Geometric desingularization of degenerate singularities in the presence of fast rotation: A new proof of known results for slow passage through Hopf bifurcations
Geometric desingularization of degenerate singularities in the presence of fast rotation: A new proof of known results for slow passage through Hopf bifurcations
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快速旋转情况下简并奇点的几何去奇异化：缓慢通过 Hopf 分岔的已知结果的新证明
DOI：
10.1016/j.indag.2015.11.005
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
M. Wechselberger
中科院分区：
文献类型：
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作者：
M. G. Hayes;T. Kaper;P. Szmolyan;M. Wechselberger
在这篇文章中，我们提出了一个新的，几何证明已知的结果缓慢通过霍普夫分岔（贝尔等人。（1989），Neishtadt（1987，1988），Shishkova（1973））。新的证明采用积分沿着一个合适的选择轮廓在复杂的时间平面，然后几何去奇异化的方法，这也被称为爆破方法。在复时间平面上选择轮廓线，使这些慢通道问题中的奇点变为幂零，然后用去奇异化方法分析双曲性的损失，而不需要快速旋转的复杂性。除了它们本身的兴趣之外，慢通道通过Hopf分歧点的新方法和已知结果的新证明也是当前的兴趣，由于延迟通过分叉的现象在分析高维非双曲点的曲线和流形的快慢系统的折叠奇点中起着越来越重要的作用。对于其中的一些问题，它也将是有用的，制定延迟稳定性损失的结果，通过选择适当的轮廓在复杂的时间平面，使奇点幂零，然后通过应用几何去奇异化方法。
In this article, we present a new, geometric proof of known results for slow passage through Hopf bifurcations (Baer et al. (1989), Neishtadt (1987, 1988), Shishkova (1973)). The new proof employs integration along a suitable choice of contour in the complex time plane and then the method of geometric desingularization, which is also known as the blow-up method. The contour in the complex time plane is chosen so that the singularities in these slow passage problems become nilpotent, and the loss of hyperbolicity can then be analyzed using the desingularization method without the complication of the fast rotation.Besides being of interest in their own right, the new method and the new proof of the known results for the slow passage through Hopf bifurcation points are also of current interest, since the phenomena of delayed passage through bifurcations plays an increasingly important role in the analysis of folded singularities in higher-dimensional fast–slow systems with curves and manifolds of nonhyperbolic points. For some of these problems, it will be useful also to formulate the delayed stability loss results by choosing appropriate contours in the complex time plane to make the singularities nilpotent and then by applying the geometric desingularization method.