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Resonances of periodic orbits in the Lorenz system

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Abstract

Usually, the physical interest of the Lorenz system is restricted to the region where its three parameters are positive. However, this famous system appears, when \(\sigma <0\), in the study of a thermosolutal convection model and in the analysis of traveling-wave solutions of the Maxwell–Bloch equations. In this context, a Takens–Bogdanov bifurcation of heteroclinic type becomes an important organizing center. It has been very recently shown that the periodic orbit born in the Hopf bifurcation of the origin undergoes a torus bifurcation. In this paper we perform a detailed numerical study of the resonances of periodic orbits in the three-parameter Lorenz system, \( \dot{x} = \sigma (y-x), \ \dot{y} = \rho x - y - xz, \ \dot{z} = -bz + xy, \) when \(\sigma <0\) and \(\rho , b >0\). The combination of numerical continuation methods and Poincaré sections of the flow provides important information of how the resonances appear and evolve giving rise to a very rich dynamical and bifurcation scenario.

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Authors and Affiliations

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

    Antonio Algaba & Manuel Merino

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

    Estanislao Gamero & Alejandro J. Rodríguez-Luis

Authors
  1. Antonio Algaba
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  2. Estanislao Gamero
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  3. Manuel Merino
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  4. Alejandro J. Rodríguez-Luis
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Correspondence to Alejandro J. Rodríguez-Luis.

Additional information

This work has been partially supported by the Ministerio de Educación y Ciencia, Plan Nacional I+D+I co-financed with FEDER funds, in the frame of the projects MTM2010-20907-C02 and 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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Algaba, A., Gamero, E., Merino, M. et al. Resonances of periodic orbits in the Lorenz system. Nonlinear Dyn 84, 2111–2136 (2016). https://doi.org/10.1007/s11071-016-2632-5

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