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arXiv 1402.7189math.DS
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分支慢流形附近的周期轨道

Periodic orbits near a bifurcating slow manifold

Kristian Uldall Kristiansen

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摘要：

研究了一类具有慢变相位的$1\frac12$ -自由度哈密顿系统的哈密尔顿分岔.本文的主要结果是，存在一个顺序的$\ln^2ε^{-1}$ -许多周期轨道，所有停留在一个$\mathcal O(ε^{1/3})$ -距离从工会的正常椭圆形慢流形发生的结果的分歧。这里，$ε\ll 1$测量时间尺度间隔。这些周期轨道主要是不稳定的。证明是基于平均的两个爆破系统，允许一个估计的奇异性的影响，结合第二Painleve方程的渐近结果。通过这些方法{持久地}获得的最小振幅的稳定轨道仍然稍微远离慢流形，其距离为$\mathcal O(ε^{1/3}\ln^{1/2}\ln ε^{-1})$阶。

英文摘要：

This paper studies a class of $1\frac12$-degree-of-freedom Hamiltonian systems with a slowly varying phase that unfolds a Hamiltonian pitchfork bifurcation. The main result of the paper is that there exists an order of $\ln^2ε^{-1}$-many periodic orbits that all stay within an $\mathcal O(ε^{1/3})$-distance from the union of the normally elliptic slow manifolds that occur as a result of the bifurcation. Here $ε\ll 1$ measures the time scale separation. These periodic orbits are predominantly unstable. The proof is based on averaging of two blowup systems, allowing one to estimate the effect of the singularity, combined with results on asymptotics of the second Painleve equation. The stable orbits of smallest amplitude that are {persistently} obtained by these methods remain slightly further away from the slow manifold being distant by an order $\mathcal O(ε^{1/3}\ln^{1/2}\ln ε^{-1})$.

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