Dynamics of the FitzHugh–Nagumo system having invariant algebraic surfaces


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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  1. 1.

    Álvarez, M., Ferragut, A., Jarque, X.: A survey on the blow up technique. Int. J. Bifurc. Chaos Appl. Sci. Eng. 21, 3103–3118 (2011)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Arioli, G., Koch, H.: Existence and stability of traveling pulse solutions of the FitzHugh–Nagumo equation. Nonlinear Anal. 113, 51–70 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Briuno, A.: Local Methods in Nonlinear Differential Equations. Springer, New York (1989)

    Book  Google Scholar 

  4. 4.

    Brunella, M., Miari, M.: Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra. J. Differ. Equ. 85, 338–366 (1990)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cima, A., Llibre, J.: Bounded polynomial vector fields. Trans. Am. Math. Soc. 318, 557–579 (1990)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dumortier, F.: Techniques in the theory of local bifurcations: blow-up, normal forms, nilpotent bifurcations, singular perturbations. In: Schlomiuk, D. (ed.) Bifurcations and periodic orbits of vector fields, pp. 19–73. Springer, New York (1993)

    Google Scholar 

  7. 7.

    Dumortier, F., Llibre, J., Artés, J.: Qualitative Theory of Planar Differential Systems. Springer, New York (2006)

    MATH  Google Scholar 

  8. 8.

    FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445 (1961)

    Article  Google Scholar 

  9. 9.

    Flores, G.: Stability analysis for the slow travelling pulse of the FitzHugh–Nagumo system. SIAM J. Math. Anal. 22, 392–399 (1991)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gantmacher, F.R.: Applications of the Theory of Matrices. Interscience Publishers Ltd., London (1959)

    MATH  Google Scholar 

  11. 11.

    Gao, W., Wang, J.: Existence of wavefronts and impulses to FitzHugh–Nagumo equations. Nonlinear Anal. 57, 667–676 (2004)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hupkes, H.J., Sandstede, B.: Stability of pulse solutions for the discrete FitzHugh–Nagumo system. Trans. Am. Math. Soc. 365, 251–301 (2013)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Jones, C.K.R.T.: Stability of the travelling wave solution of the FitzHugh–Nagumo system. Trans. Am. Math. Soc. 286, 431–469 (1984)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Liu, W., Van Vleck, E.: Turning points and traveling waves in FitzHugh–Nagumo type equations. J. Differ. Equ. 225, 381–410 (2006)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Llibre, J., Messias, M., da Silva, P.R.: Global dynamics of the Lorenz system with invariant algebraic surfaces. Int. J. Bifurc. Chaos Appl. Sci. Eng. 20, 3137–3155 (2010)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Llibre, J., Oliveira, R.D.S.: Quadratic systems with invariant straight lines of total multiplicity two having Darboux invariants. Commun. Contempl. Math., 17, 1450018, 17 (2015)

  17. 17.

    Llibre, J., Valls, C.: Analytic first integrals of the FitzHugh–Nagumo systems. Z. Angew. Math. Phys. 60, 237–245 (2009)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Llibre, J., Valls, C.: Liouvillian integrability of the FitzHugh–Nagumo systems. J. Geom. Phys. 60, 1974–1983 (2010)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)

    Article  Google Scholar 

  20. 20.

    Neumann, D.: Classification of continuous flows on \(2\)-manifolds. Proc. Am. Math. Soc. 48, 73–81 (1975)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Valls, C.: On the global dynamics of the Newell–Whitehead system. J. Nonlinear Math. Phys. 26, 569–578 (2019)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Zhang, L., Yu, J.: Invariant algebraic surfaces of the FitzHugh–Nagumo system. J. Math. Anal. Appl., 483, 123097, 19 (2020)

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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


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