Cocoon bifurcation in three-dimensional reversible vector fields
Cocoon bifurcation in three-dimensional reversible vector fields
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DOI：
10.1088/0951-7715/19/2/004
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发表时间：
2006-02
期刊：
影响因子：
1.7
通讯作者：
F. Dumortier;S. Ibáñez;H. Kokubu
中科院分区：
文献类型：
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作者：
F. Dumortier;S. Ibáñez;H. Kokubu
茧分岔是Lau（1992）在数值上观察到的一组丰富的分岔现象。j . Bifurc。混沌2 543-58)中的迈克尔逊系统，描述Kuramoto-Sivashinsky方程的行波的三维ODE系统。在可逆向量场的情况下，我们给出了一个更一般的茧分岔的主部组织中心。我们证明了在组织中心的一般展开中，在组织中心附近存在一个长度增加的异斜分岔级联，它类似于茧分岔的主体部分。我们还研究了一种称为可逆Bykov循环的异斜循环。这样的循环被认为发生在迈克尔逊系统中，也发生在约瑟夫森结的模型方程中（van den Berg等人2003非线性16 707-17）。我们推测，一个可逆的Bykov循环，在其展开时，是一个尖横向异斜链序列的积累点。在这个方向上的第一个结果，我们证明了可逆Bykov环是周期轨道的可逆一般鞍节点分岔的一个累加点，而周期轨道是尖-横向异斜链的主要成分。
The cocoon bifurcation is a set of rich bifurcation phenomena numerically observed by Lau (1992 Int. J. Bifurc. Chaos 2 543–58) in the Michelson system, a three-dimensional ODE system describing travelling waves of the Kuramoto–Sivashinsky equation. In this paper, we present an organizing centre of the principal part of the cocoon bifurcation in more general terms in the setting of reversible vector fields on . We prove that in a generic unfolding of an organizing centre called the cusp-transverse heteroclinic chain, there is a cascade of heteroclinic bifurcations with an increasing length close to the organizing centre, which resembles the principal part of the cocoon bifurcation. We also study a heteroclinic cycle called the reversible Bykov cycle. Such a cycle is believed to occur in the Michelson system, as well as in a model equation of a Josephson Junction (van den Berg et al 2003 Nonlinearity 16 707–17). We conjecture that a reversible Bykov cycle is, in its unfolding, an accumulation point of a sequence of cusp-transverse heteroclinic chains. As a first result in this direction, we show that a reversible Bykov cycle is an accumulation point of reversible generic saddle-node bifurcations of periodic orbits, the main ingredient of the cusp-transverse heteroclinic chain.