tailieunhanh - ON THE ALGEBRAIC DIFFERENCE EQUATIONS un+2 un = ψ(un+1 ) IN R+ , RELATED TO A FAMILY ∗ OF

ON THE ALGEBRAIC DIFFERENCE EQUATIONS un+2 un = ψ(un+1 ) IN R+ , RELATED TO A FAMILY ∗ OF ELLIPTIC QUARTICS IN THE PLANE G. BASTIEN AND M. ROGALSKI Received 20 October 2004 and in revised form 27 January 2005 We continue the study of algebraic difference equations of the type un+2 un = ψ(un+1 ), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics Q(K) of the plane. We prove, as in “on some algebraic difference equations un+2 un = ψ(un+1 ) in R+ , related to families of. | ON THE ALGEBRAIC DIFFERENCE EQUATIONS un 2un V un 1 IN R RELATED TO A FAMILY OF eLliptiC Quartics in the plane G. BASTIEN AND M. ROGALSKI Received 20 October 2004 and in revised form 27 January 2005 We continue the study of algebraic difference equations of the type un 2un y un 1 which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics Q K of the plane. We prove as in on some algebraic difference equations un 2un y un 1 in RỊ related to families of conics or cubics generalization of the Lyness sequences 2004 that the solutions Mn un 1 un are persistent and bounded move on the positive component Q0 K of the quartic Q K which passes through M0 and diverge if M0 is not the equilibrium which is locally stable. In fact we study the dynamical system F x y a bx cx2 y c dx x2 x a b c d e R 4 a b 0 b c d 0 in RỊ2 and show that its restriction to Q0 K is conjugated to a rotation on the circle. We give the possible periods of solutions and study their global behavior such as the density of initial periodic points the density of trajectories in some curves and a form of sensitivity to initial conditions. We prove a dichotomy between a form of pointwise chaotic behavior and the existence of a common minimal period to all nonconstant orbits of F. 1. Introduction In 4 we study the difference equations un 2un a bun 1 u2n 1 a bun 1 cu2n 1 un 2un c un 1 which generalize the Lyness difference equations un 2un a un 1 see 2 7 8 9 . The first of these equations is related to a family of conics and the second to a family of cubics whose Lyness cubics are particular cases . The results of 4 in the two cases are analogous to the results obtained in 3 about the global behavior of the solutions of Lyness difference equation. In the present paper we will study the difference equation un 2un a bun 1 cu2n 1 c dun 1 u2n 1 Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 3 2005 227-261 DOI