Acta Mathematica Sinica, English Series

, Volume 34, Issue 6, pp 947–958 | Cite as

On the Darboux integrability of the Hindmarsh–Rose burster

  • Jaume Llibre
  • Clàudia Valls


We study the Hindmarsh–Rose burster which can be described by the differential system
$$\dot x = y - {x^3} + b{x^2} + I - z,\dot y = 1 - 5{x^2} - y,\dot z = \mu \left( {s\left( {x - {x_0}} \right) - z} \right),$$
where b, I, μ, s, x0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist.


Polynomial integrability rational integrability Darboux polynomials Darboux first integrals invariant algebraic surfaces exponential factors Hindmarsh–Rose burster 

MR(2010) Subject Classification

34C05 34A34 34C14 


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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, Barcelona, CataloniaSpain
  2. 2.Departamento de Matemática, Instituto Superior TécnicoUniversidade Técnica de LisboaLisboaPortugal

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