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Comments on “Asymptotically stable equilibrium points in new chaotic systems”
Comentarios sobre “Puntos de equilibrio asintóticamente estables en nuevos sistemas caóticos”
A. Algaba1
F. Fernández-Sánchez 2
M. Merino1
A.J. Rodríguez-Luis2
*
1Departamento de Ciencias Integradas, Centro de Investigación de Física Teórica y Matemática FIMAT, Universidad de Huelva, Huelva. España.
2 Departamento de Matemática Aplicada II, E.T.S. Ingenieros, Universidad de Sevilla, Sevilla España.
Abstract
In the commented paper ten nonlinear chaotic systems are presented. Authors state that these systems do not exhibit Shilnikov chaos. Unfortunately, this assertion is not correctly proved because they use an erroneous theorem from the literature.
Keywords: chaotic systems; asymptotically stable equilibrium; non-existence of Shilnikov chaos; Lyapunov exponents
Resumen
En el trabajo comentado, los autores presentan diez sistemas autónomos no lineales caóticos, de los que afirman que no tienen caos en el sentido de Shilnikov. Desgraciadamente, esta afirmación carece de fundamento pues utilizan un teorema erróneo de la literatura.
Palabras clave: sistemas caóticos; equilibrio asintóticamente estable; no existencia del caos de Shilnikov; exponentes de Lyapunov
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
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.
[ Links ]
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.
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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.
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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.
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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.
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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.
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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.
[ Links ]
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.
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