Casas-García, Quezada-Téllez, Carrillo-Moreno, Flores-Godoy, and Fernández-Anaya: Respuesta a: Comentarios sobre “Puntos de equilibrio asintóticamente estables en nuevos sistemas caóticos”



We would like to acknowledge the information about Theorem 1 from (Elhadj and Sprott, 2012), used in our article (Casas-García et al. 2016) is incorrect, as it was shown in (Algaba et al. 2013). This opens up the possibility that the systems that we find have homoclinic or heteroclinic orbits, and may appears chaos in the Shilnikov sense, making its dynamics more interesting, and opening the possibility of deeper analysis for these systems.

On the other hand, we would like to point out that the fundamental contribution of our paper lies in the fact of finding ten simple, three-dimensional dynamic systems with nonlinear quadratic terms, which have an asymptotically stable equilibrium point and present chaos, which was clearly shown in the our work. In addition, the other relevant contribution is the method to find the systems, using the Monte Carlo method, which was first proposed in (Carrillo et al. 2013) for this type of search for chaotic systems with special properties, and later used by (Sprott and Xiong, 2015).

In addition, we emphasize that the mention of the dynamic systems of Chen and Lü in the introduction of our article is only to establish some pioneering work in this field of nonlinear dynamics, which is usual in many articles in this area. We do not use them in the rest of the article.

Finally, for a broader discussion of comments similar to those made in our paper, we recommend viewing the comments (Algaba et al. 2012a), and replies (Zuo-Huan Zheng and Guanrong Chen, 2012), (Guochang et al. 2012). For a more recent independent analysis see (Leonov and Kuznetsov, 2015).

Acknowledgments

This work has been partially supported by the Fomento de Investigación y Cultura, A. C. Patronato Económico y de Desarrollo, and by the Dirección de Investigación both of the Universidad Iberoamericana Ciudad de México.

References

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A. Algaba F. Fernández-Sánchez M. Merino A. J. Rodríguez-Luis 2013Non-existence of Shilnikov chaos in continuous-time systemsApplied Mathematics and MechanicsEnglish34911751176

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Algaba A. , Fernández-Sánchez F. , Merino M. , and Rodríguez-Luis A. J. (2012a) “Comment on ‘Existence of heteroclinic orbits of the Shilnikov type in a 3D quadratic autonomous chaotic system’ [J. Math. Anal. Appl. 315, 106-119 (2006)]”. J. Math. Anal. Appl., 392, 99-101.

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