Skip to main content
SpringerLink
Log in

Integrability of the Rucklidge system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We study the Darboux and the analytic integrability of the Rucklidge system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Christopher, C., Llibre, J., Pereira, J.V.: Multiplicity of invariant algebraic curves in polynomial vector fields. Pacific J. Math. 229, 63–117 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Darboux, G.: Mèmoire sur les équations différentielles algébraiques du premier ordre et du premier degré (Métanges), Bull. Sci. Math. 2éme série 2, 60–96; 123–144; 151–200 (1878)

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

  4. Irving, R.S.: Integers, Polynomials and Rings. Springer-Verlag, New York (2009)

    Google Scholar 

  5. Llibre, J., Zhang, X.: On the Darboux integrability of the polynomial differential systems. Qual. Theory Dyn. Syst. 11, 129–144 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Llibre, J., Zhang, X.: Darboux theory of integrability for polynomial vector fields in \({\mathbb{R}}^n\) taking into account the multiplicity at infinity. Bull. Sci. Math. 133, 765–778 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Li, W., Llibre, J., Zhang, X.: Local first integrals of differential systems and diffeomorphisms. Angew. Math. Phys. 54, 235–255 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hu, Z., Aldazharova, M., Aldibekov, T.M., Romanovski, V.: Integrability of 3-dim polynomial systems with three invariant planes. Nonlinear Dyn. 74, 1077–1092 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Rucklidge, A.M.: Chaos in models of double convection. J. Fluid Mech. 237, 209–229 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Svetoslav, N.: First Lyapunov value and bifurcation behavior of specific class of three-dimensional systems. Int. J. Bifurcation Chaos 14, 2811–2823 (2004)

    Article  MATH  Google Scholar 

  11. Wang, X.: Si’lnikov chaos and Hopf bifurcation analysis of Rucklidge system. Chaos Solitions Fractals 42, 2208–2217 (2009)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The first author is partially supported by Fapesp Grant Number 2013/15941-5. The second author is partially supported by a MINECO/FEDER Grant MTM2008-03437, a CIRIT Grant Number 2009SGR-410, by ICREA Academia, and Grant FP7-PEOPLE-2012-IRSES 318999, and FEDER/UNAB10-4E-378. Both, first and second authors, are also supported by the joint project CAPES-MECD Grant PHB-2009-0025-PC and by FP7-PEOPLE-2012-IRSES 316338. The third author has been supported by FCT (Grant PTDC/MAT/117106/2010 and through CAMGSD).

Author information

Authors and Affiliations

  1. Centro de Matematica Computação e Cognição, Universidade Federal do ABC, Santo Andre, SP, 09210-170, Brazil

    Maurício F. S. Lima

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

    Jaume Llibre

  3. Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal

    Clàudia Valls

Authors
  1. Maurício F. S. Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaume Llibre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Clàudia Valls
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maurício F. S. Lima.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lima, M.F.S., Llibre, J. & Valls, C. Integrability of the Rucklidge system. Nonlinear Dyn 77, 1441–1453 (2014). https://doi.org/10.1007/s11071-014-1389-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1389-y

Keywords

Search

Navigation