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Invariant Algebraic Surfaces and Impasses

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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.

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Funding

Paulo R. da Silva is partially supported by CAPES and FAPESP. Otavio H. Perez is partially supported by Sao Paulo Research Foundation (FAPESP) Grant 2016/22310-0, and by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

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  1. Institute of Biosciences, Humanities and Exact Sciences, São Paulo State University (Unesp), Rua C. Colombo, 2265, S. J. Rio Preto, São Paulo, 15054–000, Brazil

    Paulo Ricardo da Silva & Otavio Henrique Perez

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  1. Paulo Ricardo da Silva
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  2. Otavio Henrique Perez
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Correspondence to Paulo Ricardo da Silva.

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da Silva, P.R., Perez, O.H. Invariant Algebraic Surfaces and Impasses. Qual. Theory Dyn. Syst. 20, 23 (2021). https://doi.org/10.1007/s12346-021-00465-x

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