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Nonlinear Systems, Vol. 1 pp 3–36Cite as

A Review on Some Bifurcations in the Lorenz System

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Part of the book series: Understanding Complex Systems ((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.

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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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  1. Departamento de Ciencias Integradas, Centro de Investigación de Física Teórica y Matemática FIMAT, Universidad de Huelva, 21071, Huelva, Spain

    Antonio Algaba, M. Cinta Domínguez-Moreno & Manuel Merino

  2. Departamento de Matemática Aplicada II, E.S. Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Sevilla, Spain

    Alejandro J. Rodríguez-Luis

Authors
  1. Antonio Algaba
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  2. M. Cinta Domínguez-Moreno
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  3. Manuel Merino
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  4. Alejandro J. Rodríguez-Luis
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Corresponding author

Correspondence to M. Cinta Domínguez-Moreno .

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Editors and Affiliations

  1. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Victoriano Carmona

  2. Departamento de Física Aplicada I, Universidad de Sevilla, Sevilla, Spain

    Jesús Cuevas-Maraver

  3. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Fernando Fernández-Sánchez

  4. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Elisabeth García- Medina

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Algaba, A., Domínguez-Moreno, M.C., Merino, M., Rodríguez-Luis, A.J. (2018). A Review on Some Bifurcations in the Lorenz System. In: Carmona, V., Cuevas-Maraver, J., Fernández-Sánchez, F., García- Medina, E. (eds) Nonlinear Systems, Vol. 1. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-66766-9_1

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