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A Review on Some Bifurcations in the Lorenz System

  • Antonio Algaba
  • M. Cinta Domínguez-Moreno
  • Manuel Merino
  • Alejandro J. Rodríguez-Luis
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

In this chapter, we review some bifurcations exhibited by the classical Lorenz system, where the parameters can have any real value. Analytical results on the pitchfork, Hopf and Takens–Bogdanov bifurcations of the origin, as well as the Hopf bifurcation of the nontrivial equilibria, are summarized. These results serve as a guide for the numerical study that reveals other important organizing centers of the dynamics: Takens–Bogdanov bifurcations of periodic orbits, torus bifurcations and the resonances associated, homoclinic and heteroclinic connections with several degeneracies, etc. We also point out that the analysis of the Hopf-pitchfork and the triple-zero bifurcations of the origin cannot be performed with the usual tools and propose a way to carry out this study avoiding the structural singularities exhibited by the Lorenz system.

Keywords

Lorenz Bifurcation Pitchfork Hopf Takens–Bogdanov Torus Resonances Hopf-pitchfork Triple-zero 

Notes

Acknowledgements

This work has been partially supported by the Ministerio de Economía y Competitividad, Plan Nacional I+D+I co-financed with FEDER funds, in the frame of the project MTM2014-56272-C2, and by the Consejería de Economía, Innovación, Ciencia y Empleo de la Junta de Andalucía (FQM-276, TIC-0130 and P12-FQM-1658).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Antonio Algaba
    • 1
  • M. Cinta Domínguez-Moreno
    • 1
  • Manuel Merino
    • 1
  • Alejandro J. Rodríguez-Luis
    • 2
  1. 1.Departamento de Ciencias Integradas, Centro de Investigación de Física Teórica y Matemática FIMATUniversidad de HuelvaHuelvaSpain
  2. 2.Departamento de Matemática Aplicada II, E.S. IngenierosUniversidad de SevillaSevillaSpain

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