CUBO A Mathematical Journal Vol.14, N^o'02, (111-152). June 2012

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On the global behavior of

 

E.A. Grove*, E. Lapierre** and W. Tikjha***

*Department of Mathematics, University of Rhode Island, Kingston, Rhode Island, 02881-0816, USA email: grove@math.uri.edu

**Department ofMathematics, Johnson and Wales University, Providence, Rhode Island02903, USA. email: elapierre@jwu.edu

***Faculty ofScience andTechnology, Pihulsongkram Rajabhat University, Muang District, Phitsanuloke,65000, Thailand email: wirot_tik@yahoo.com

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ABSTRACT

In this paper we consider the system of piecewise linear difference equations in the title, where the initial conditions x₀ and y₀ are real numbers. We show that there exists a unique equilibrium solution and exactly two prime period-3 solutions, and that except for the unique equilibrium solution, every solution of the system is eventually one of the two prime period-3 solutions.

Keywords and Phrases: Periodic solution; systems of piecewise linear difference equations 2010 AMS Mathematics Subject Classification: 39A10, 65Q10.

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RESUMEN

En este artículo consideramos el sistema de ecuaciones en diferencia lineales por partes indicado en el título, donde las condiciones iniciales x₀ e y₀ son números reales. Demostramos que existe una única solución de equilibrio y exactamente dos soluciones de período 3-primo, y que exceptuando la soluciún única de equilibrio, toda solucion del sistema es eventualmente una de las dos soluciones de periodo 3-primo.

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References

[1] E. Camouzis, and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, New York, 2008.         [ Links ]

[2] R.L. Devaney, A piecewise linear model of the the zones of instability of an area-preserving map, Physica 10D (1984), 387-393.         [ Links ]

[3] M.R.S. Kulenovic, and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman & Hall/CRC, New York, 2002.         [ Links ]

[4] H.O. Peitgen and D. Saupe, (eds.) The Science of Fractal Images, Springer-Verlog, New York,1991.         [ Links ]

[5] W. Tikjha, Y. Lenbury, and E. G. Lapierre, On the Global Character of the System of Piecewise Linear Difference Equations , Advances in Difference Equations, ,Volume 2010 (2010), Article ID 573281.         [ Links ]

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Received:November 2011.Revised:November 2011.