Stability and bifurcations analysis in delay differential equations with applications

时滞微分方程的稳定性和分岔分析及其应用

• 批准号: 261357-2007
• 负责人: Yuan, Yuan
• 金额: $ 0.87万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=394973
• 关键词: Stability; bifurcations; analysis; delay; differential

The dynamical systems with time delay are of great theoretical interest and form an important class with regards to their applications. Mathematically, these systems are represented by delay differential equations (DDEs). DDEs are used extensively in the modeling of a multitude phenomena in biological sciences, physics, engineering, economics, etc. Stability and bifurcation are the most significant properties in dynamical system since an individual predictable process can be physically realized only if it is stable or quasistable in the corresponding natural sense. Center manifold theory and normal form (NF) method are of fundamental importance in the study of nonlinear dynamical systems, and have been applied in finite-dimensional ordinary differential equations (ODEs) broadly. Compared with the great amount of the publications in the study of stability, bifurcation and NF computation for ODEs, there are only few results in that of DDEs. In order to predict and control the long term behavior of real world models, to the mathematicians working in dynamical system, the challenging problems include how time delay and the system parameters affect the stability of the system, and what kind of bifurcations will occur when the stability is destroyed with the variation of delay and parameters. Therefore, it is necessary to extend the theory and methodology to study DDEs deeply and widely, as well as a need to consider the efficiency of the computational methods. This proposal is concerned with developing efficient computational methods for reducing the center manifold and computing the NF of DDEs, then analyzing the stability and bifurcations as the parameters and delays vary, although other aspects will also be involved. It is anticipated that this proposed research will increase our understanding of the dynamical behavior if a system cooperates with time delays. The methodologies developed in this proposed research will have very high potential impact on the nonlinear dynamics community, which will not only strengthen the foundation for theoretical development in a large class of DDEs, but also provide a practical tool for solving real complex dynamical systems.

时滞动力系统是一类具有重要应用价值的理论问题。在数学上，这些系统可以用时滞微分方程(DDES)来表示。常微分方程组被广泛应用于生物科学、物理、工程、经济等众多现象的建模中。稳定性和分叉是动力系统中最重要的性质，因为只有当个体可预测的过程在相应的自然意义上是稳定或准静态的时，才能在物理上实现该过程。中心流形理论和规范型方法是研究非线性动力系统的重要方法，在有限维常微分方程组中得到了广泛的应用。与研究常微分方程组稳定性、分叉和核因子计算的大量文献相比，关于常微分方程组稳定性、分叉和核因子计算的结果很少。为了预测和控制真实世界模型的长期行为，对于从事动态系统工作的数学家来说，具有挑战性的问题包括时滞和系统参数如何影响系统的稳定性，以及当稳定性随着时滞和参数的变化而被破坏时会出现什么样的分叉。因此，有必要扩展理论和方法以深入和广泛地研究微分方程组，并需要考虑计算方法的效率。这项建议致力于开发有效的计算方法来减少中心流形和计算偏微分方程组的核函数，然后分析随着参数和延迟的变化而产生的稳定性和分叉，尽管还将涉及其他方面。预计这项研究将增加我们对时滞系统合作时的动力学行为的理解。所提出的研究方法将对非线性动力学领域产生很大的潜在影响，这不仅为一大类时滞系统的理论发展奠定了基础，也为解决实际的复杂动力系统提供了实用的工具。