On the persistence of quasi-periodic invariant tori for double Hopf bifurcation of vector fields
On the persistence of quasi-periodic invariant tori for double Hopf bifurcation of vector fields
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DOI：
10.1016/j.jde.2016.01.025
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发表时间：
2016-05
影响因子：
2.4
通讯作者：
Xuemei Li
中科院分区：
文献类型：
--
作者：
Xuemei Li
我们使用 KAM 方法分析了双 Hopf (Hopf-Hopf) 分岔的准周期不变 2- 和 3-tori 的持久性。我们证明，在分叉点足够小的邻域中，整个系统对于大多数参数集都具有准周期 2-环面，其中截断的范式具有 2-环面。在适当的条件下，我们发现整个系统在其截断正规形式的 Hopf 分岔曲线附近并沿着分岔方向也具有准周期 3-环面，并且这些 3-环面从不变的 2-环面分叉。我们还给出了基于截断范式系数的准周期不变2-环和3-环存在性的具体公式。
We analyze the persistence of quasi-periodic invariant 2- and 3-tori for the double Hopf (Hopf–Hopf) bifurcation by using the KAM method. We prove that in a sufficiently small neighborhood of the bifurcation point, the full system has quasi-periodic 2-tori for most of the parameter sets where its truncated normal form possesses 2-tori. Under appropriate conditions we obtain that the full system also has quasi-periodic 3-tori for most parameters near the Hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation, and these 3-tori bifurcate from invariant 2-tori. We also give concrete formulas on the existence of quasi-periodic invariant 2- and 3-tori, which are based on coefficients of the truncated normal form.