Dynamics of the FitzHugh–Nagumo system having invariant algebraic surfaces

Abstract

In this paper, we study the dynamics of the FitzHugh–Nagumo system \(\dot{x}=z,\;\dot{y}=b\left( x-dy\right) ,\;\dot{z}=x\left( x-1\right) \left( x-a\right) +y+cz\) having invariant algebraic surfaces. This system has four different types of invariant algebraic surfaces. The dynamics of the FitzHugh–Nagumo system having two of these classes of invariant algebraic surfaces have been characterized in Valls (J Nonlinear Math Phys 26:569–578, 2019). Using the quasi-homogeneous directional blow-up and the Poincaré compactification, we describe the dynamics of the FitzHugh–Nagumo system having the two remaining classes of invariant algebraic surfaces. Moreover, for these FitzHugh–Nagumo systems we prove that they do not have limit cycles.

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Acknowledgements

We thank the reviewer his/her comments that help us to improve the presentation of this paper.

J. Llibre is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación Grants M TM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR1617 and the H2020 European Research Council grant MSCA-RISE-2017-777911. Y. Tian is partially supported by the National Natural Science Foundation of China (Nos. 11971495 and 11801582), China Scholarship Council (No. 201906380022) and Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515011239).

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Correspondence to Yuzhou Tian.

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Llibre, J., Tian, Y. Dynamics of the FitzHugh–Nagumo system having invariant algebraic surfaces. Z. Angew. Math. Phys. 72, 15 (2021). https://doi.org/10.1007/s00033-020-01450-1

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Mathematics Subject Classification

  • Primary 37C10
  • Secondary 34C05
  • 37C70

Keywords

  • Global dynamics
  • FitzHugh–Nagumo system
  • Invariant algebraic surface
  • Poincaré compactification