几乎周期振子等。拟议的研究旨在为概周期微分方程和格子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中的复杂行为。

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.