Gloval Bifurcation & Asymptotics Behavior in Nonlinear Differential Equations

501497一般或抛物型偏微分方程的Ward系统可用于许多物理系统的建模。微分方程通常包含参数，其值对应于系统的物理性质。这样的系统在适当的相空间中产生一个动力系统。为了理解一个动力系统，我们应该确定它的紧不变集，它包含非瞬态解，如平衡解、周期解或更复杂的解。确定瞬态解相对于非瞬态解的行为也很重要。最后，我们应该知道所有这些是如何依赖于参数的。例如，一个参数的变化可能导致一个稳定或双曲平衡状态分岔成一个或多个时间周期解，或具有更复杂时间结构的解。Conley同伦指标理论包含了可以用来证明动力系统中紧孤立不变量集的存在性和确定其性质的概念和方法。与拓扑度一样，索引是局部常数，如果索引随着参数穿过值而变化，则隐含分岔。本项目的一个主要部分是应用Conley同伦指标方法研究反应扩散方程系统的分岔问题。研究人员将：(1)继续研究无限维动力系统，特别是非线性反应扩散系统中由平凡解分叉的全有界解的连续体。Ward最近利用Conley同伦指标方法证明了这种分岔连续体存在的条件。该项目的两个主要目标是充分描述这些连续体的整体方面，并将结果应用于梯度系统平衡体的整体分岔研究。(2)研究群体生物学和化学中反应扩散模型的分岔和持续性。物理上有意义的解决方案必须具有非负分量。在此基础上，研究了具有不变非负锥的非线性常抛物型偏微分系统的同伦指标分岔理论。（III）建立在他们早期关于非自治微分方程的工作基础上。特别是，他们将继续发展基于同伦指数的非线性时相关常微分和抛物型偏微分系统理论。在这种情况下，动力系统不是在初始值空间中生成的，但可以将斜积流联系起来，并使用康利理论来研究其不变集。从这个可以得到关于原始系统的信息。研究人员将集中于存在、动力学和分岔现象。(4)研究具有非线性边界条件的半线性抛物型偏微分方程的相关问题，如周期解的存在性和渐近稳定性，以及此类方程线性部分的谱与非线性之间的相互作用的影响。微分方程模拟物理现象，如机械系统、化学反应和疾病的传播。然后，初始状态（例如，受感染个体的初始数量）根据方程组所表达的系统规律随时间演变。这就构成了一个动力系统。动力系统理论的一个目标是描述系统的状态如何演化。因此，湖中某一特定物种的鱼的数量可能趋向于稳定、固定的种群，而在任何特定时间感染一种传染性疾病（如水痘）的个体数量可能表现出周期性的行为，每六、七年达到峰值。为了理解一个复杂的动力系统，人们应该确定非瞬态或长期现象，如平衡、时间周期状态和其他可能具有更复杂性质的现象，确定瞬态解和非瞬态解之间的关系，以及这一切如何取决于系统参数。数学模型依赖于物理系统的某些参数，例如研究湖中鱼类种群时湖泊的体积或化学反应的扩散速率。一个系统的长期行为的性质可能取决于它的参数值。如果参数改变，非瞬态解的数量或结构也会改变。这种与参数相关的变化就是分岔。另一种分岔发生在瞬态行为发生变化时，如当平衡点（非瞬态）随着参数变化而失去稳定性时。非瞬态解对应于动力系统中的不变集。同伦指标理论是Charles Conley等人为了分析动力系统中孤立不变量集的存在性和性质而提出的。它也可以用来表明分岔必须发生在某些参数值。本课题的一个主要部分是将Conley同伦指标方法应用于反应扩散方程系统的分岔现象。偏微分方程的反应扩散系统将扩散现象，如化学物质在介质中的扩散，与反应速率（如化学反应）联系起来。它们是许多现象的数学模型，包括化学反应、人口密度和气候。研究人员将：(I)继续研究非线性反应扩散系统中从平衡解分叉的有界解的连续体（连接在一起的解族）。Ward最近利用Conley同伦指标方法找到了这种分岔连续存在的条件。该项目的两个主要目标是全面描述这些分岔解族的全局方面，并将这些结果应用于研究梯度系统（从长远来看，将达到平衡状态的系统）中平衡的全局分岔。(2)研究群体生物学和化学中反应扩散模型的分岔和持续性。（III）建立在他们早期关于时变微分方程系统的工作的基础上。这样的系统表现出与上面讨论的时间无关系统不同的现象。研究人员将继续发展基于同伦指数的非线性时相关常抛物型偏微分系统理论。* * *

501497 Ward Systems of ordinary or parabolic partial differential equations serve in modeling many physical systems. The differential equations typically contain parameters whose values correspond to physical properties of the system. Such a system generates a dynamical system in an appropriate phase space. To understand a dynamical system we should determine its compact invariant sets, which contain the non-transient solutions such as equilibria, periodic solutions, or ones of greater complexity. It is important also to determine the behavior of the transient solutions with respect to the non-transients. Finally, one should know how all of this depends upon the parameters. For example, a parameter change may induce a stable or hyperbolic equilibrium state to bifurcate into one or more time-periodic solutions, or ones with a more complex time structure. The Conley homotopy index theory contains concepts and methods that can be used to prove existence, and determine the properties, of compact isolated invariant sets in a dynamical system. Like topological degree the index is locally constant, and if the index changes as a parameter crosses a value, a bifurcation is implied. A major part of this project is the application of Conley homotopy index methods to study bifurcation in systems of reaction-diffusion equations. The investigators will: (I) Continue investigations of continua of full bounded solutions bifurcating from trivial solutions in infinite dimensional dynamical systems, especially in nonlinear reaction- diffusion systems. Using Conley homotopy index methods Ward recently demonstrated conditions for the existence of such bifurcating continua. Two of the main goals of this project are to fully describe the global aspects of these continua, and to apply the results to the study of global bifurcation of equilibria in gradient systems. (II) Study bifurcation and persistence in reaction-diffusion models arising in population biology an d chemistry. Physically meaningful solutions must have non-negative components. In this connection the investigators propose the development of a homotopy index bifurcation theory for nonlinear ordinary and parabolic partial differential systems which have an invariant non-negative cone. (III) Build on their earlier work on nonautonomous differential equations. In particular, they will continue the development of a homotopy index based theory for nonlinear time-dependent ordinary and parabolic partial differential systems. In this case a dynamical system is not generated in the space of initial values, but one can associate a skew-product flow and use the Conley theory to study its invariant sets. From this one can derive information regarding the original system. The investigators will focus on existence, dynamics, and bifurcation phenomena. (IV) Study related problems for semilinear parabolic partial differential equations with nonlinear boundary conditions, such the existence and asymptotic stability of periodic solutions, and the effect of interactions between the spectrum of the linear part of such equations with the nonlinearities. %%% Differential equations model physical phenomena such as mechanical systems, chemical reactions, and the spread of disease. An initial state (e.g., initial number of infected individuals) then evolves over time according to the laws of the system expressed by its equations. This constitutes a dynamical system. A goal of dynamical systems theory is to describe how the states of the system evolve. Thus, the number of fish of a given species in a lake might tend toward a steady, fixed population, while the number of individuals infected at any given time with a transmittable disease, such as chicken pox, may exhibit a periodic behavior, peaking every six or seven years. To understand a complex dynamical system one should determine the non-transient or long-term phenomena such as equilibria, time-periodic regimes, and others perhaps of a more complex nature, the relation between the transient solutions and the non-transients, and how this all depends on system parameters. Mathematical models depend upon certain parameters fom the physical system, such as the volume of a lake in a study of its fish population or diffusion rates in a chemical reaction. The nature of the long-term behavior of a system may depend upon its parameter values. If the parameters change, the number or structure of the non-transient solutions can change. Such a parameter dependent change is a bifurcation. Another kind of bifurcation occurs when the behavior of the transients changes, as when an equilibrium point (a non-transient) loses stablity with a parameter change. Non-transient solutions correspond to invariant sets in the dynamical system. The homotopy index theory was developed by Charles Conley and others to analyze the existence and properties of isolated invariant sets in dynamical systems. It also can be used to show that bifurcations must take place at certain parameter values. A main part of this project involves the application of Conley homotopy index methods to bifurcation phenomena in systems of reaction-diffusion equations. Reaction-diffusion systems of partial differential equations relate diffusion phenomena, such as diffusion of a chemical in a medium, with reaction rates, as in a chemical reaction. They serve as the mathematical models of a great many phenomena, including chemical reactions, population densities, and climate. The investigators will: (I) Continue investigations of continua (families of solutions joined together) of bounded solutions bifurcating from equilibrium solutions in nonlinear reaction-diffusion systems. By using Conley homotopy index methods Ward recently found conditions for the existence of such bifurcating continua. Two of the main goals of this project are to fully describe the global aspects of these families of bifurcat ing solutions, and to apply these results to study global bifurcation of equilibria in gradient systems (systems which, in the long run, will reach an equilibrium state). (II) Study bifurcation and persistence in reaction-diffusion models arising in population biology and chemistry. (III) Build on their earlier work on time-varying systems of differential equations. Such systems exhibit phenomena different from the time-independent systems discussed above. The investigators will continue to develop a homotopy index based theory for nonlinear time-dependent ordinary and parabolic partial differential systems. ***