Almost Periodic Differential Equations and Lattice Dynamical Systems

小行星9704245 提出了两个方面的研究：概周期微分方程 和晶格动力系统。最近，首席研究员和一位 合作者系统地研究了各种类型的动态 概周期微分方程基本的动力学问题， 由于存在概周期解和概自守解， 有界解的渐近行为，Ω极限集的结构， 等等都受到了相当大的关注。 校长 研究人员和她的合作者还研究了一些 几乎周期微分方程源于生物学和物理学， 利用一般的方法和理论建立在原则 调查员和易一起工作 首席研究员计划 继续她在几乎周期微分方程方面的研究。 在 特别是，她打算探索各种几乎 周期性种群模型，如迁移选择模型和多个 物种竞争模型;研究几乎周期强迫振子 例如货车范德波尔振荡器和约瑟夫森结;考虑 空间概周期抛物方程;找到条件上的 几乎周期动力学的存在;并研究几乎 一般微分方程中的自守动力学。 在她最近的作品中 在与合作者的合作中，她研究了 驻波和行波以及混沌动力学的存在 晶格动力系统 在这些结果中， 发现了耦合映象格子中的混沌动力学， 方法，它不同于但类似于传统的移动 坐标法在偏微分方程，介绍了处理 与行波;和各种动力学，如空间混沌， 传播故障、行波等，存在于离散的 南云方程上述工作提供了一些洞察复杂的 晶格动力系统的行为 继续她的承诺， 格动力系统q，她想研究以下问题 在不久的将来：其他可能的路线，如分叉到混沌 动力学;行波的存在性和稳定性以及 同步 此外，她还打算继续分析 离散Nagumo方程中的动力学;并探索 格点种群模型 概周期微分方程和格动力系统 被广泛用作许多物理和生物问题的模型。 为 例如，在单个物种的种群动态中， 该物种由几乎周期反应扩散方程描述，如果 一个连续的环境是有人居住的，季节性变化是占 (note季节变化可能不是完全周期性的， 周期性的），并且存在另一种固有的周期性变化。 如果物种 栖息在一个斑块环境中，那么它的动力学可以用一个格来描述， 常微分方程 此外，如果每个人的 物种以离散时间的方式从一个斑块迁移到另一个斑块， 动力学的特征在于耦合的映射点阵。 许多其他 例如电路、气候动力学、图像处理 和模式识别，材料科学等，因此，它是伟大的 研究概周期微分方程和格重要性 动力系统 虽然已经做了很多研究，但许多情况 在这两个领域出现的问题还远未得到很好的理解。 校长 研究人员建议在这两个领域进行研究。 特别是基于 根据她以前的经验，她计划研究各种模式， 物理学和生物学，如竞争模型，几乎周期性地强迫 振荡器等。拟议的研究旨在成为一个数学 对概周期的定性和定量理论的贡献 微分方程和格动力系统，从而更好地 了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.