Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
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二次保体积映射：不变圆和分岔
DOI：
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发表时间：
2008
影响因子：
2.1
通讯作者：
J. Meiss
中科院分区：
文献类型：
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作者：
H. Dullin;J. Meiss
研究了$mathbb{R}^3$的保体积微分同态的五参数二次族的动力学性质。该族是具有3 - 1乘法器的不动点分岔的展开正规形式，也是具有二次逆的二次三维映射的一般形式。这张图的许多非平凡动力学发生在它的两个固定点是鞍形焦点，与二维稳定和不稳定流形相交，形成一个球形“涡泡”。我们证明这种情况发生在鞍中心neimmark - sacker （SCNS）分岔附近，该分岔至少在其正常形式下也会产生一个椭圆不变圆。我们开发了一种简单的算法来精确地计算这些椭圆不变圆及其纵向和横向旋转数，并用它来研究它们的分岔，并通过旋转数之间的共振来对它们进行分类。特别是，纵向旋转数的有理值会产生一串pe…
We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of $mathbb{R}^3$. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and is also the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical “vortex-bubble.” We show that this occurs near a saddle-center-Neimark–Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pe...