Abstract
We want to draw attention to the system of first order piecewise linear difference equations of two equations \(x_{n+1} = |x_n| - y_n - b\) and \(y_{n+1} = x_n - |y_n| - d\), \(n=0,1,2,..., (x_0, y_0)\in \textbf{R}^2\), where the parameters b and d are any positive real numbers. We will show that this system has interesting behavior compared to other similar systems. We show that there exists an unstable equilibrium \((d,-b)\). It has been shown that there are no solutions with period 2 and 3, but depending on the values of parameters b and d there are solutions with periods 5, 6, 7, 11, 12, 13, 16, 17, 18, 19, 20, 24, 25, 27, 30, 36. We have a hypothesis that all solutions are eventually periodic solutions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aiewcharoen, B., Boonklurb, R., Konglawan, N.: Global and local behavior of the system of piecewise linear difference equations \(x_{n+1} = |x_n| - y_n -b\) and \(y_{n+1} = x_n - | y_n| +1\) where \(b\ge 4\). Math. 9(12), 1390 (2021). https://doi.org/10.3390/math9121390
Bula, I.: Periodic solutions of the second order quadratic rational difference equation \(x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}}\) . 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 and Statistics, vol. 180, pp. 29–47. Springer, Berlin, Heidelberg (2016)
Devaney, R.L.: A piecewise linear model of the the zones of instability of an area-preserving map. Phys. D 10, 387–393 (1984)
Grove, E.A., Ladas, G: Periodicities in Nonlinear Difference Equations. Chapman Hall, New York (2005)
Grove, E.A., Lapierre, E., Tikjha, W.: On the global behavior of \(x_{n+1} = |x_n| - y_n -1\) and \(y_{n+1} = x_n + | y_n|\). CUBO 14(2), 111–152 (2012)
Koeddit, S., Tikjha, W.: Periodic solution of a piecewise linear system of difference equations with initial condition in positive X-axis. Burapha Sci. J. 26(3), 12 (2021)
Krisuk, S., Soprom, K., Pantain, J., Tikjha, W.: Equilibrium solution on a two-dimensional piecewise linear map. KKU Sci. J. 50(3), 223–230 (2022)
Kulenovic, M.R.S., Merino, O.: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/CRC, New York (2002)
Lapierre, E.: On the global behaviour of a systems of piecewise linear difference equations, 179 p. Doctoral Dissertation, University of Rhode Island (2013)
Lapierre, E., Tikjha, W.: On the global behaviour of some systems of difference equations. Int. J. Dyn. Syst. Differ. Equ. 11, 341–358 (2021)
Lozi, R.: Un attracteur etrange du type attracteur de Henon. J. Phys. 39, 9–10 (1978)
Tikjha, W., Lapierre, E.G., Lenbury, Y.: On the global character of the system of piecewise linear difference equations \(x_{n+1} = |x_n| - y_n -1\) and \(y_{n+1} = x_n - | y_n|\). Adv. Differ. Equ. 14 (2010). Article ID 573281
Tikjha, W.: Boundedness of some rational systems of difference equations and the global character of some piecewise linear system of difference equations. Doctoral Dissertation, Department of Mathematics, Mahidol University, Bangkok (2011)
Tikjha, W., Lapierre, E.G., Lenbury, Y.: Periodic solutions of a generalized system of piecewise linear difference equations. Adv. Differ. Equ. 15 (2015). Paper No. 248
Tikjha, W., **tanasonti, S., Lenbury, Y.: Periodic behavior of solutions of a certain piecewise linear system of difference equations. Thai J. Math. 13(1) (2015)
Tikjha, W., Lapierre, E., Sitthiwirattham, T.: The stable equilibrium of a system of piecewise linear difference equations. Adv. Differ. Equ. 10 (2017). Paper No. 67
Tikjha, W., Lapierre, E.: On the periodic behaviour of a system of piecewise linear difference equations. In: Elaydi, S., Hamaya, Y., Matsunaga, H., Pötzsche, C. (eds.) Advances in Difference Equations and Discrete Dynamical Systems, ICDEA 2016. Springer Proceedings in Mathematics and Statistics, vol. 212, pp. 275–282. Springer, Singapore
Tikjha, W., Lapierre, E.: Periodic solutions of a system of piecewise linear difference equations. Kyungpook Math. J. 60, 401–413 (2020)
Tikjha, W., Piasu, K.: A necessary condition for eventually equilibrium or periodic to a system of difference equations. J. Comput. Anal. Appl. 28(2), 254–261 (2020)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bula, I., Sīle, A. (2024). About a System of Piecewise Linear Difference Equations with Many Periodic Solutions. In: Olaru, S., Cushing, J., Elaydi, S., Lozi, R. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2022. Springer Proceedings in Mathematics & Statistics, vol 444. Springer, Cham. https://doi.org/10.1007/978-3-031-51049-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-031-51049-6_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-51048-9
Online ISBN: 978-3-031-51049-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)