In this paper by using the Poincaré compactification in ℝ3 we make a global analysis of the model x′ = z, y′ = b(x – dy), z′ = x(x2 – 1)+ y + cz with b ϵ ℝ and c, d ϵ ℝ+, here known as the three-dimensional Newell–Whitehead system. We give the complete description of its dynamics on the sphere at infinity. For some values of the parameters this system has invariant algebraic surfaces and for these values we provide the dynamics of the system restricted to these surfaces and its global phase portrait in the Poincaré ball. We also include the description of the α-limit and ω-limit set of its orbits in the Poincaré ball including its boundary, that is, in the compactification of ℝ3 with the sphere at the infinity. We recall that the restricted systems are not analytic and so in this paper we overcome this difficulty by using the blow-up technique.

中文翻译：

---

关于 Newell-Whitehead 系统的全局动力学

在本文中，通过使用ℝ3 中的庞加莱紧缩，我们对模型 x′ = z, y′ = b(x – dy), z′ = x(x2 – 1)+ y + cz 进行全局分析，其中 b ϵ ℝ和 c, d ϵ ℝ+，这里称为三维 Newell-Whitehead 系统。我们给出了它在无穷远球体上的动力学的完整描述。对于某些参数值，该系统具有不变的代数曲面，对于这些值，我们提供了系统的动力学限制在这些曲面及其在庞加莱球中的全局相图。我们还描述了庞加莱球中其轨道的 α 极限和 ω 极限集，包括其边界，即 ℝ3 与无穷远球体的紧化。我们记得受限系统不是解析系统，因此在本文中我们通过使用膨胀技术克服了这个困难。