Bifurcation Analysis and Its Applications
Bifurcation Analysis and Its Applications
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分岔分析及其应用
DOI：
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发表时间：
2012
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通讯作者：
Canan Çelik Karaaslanlı
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作者：
Canan Çelik Karaaslanlı
涉及微分方程的连续动力系统大多包含参数。参数的微小变化可能会对解决方案产生重大影响。本章感兴趣的主要问题是：如何延续动力系统关于参数的平衡和周期轨道？如何计算参数空间中平衡点和极限环的稳定边界？如何预测在这些平衡点上发生的系统行为（分叉）的质的变化？这一章也将涵盖在平衡和周期轨道方面的分叉的分类。特别是当分岔参数超过某一临界值时，系统将出现一种特殊的分岔现象，称为“Hopf分岔”，即从稳定平衡点开始，系统的周期轨道的发展。由于平衡点的分支理论是基于中心流形约化和Poincare范式的，因此数学模型的分支方向也将使用该理论来解释。最后，通过介绍几个软件包和数值方法，本章还将涵盖的技术，以确定和继续在一些控制参数的动力系统的周期轨道的所有局部分支和相关的规范形计算结合中心流形定理，包括周期规范形的周期轨道。
Continuous dynamical systems that involve differential equations mostly contain parameters. It can happen that a slight variation in a parameter can have significant impact on the solution. The main questions of interest in this chapter are: How to continue equilibria and periodic orbits of dynamical systems with respect to a parameter? How to compute stability boundaries of equilibria and limit cycles in the parameter space? How to predict qualitative changes in system’s behavior (bifurcations) occurring at these equilibrium points? This chapter will also cover the classification of bifurcations in terms of equilibria and periodic orbits. Especially it will present the specific bifurcation called ”Hopf bifurcation” which refers to the development of periodic orbits from stable equilibrium point, as a bifurcation parameter crosses a critical value. Since the theory of bifurcation from equilibria based on center manifold reduction and Poincare-Normal forms, the direction of bifurcations for the mathematical models will also be explained using this theory. Finally, by introducing several software packages and numerical methods this chapter will also cover the techniques to determine and continue in some control parameters all local bifurcations of periodic orbits of dynamical systems and relevant normal form computations combined with the center manifold theorem, including periodic normal forms for periodic orbits.