Abstract
In this paper we continue and complete with numerical examples, the study of chaos, in Li–Yorke and Marotto sense of difference equations of order two polynomials and rational, initiated in Balibrea and Cascales (Li-Yorke chaos in difference equations of order two polynomials and rationals [3]). Such equations arise, for example, in the application of the Newton’s method to polynomials equations, models of population dynamics and control problems. We use and review Marotto’s ideas from Marotto (J. Math. Anal. Appl. 63:199, 1978 [14]) based on a subtle study of the dynamics near a special kind of equilibrium (snap-back repeller). We claim that in the second order setting, there are cases which can be seen as two dimensional perturbations of equations of first order which are easier to handle. In this sense we continue applying results from Marotto (J. Math. Anal. Appl. 72:716–729, 1979 [15]). As an example we compute numerically the snap-back repellers of the inverse logistic first order equation whose perturbation allow to obtain some rational two order equations with chaotic dynamics. We apply the stated results and conjectures to the rational competition model of Hassell and Comins (Theor. Popul. Biol. 9:202–221, 1976 [6]).
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Acknowledgments
This paper has been partially supported by Grants MTM2011-232211 and CGL2008-05688-C02-02 from Ministerio de Ciencia e Innovación (Spain), Project 08667/PI-08 Fundación Séneca de la Comunidad Autónoma de Murcia (Spain) and Grant PEII09-0220-0222 from Junta de Comunidades de Castilla La Mancha (Spain).
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Balibrea, F., Cascales, A. (2016). Li–Yorke Chaos in Perturbed Rational Difference Equations. In: Alsedà i Soler, L., Cushing, J., Elaydi, S., Pinto, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2012. Springer Proceedings in Mathematics & Statistics, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52927-0_4
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