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Unfolding a Bykov Attractor: From an Attracting Torus to Strange Attractors
Journal of Dynamics and Differential Equations ( IF 1.3 ) Pub Date : 2020-07-04 , DOI: 10.1007/s10884-020-09858-z
Alexandre A. P. Rodrigues

In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different Morse indices. After slightly increasing both parameters, while keeping the one-dimensional connections unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the equilibria do not intersect. Under some conditions on the parameters and on the eigenvalues of the linearisation of the vector field at the saddle-foci, we prove the existence of many complicated dynamical objects, ranging from an attracting quasi-periodic torus to Hénon-like strange attractors, as a consequence of the Torus-Breakdown Theory. The mechanism for the creation of horseshoes and strange attractors is also discussed. Theoretical results are applied to show the occurrence of strange attractors in some analytic unfoldings of a Hopf-zero singularity.



中文翻译:

展开Bykov吸引子:从吸引环到奇怪吸引子

在本文中,我们为作用于三维球体的两参数微分方程族提供了一种奇异吸引子出现的综合机制。当两个参数均为零时,其流量表现出由两个一维连接和两个具有不同摩尔斯指数的双曲线鞍形焦点之间的一个二维分离线构成的吸引异宿网络(Bykov网络)。在略微增加两个参数之后,在保持一维连接不变的情况下,我们将注意力集中在平衡的二维不变流形不相交的情况下。在某些条件下,在鞍形焦点上矢量场线性化的参数和特征值上,我们证明了存在许多复杂的动力学对象,环面崩溃理论。还讨论了创建马蹄铁和奇怪吸引子的机制。理论结果被用于显示在Hopf零奇点的某些解析展开中奇怪吸引子的出现。

更新日期:2020-07-05
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