Bifurcation theory, computation and applications

分岔理论、计算与应用

• 批准号: RGPIN-2015-06210
• 负责人: Yu, Pei
• 金额: $ 1.46万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677084
• 关键词: Bifurcation; theory; computation; applications

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.*****

在科学和工程中产生的许多问题可以通过动态发展的数学系统来建模。其中，非线性微分方程被广泛用于描述复杂现象，例如机械，天气模式，动物的振动，动物运动，人口增长，金融市场，疾病的传播，经常性感染等。这种动态过程经常表现出定性变化，称为繁殖，诸如地震，人口爆炸，疾病崩溃，疾病崩溃，疾病，爆发，以及等等。分叉理论是研究分叉现象的强大工具，并解释系统参数的小变化如何导致这些定性变化。分叉理论的方法使我们能够确定发生分叉发生的参数值，并预测分叉点附近系统行为的变化。为了促进分叉理论在解决现实世界中的复杂问题中的应用，这些问题通常是高维和非线性的，需要有效的算法和计算机程序的发展来降低系统的复杂性。中心歧管理论和正常形式理论将通过提取“密钥”信息（或术语）形成简单系统，同时保持系统的定性behaivour在分叉点附近保持不变，从而极大地简化系统方程。该建议的研究计划有三个目标。其中之一是为一般延迟微分方程的参数正常形式开发理论和有效的计算方法，可以轻松地用于查找复杂系统的周期性解决方案（极限周期）并确定其稳定性。其他两个目标集中在应用程序上。我们打算将动态系统和分叉理论的方法与有效的计算工具一起分析特定生物学过程的模型：HIV-1治疗和经常感染。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叶片的新机制。我们还希望将分叉理论和正常形式应用于延迟差分系统，以研究带有延迟的HIV-1模型中的BLIP现象。这些应用不仅对于提高科学理解的意义很重要，而且进步可能会带来实际的临床益处。此外，我们针对这些特定应用程序开发的理论和方法将对其他学科的许多应用有用，如上所述。*****