The Dynamics of a Polynomial Differential System in $$\mathbb {R}^3$$

Abstract We study aspects of the global dynamics of a family of polynomial differential systems in $$\mathbb {R}^3$$ related to a chaotic system with non-isolated equilibria. Using the Poincaré compactification, we analyze the dynamics at infinity and determine the qualitatively distinct configurations arising for the parameter family. Under a suitable restriction on the parameters, the system admits a family of invariant surfaces, allowing the three-dimensional dynamics to be studied through the corresponding planar systems. We characterize the phase portraits on these invariant surfaces by combining the local analysis of finite and infinite equilibria with the global constraints imposed by the invariant surfaces. When the local analysis does not uniquely determine the global configuration, computational investigations using the P4 program are employed to identify concrete phase portraits compatible with the analytically obtained conditions. The results provide an analytical characterization of several global structures of the system and illustrate a methodology that can be applied to the study of polynomial differential systems in $$\mathbb {R}^2$$ and $$\mathbb {R}^3$$ .