The roles of conic sections and elliptic curves in the global dynamics of a class of planar systems of rational difference equations

Sukanya Basu

doi:10.1186/1687-1847-2013-292

Abstract

Consider the class of planar systems of first-order rational difference equations 1´ where , and the parameters are nonnegative and such that both terms in the right-hand side of (1′) are nonlinear. In this paper, we prove the following discretized Poincare-Bendixson theorem for the class of systems (1′). If the map associated to system (1′) is bounded, then the following statements are true: In particular, system (1′) cannot exhibit chaos when its associated map is bounded. Moreover, we show that if both equilibrium curves of (1′) are reducible conics and the map associated to system (1′) is unbounded, then every solution converges to one of up to infinitely many equilibria or to or . MSC:39A05, 39A11.

Highlights

Introduction and main theoremConsider the system of first-order rational difference equations with nonnegative parameters xn+α +β xn +γ yn A +B xn +C yn yn+α +β xn +γ yn A +B xn +C yn, n =, . . . , (x, y ) ∈ R, ( )where R = {(x, y) ∈ [, ∞) : Aix + Biy + Ci =, i =, }, and the parameters are nonnegative and such that both terms in the right-hand side of ( ) are nonlinear
General solutions of planar linear discrete systems with constant coefficients and weak delays were studied by Diblík and Halfarová in [ ] and [ ]
The authors were able to use this result along with the results proved by Kulenović and Merino in [ ] to give a complete qualitative description of the global dynamics of (LG- )

Summary

Introduction

Theorem If the map associated to system ( ) is bounded with parameters in P, the following is true: (i) If both equilibrium curves of ( ) are reducible conics, that is, if i. Respectively deal with the number of nonnegative equilibria, local stability of equilibria, existence and uniqueness of minimal period-two solutions, and global behavior of solutions of system ( ) when both equilibrium curves are irreducible conics.

Objectives

Results

Conclusion