Bifurcation theory, computation and applications

科学和工程中出现的许多问题都可以用随时间动态演化的数学系统来建模。其中，非线性微分方程被广泛用于描述复杂的现象，如机械振动、天气模式、动物运动、人口增长、金融市场、疾病传播、反复感染等，这些动力学过程往往表现出质变，称为分叉，如地震、人口爆炸、股市崩盘、分叉理论是研究分叉现象和解释系统参数的微小变化如何导致这些定性变化的有力工具。分叉理论的方法使我们能够识别分叉发生的参数值，并预测系统在分叉点附近的行为变化。为了使分支理论更好地应用于解决现实世界中的复杂问题，这些问题通常是高维的、非线性的，需要开发有效的算法和计算机程序来降低系统的复杂性。中心流形理论和规范形理论将通过提取“关键”信息（或项）来极大地简化系统方程以形成简单的系统，同时保持系统在分叉点附近的定性性质不变。这项研究计划有三个目标。其中之一是发展一般延迟微分方程的参数规范形的理论和有效的计算方法，这可以很容易地用来找到，例如，周期解（极限环）的复杂系统，并确定其稳定性。另外两个目标则侧重于应用。我们打算应用动力系统和分支理论的方法与有效的计算工具来分析特定的生物过程的模型：HIV-1治疗，和复发性感染。特别是，我们将进一步研究我们早期开发的HIV-1治疗模型，以找到可用于识别其他类型病毒信号的新机制。我们还想应用时滞微分系统的分支理论和规范形来研究HIV-1时滞模型中的blips现象。这些应用不仅对提高科学认识有重要意义，而且进展可能产生实际的临床效益。此外，我们为这些特定应用开发的理论和方法将对上述其他学科的许多应用有用。

Many problems arising in science and engineering can be modelled by mathematical systems that dynamically evolve in time. Among them, nonlinear differential equations are widely used to describe complex phenomena, such as vibration of machinery, weather patter, animal locomotion, population growth, financial markets, spread of diseases, recurrent infection, etc. Such dynamical processes often exhibit qualitative changes, called bifurcations, like earthquakes, population explosions, stock market crashes, disease outbreaks and so on. Bifurcation theory is a powerful tool to study bifurcation phenomena and explain how small changes in the system parameters can lead to these qualitative changes. The methods of bifurcation theory enable us to identify the parameter values where bifurcations occur, and to predict the changes in behaviour of the system near the bifurcation points. To facilitate the application of bifurcation theory in solving real-world complex problems, which are usually high dimensional and nonlinear, the developments of efficient algorithms and computer programs to reduce system complexity are required. Center manifold theory and normal form theory will be used to greatly simplify system equations by extracting the "key" information (or terms) to form a simple system, while keeping the qualitative behaivour of the system unchanged near bifurcation points. This proposed research program has three objectives. One of them is to develop the theory and efficient computational methods for the parametric normal forms of general delay differential equations, which can be easily used to find, for example, periodic solutions (limit cycles) of a complex system and determine their stability. The other two objectives are focused on applications. We intend to apply the methods of dynamical systems and bifurcation theory with the efficient computational tools to analyze models of specific biological processes: HIV-1 therapy, and recurrent infection. In particular, we shall further study our early developed HIV-1 therapy model to find new mechanisms which can be used to identify other types of viral blips. We also want to apply the bifurcation theory and normal forms for delay differential systems to investigate the blips phenomenon in HIV-1 models with delays. These applications are not only significant for increasing scientific understanding, but progress may yield practical, clinical benefits. In addition, the theory and methodology we develop for these specific applications will be useful for many applications from other disciplines, as mentioned above.