论文标题
在折叠节点的干草叉分叉和其他无限时间可逆连接问题中
On the pitchfork bifurcation of the folded node and other unbounded time-reversible connection problems in $\mathbb R^3$
论文作者
论文摘要
在本文中，我们重新访问了折叠式的折叠节点和二次乳清的分叉，共鸣$μ\ in \ mathbb n $。特别是，我们首次证明了干草叉分叉的甚至$μ$。我们的方法依赖于\ cite {wechselberger2002a}的梅尔尼科夫方法的时间可逆版本，该方法在\ cite {wechselberger_existence_2005}中使用，以证明所有$μ$ $ $ $ $ $ $ $ $ $ $的cricentical crication crication crication crication crication crifiction crigital。众所周知，由跨批评和干草叉分叉产生的次要乳清仅适用于每个跨临界分叉的一侧的fenichel慢歧管，所有$ 0 <ε\ ll 1 $。在本文中，我们依靠正常形式的对称性和折叠线的单独爆炸提供了一个新的几何解释。我们还表明，我们评估折叠节点的Melnikov积分的方法 - 基于通过高阶变量方程对不变流形的局部表征，并将其简化为不均匀的Weber方程，适用于一般，典型，时间可交流，无限的，无限的连接问题，$ \ \ rathbbb r^3 $。我们通过使用我们的方法来介绍Falkner-Skan方程和Nosé方程中的周期性轨道分叉的新证明来结束本文。
In this paper, we revisit the folded node and the bifurcations of secondary canards at resonances $μ\in \mathbb N$. In particular, we prove for the first time that pitchfork bifurcations occur at all even values of $μ$. Our approach relies on a time-reversible version of the Melnikov approach in \cite{wechselberger2002a}, used in \cite{wechselberger_existence_2005} to prove the transcritical bifurcations for all odd values of $μ$. It is known that the secondary canards produced by the transcritical and the pitchfork bifurcations only reach the Fenichel slow manifolds on one side of each transcritical bifurcation for all $0<ε\ll 1$. In this paper, we provide a new geometric explanation for this fact, relying on the symmetry of the normal form and a separate blowup of the fold lines. We also show that our approach for evaluating the Melnikov integrals of the folded node -- based upon local characterization of the invariant manifolds by higher order variational equations and reducing these to an inhomogeneous Weber equation -- applies to general, quadratic, time-reversible, unbounded connection problems in $\mathbb R^3$. We conclude the paper by using our approach to present a new proof of the bifurcation of periodic orbits from infinity in the Falkner-Skan equation and the Nosé equations.
