Invariant Algebraic Surfaces and Impasses

Abstract

Polynomial vector fields $$X:\mathbb {R}^3\rightarrow \mathbb {R}^3$$ that have invariant algebraic surfaces of the form $$\begin{aligned} M = \{ f(x,y)z - g(x,y) = 0\} \end{aligned}$$ are considered. We prove that trajectories of X on M are solutions of a constrained differential system having $$\mathcal {I}=\{f(x,y)=0\}$$ as impasse curve. The main goal of the paper is to study the flow on M near points that are projected on typical impasse singularities. The Falkner–Skan equation (Llibre and Valls in Comput Fluids 86:71–76, 2013), the Lorenz system (Llibre and Zhang in J Math Phys 43:1622–1645, 2002) and the Chen system (Lu and Zhang in Int J Bifurc Chaos 17–8:2739–2748, 2007) are some of the well-known polynomial systems that fit our hypotheses.