Global Dynamics of Cubic Second Order Difference Equation in the First Quadrant

Authors

Jasmin Bektešević
Mustafa Kulenovic, University of Rhode IslandFollow
Esmir Pilav

Document Type

Article

Date of Original Version

6-11-2015

Abstract

We investigate the global behavior of a cubic second order difference equation x_n+1=Ax³_n+ Bx ²_nx_n−1 + Cx_nx²_n−1 + Dx³_n−1 + Ex²_n + Fx_nx_n−1 + Gx²_n−1 + Hx_n + Ix_n−1 + J, n = 0, 1, …, with nonnegative parameters and initial conditions. We establish the relations for the local stability of equilibriums and the existence of period-two solutions. We then use this result to give global behavior results for special ranges of the parameters and determine the basins of attraction of all equilibrium points. We give a class of examples of second order difference equations with quadratic terms for which a discrete version of the 16th Hilbert problem does not hold. We also give the class of second order difference equations with quadratic terms for which the Julia set can be found explicitly and represent a planar quadratic curve.

Citation/Publisher Attribution

Bektešević, J., Kulenović, M. R.S., & Pilav, E. (2015). Global dynamics of cubic second order difference equation in the first quadrant. Advances in Difference Equations, 2015, 176. doi: 10.1186/s13662-015-0503-x

Available at: http://dx.doi.org/10.1186/s13662-015-0503-x

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.