论文标题
严格验证HOPF分叉通过降低和延续
Rigorous verification of Hopf bifurcations via desingularization and continuation
论文作者
论文摘要
在本文中，我们提出了一种严格验证Hopf分叉以及ODES系统中周期性轨道的鞍节分叉的一般方法。通过分析估计和计算机辅助计算的组合，我们遵循循环的溶液曲线，并沿途检查单个非等级鞍形节点分叉发生。同样，我们严格地继续循环从HOPF分叉开始的循环曲线。我们使用爆炸分析来使HOPF分叉点附近的延续问题正规化。这将经过验证的持续方法的适用性扩展到了分叉问题的数学严格计算研究。
In this paper we present a general approach to rigorously validate Hopf bifurcations as well as saddle-node bifurcations of periodic orbits in systems of ODEs. By a combination of analytic estimates and computer-assisted calculations, we follow solution curves of cycles through folds, checking along the way that a single nondegenerate saddle-node bifurcation occurs. Similarly, we rigorously continue solution curves of cycles starting from their onset at a Hopf bifurcation. We use a blowup analysis to regularize the continuation problem near the Hopf bifurcation point. This extends the applicability of validated continuation methods to the mathematically rigorous computational study of bifurcation problems.
