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On the Pitchfork Bifurcation of the Folded Node and Other Unbounded Time-Reversible Connection Problems in $\mathbb R^3$
SIAM Journal on Applied Dynamical Systems ( IF 1.7 ) Pub Date : 2020-09-15 , DOI: 10.1137/20m1326180
Kristian U. Kristiansen

SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2059-2102, January 2020.
In this paper, we revisit the folded node and the bifurcations of secondary canards at resonances $\mu\in \mathbb N$. In particular, we prove for the first time that pitchfork bifurcations occur at all even values of $\mu$. Our approach relies on a time-reversible version of the Melnikov approach in [M. Wechselberger, Dynam. Syst., 17 (2002), pp. 215--233] used in [M. Wechselberger, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101--139] to prove the transcritical bifurcations for all odd values of $\mu$. 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 $0<\epsilon\ll 1$. 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.


中文翻译:

关于$ \ mathbb R ^ 3 $中折叠节点的干草叉分叉和其他无界的时间可逆连接问题

SIAM应用动力系统杂志,第19卷第3期,第2059-2102页,2020年1月。
在本文中,我们将重新讨论共振点$ \ mu \ in \ mathbb N $中折叠后的节点和二级鸭嘴的分叉。特别是,我们首次证明在$ \ mu $的所有偶数值上都会出现干草叉分叉。我们的方法依赖于[M. 威南(Wechselberger),Dynam。Syst。,17(2002),pp.215--233] Wechselberger,SIAM J. Appl。达因 Syst。,4(2005),pp.101--139]证明了\ mu $的所有奇数值的跨临界分叉。已知由跨临界和干草叉分叉产生的次级鸭囊仅到达每个跨临界分叉的一侧上的费尼切尔慢流形,所有$ 0 <ε> 11 $。在本文中,我们为此提供了新的几何解释,依靠正常形式的对称性和折叠线的单独爆炸。我们还表明,我们的评估折叠节点梅尔尼科夫积分的方法-基于不变量流形的局部特征,通过高阶变分方程并将其简化为不均匀的韦伯方程-适用于一般的,二次的,时间为$ \ mathbb R ^ 3 $中的可逆,无界连接问题。我们通过使用我们的方法得出本文的结论,以提出关于Falkner-Skan方程中无穷远的周期性轨道分叉的新证明。$ \ mathbb R ^ 3 $中的二次,时间可逆,无界连接问题。我们通过使用我们的方法来总结本文,以提出关于Falkner-Skan方程中无穷远的周期性轨道分叉的新证明。$ \ mathbb R ^ 3 $中的二次,时间可逆,无界连接问题。我们通过使用我们的方法得出本文的结论,以提出关于Falkner-Skan方程中无穷远的周期性轨道分叉的新证明。
更新日期:2020-09-15
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