This paper reports numerical experiments done on a two-parameter family of vector fields which unfold an attracting heteroclinic cycle linking two saddle-foci. We investigated both local and global bifurcations due to symmetry breaking in order to detect either hyperbolic or chaotic dynamics. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is still out of reach, using a combination of theoretical tools and computer simulations we have uncovered some complex patterns. We have selected suitable initial conditions to analyze the bifurcation diagrams, and regarding these solutions we have located: (a) an open domain of parameters with regular dynamics; (b) infinitely many parabolic-type curves associated to homoclinic Shilnikov cycles which act as organizing centers; (c) a crisis region related to the destruction or creation of chaotic attractors; (d) a large Lebesgue measure set of parameters where chaotic regimes are dominant, though sinks and chaotic attractors may coexist, and in whose complement we observe shrimps.

中文翻译：

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贝科夫循环附近的实验可访问轨道

本文报告了对两个参数向量场族进行的数值实验，该向量场展开了一个连接两个鞍状焦点的吸引异宿循环。我们研究了由于对称性破坏引起的局部和全局分岔，以检测双曲线或混沌动力学。尽管对相应的分岔图和动态变化背后的机制的完整理解仍然遥不可及，但结合理论工具和计算机模拟，我们已经发现了一些复杂的模式。我们选择了合适的初始条件来分析分岔图，关于这些解决方案，我们找到了：（a）具有规则动力学的开放参数域；(b) 与作为组织中心的同宿 Shilnikov 循环相关的无数抛物线型曲线；(c) 与混乱吸引子的破坏或产生有关的危机区域；(d) 大量 Lebesgue 测量参数集，其中混沌状态占主导地位，尽管汇和混沌吸引子可能共存，并且我们观察虾的补充。