Periodic solutions of a generalized system of piecewise linear difference equations

Wirot Tikjha,Evelina Lapierre,Yongwimon Lenbury

doi:10.1186/s13662-015-0585-5

Wirot Tikjha, Evelina Lapierre + Show 1 more

Open Access

https://doi.org/10.1186/s13662-015-0585-5

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Abstract

In this article we consider the global behavior of the following system of piecewise linear difference equations: $x_{n+1}=|x_{n}|-y_{n}-b$ and $y_{n+1} = x_{n}+|y_{n}|$ where b is any positive real number and the initial condition $(x_{0}, y_{0})$ is an element of $R^{2}$ . By mathematical induction and direct computations we show that the solution to the system is eventually one of two particular prime period 3 solutions or the unique equilibrium solution.

Highlights

There has been a surge of interest in systems of both rational and piecewise linear difference equations due to their practical applications in evolutionary biology, neural networks, and ecology [ – ]
In this paper we consider the behavior of the generalized system of piecewise linear difference equations, xn+ = |xn| – yn – b, n =, . . . , yn+ = xn + |yn|, where the parameter b ∈ (, ∞) and the initial condition (x, y ) ∈ R
4 Discussion and conclusion The system of piecewise linear difference equations examined in this paper was created as a prototype to understand the global behavior of systems like the Lozi equation

Summary

Introduction

There has been a surge of interest in systems of both rational and piecewise linear difference equations due to their practical applications in evolutionary biology, neural networks, and ecology [ – ]. Is the equilibrium solution or eventually prime period cycle P or P . Lemma Let {(xn, yn)}∞ n= be a solution to system ( ) and suppose that there exists an integer N ≥ such that yN = –xN – b ≥ .

Results

Conclusion