The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon’s system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different regions in each of which the system exhibits qualitatively the same behavior. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist.

中文翻译：

---

狄克逊系统没有混乱

所谓的 Dixon 系统经常被引用为二维（连续）动态系统的示例，如果它的两个参数在某个域中取值，则它表现出混沌行为。我们首先提供一个严格的证明，证明 Dixon 系统中没有混沌。然后我们对系统进行了完整的分岔分析，表明参数空间可以分解为 16 个不同的区域，在每个区域中系统表现出质量上相同的行为。特别是，我们证明了在某些区域存在两个具有无限多同宿轨道的椭圆扇区。