A model for the nonautonomous Hopf bifurcation
A model for the nonautonomous Hopf bifurcation
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DOI：
10.1088/0951-7715/28/7/2587
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发表时间：
2013-05
期刊：
影响因子：
1.7
通讯作者：
V. Anagnostopoulou;T. Jäger;G. Keller
中科院分区：
文献类型：
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作者：
V. Anagnostopoulou;T. Jäger;G. Keller
受Grebogi等人（1984 Physica D 13 261-8）的一个例子的启发，我们研究了Arnold（1998 Random Dynamical Systems（柏林：Springer））提出的一类模型系统，它们表现出非自治Hopf分岔的完整两步情形.这些模型的特定结构允许对分叉模式进行严格和彻底的分析。特别是，我们证明了存在一个不变的“广义环面”分裂了以前稳定的中心流形后的第二个分歧点。在两种不同的设置中描述了该场景。首先，我们考虑确定性强迫模型，它可以被视为一个紧凑的产品空间上的连续斜积系统。其次，我们处理随机强迫系统，这导致在一个测度保持基变换的斜积。在随机情形下，需要随机动力系统的半一致遍历定理，以弥补紧致性的不足。
Inspired by an example of Grebogi et al (1984 Physica D 13 261–8), we study a class of model systems which exhibit the full two-step scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold (1998 Random Dynamical Systems (Berlin: Springer)). The specific structure of these models allows a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant ‘generalised torus’ splitting off a previously stable central manifold after the second bifurcation point. The scenario is described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew product systems on a compact product space. Secondly, we treat randomly forced systems, which lead to skew products over a measure-preserving base transformation. In the random case, a semiuniform ergodic theorem for random dynamical systems is required, to make up for the lack of compactness.