Algaba, Fernández-Sánchez, Merino, and Rodríguez-Luis: Comentarios sobre “Puntos de equilibrio asintóticamente estables en nuevos sistemas caóticos”



In the commented paper (Casas-García et al., 2016) the authors analyze ten nonlinear chaotic systems. As an important feature, they affirm that these systems do not exhibit Shilnikov chaos (see Abstract). To demonstrate this fact, first they copy, in Sect. 2, Definitions 1-3 and Theorem 1 from (Elhadj and Sprott, 2012) and state: Theorem 1 characterizes the conditions under which a system does not present homoclinic and heteroclinic orbits. From this information we can identify systems that present Smale’s horseshoe behavior. Second, they assert in Sect. 3: From Definitions 1-3 and Theorem 1 we know that the chaos presented is not of the Smale horseshoe type due to the fact that the systems does not contain homoclinic or heteroclinic orbits.

However, as we clearly demonstrate in (Algaba et al., 2013a), Theorem 1 is erroneous. Consequently, some of the ten systems considered might have homoclinic or heteroclinic orbits and then they might exhibit Shilnikov chaos. Therefore, the sentence stated in the Conclusions, The chaos behaviour of the studied systems is not of the class of Smale’s horseshoe type, due to their orbits are not either homoclinic nor heteroclinic in the sense Shilnikov, has not scientific basis.

We would like to add a last comment. Chen’s and Lü’s systems, cited in the commented paper, are only particular cases of the Lorenz system as it is demonstrated in (Algaba et al., 2013b, 2013c), by using a linear scaling in time and state variables. This fact is illustrated in (Algaba et al., 2014, 2015, 2016).

Acknowledgments

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 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).

References

1

Algaba A., Fernández-Sánchez F., Merino M. and Rodríguez-Luis A.J. (2013a). Comments on “Non-existence of Shilnikov chaos in continuous-time systems”, Applied Mathematics and Mechanics (English Edition), 34(9), 1175-1176.

A. Algaba F. Fernández-Sánchez M. Merino A.J. Rodríguez-Luis 2013Comments on “Non-existence of Shilnikov chaos in continuous-time systems”Applied Mathematics and MechanicsEdition34911751176

2

Algaba A. , Fernández-Sánchez F. , Merino M. and Rodríguez-Luis A.J. (2013b). Chen's attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz system, Chaos 23, 033108.

A. Algaba F. Fernández-Sánchez M. Merino A.J. Rodríguez-Luis 2013Chen's attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz systemChaos23033108

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Algaba A. , Fernández-Sánchez F. , Merino M. and Rodríguez-Luis A.J. (2013c). The Lü system is a particular case of the Lorenz system. Physics Letters A 377, 2771-2776.

A. Algaba F. Fernández-Sánchez M. Merino A.J. Rodríguez-Luis 2013The Lü system is a particular case of the Lorenz systemPhysics LettersA 37727712776

4

Algaba A. , Fernández-Sánchez F. , Merino M. and Rodríguez-Luis A.J. (2014). Centers on center manifolds in the Lorenz, Chen and Lü systems. Communications in Nonlinear Science and Numerical Simulation 19, 772-775.

A. Algaba F. Fernández-Sánchez M. Merino A.J. Rodríguez-Luis 2014Centers on center manifolds in the Lorenz, Chen and Lü systemsCommunications in Nonlinear Science and Numerical Simulation19772775

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Algaba A. , Domínguez-Moreno M.C., Merino M. and Rodríguez-Luis A.J. (2015). Study of the Hopf bifurcation in the Lorenz, Chen and Lü systems. Nonlinear Dynamics 79, 885-902.

A. Algaba M. Merino A.J. Rodríguez-Luis 2015Study of the Hopf bifurcation in the Lorenz, Chen and Lü systemsNonlinear Dynamics79885902

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Algaba A. , Domínguez-Moreno M.C., Merino M. and Rodríguez-Luis A.J. (2016). Takens-Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz system. . Communications in Nonlinear Science and Numerical Simulation 30, 328-343.

A. Algaba M. Merino A.J. Rodríguez-Luis 2016Takens-Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz systemCommunications in Nonlinear Science and Numerical Simulation30328343

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Casas-García K., Quezada-Téllez L.A., Carrillo-Moreno S., Flores-Godoy J.J. and Fernández-Anaya G. (2016) Asymptotically stable equilibrium points in new chaotic systems, Nova Scientia 8(16), 41-58.

K. Casas-García L.A. Quezada-Téllez S. Carrillo-Moreno J.J. Flores-Godoy G. Fernández-Anaya 2016Asymptotically stable equilibrium points in new chaotic systemsNova Scientia8164158

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Elhadj Z. and Sprott J.C. (2012). Non-existence of Shilnikov chaos in continuous-time systems. Applied Mathematics and Mechanics (English Edition), 33(3), 371-374.

Z. Elhadj J.C. Sprott 2012Non-existence of Shilnikov chaos in continuous-time systemsApplied Mathematics and MechanicsEdition333371374



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