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

ON THE ALGEBRAIC DIFFERENCE EQUATIONS u n+2 u n = ψ(u n+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 u n+2 u n = ψ(u n+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 alge- braic difference equations u n+2 u n = ψ(u n+1 )inR + ∗ , related to families of conics or cubics: generalization of the Lyness’ sequences” (2004), that the solutions M n = (u n+1 ,u n )are persistent and bounded, move on the positive component Q 0 (K) of the quartic Q(K) which passes through M 0 ,anddivergeifM 0 is not the equilibrium, which is locally sta- ble. In fact, we study the dynamical system F(x, y) = ((a+ bx + cx 2 )/y(c + dx + x 2 ),x), (a,b,c,d) ∈ R + 4 , a + b>0, b + c + d>0, in R + ∗ 2 , and show that its restriction to Q 0 (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 trajecto- ries in some cur ves, 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 u n+2 u n = a + bu n+1 + u 2 n+1 , u n+2 u n = a +bu n+1 + cu 2 n+1 c + u n+1 (1.1) which generalize the Lyness’ difference equations u n+2 u n = a + u n+1 (see [2, 7, 8, 9]). The first of these equations is related to a family of conics, and t he second to a family of cubics (whose Lyness’ cubics are par ticular 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 u n+2 u n = a +bu n+1 + cu 2 n+1 c + du n+1 + u 2 n+1 . (1.2) Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:3 (2005) 227–261 DOI: 10.1155/ADE.2005.227 228 Difference equations related to elliptic quartics The dynamical system in R + ∗ 2 which represents this difference equation is F(x, y) =  a +bx + cx 2 y  c + dx + x 2  ,x  . (1.3) It is well defined as a homeomorphism of R + ∗ 2 when a,b,c,d ≥ 0anda + b + c>0, as we always assume. We have M n+1 =  u n+2 ,u n+1  = F  M n  = F  u n+1 ,u n  . (1.4) There is an invariant function G(x, y) = xy+ d(x + y)+c  x y + y x  + b  1 x + 1 y  + a xy , (1.5) which satisfies G ◦F =G, and thus G(u n+1 ,u n ) is constant on every solution of (1.2). If K = G(u 1 ,u 0 ), the quartic Q(K) with equation G(x, y) =K,or x 2 y 2 + dxy(x + y)+c  x 2 + y 2  + b(x + y)+a −Kxy= 0, (1.6) passes through M 0 . The quartics Q(K) are invariant on the action of F, and thus the points M n move on the quartic passing through M 0 , more precisely on its positive component Q 0 (K). The map F has a geometrical interpretation. If M ∈R + ∗ 2 ,letM  be the second point of the quartic Q(K) which passes through M whose first coordinate is the same as those of M (there is only one such point M  because the point at the vertical infinity is a double point of the quartic). The image F(M) is the symmetric point of M  with respect to the diagonal x = y. For all this results, we refer to [4]. In Section 2, we give a general topological result useful for our study, which extends a result of [4], and we define a general property of weak chaotic behavior, whose proof for (1.2) is the goal of this paper. In Section 3, we use this result to show that the solutions of difference equation (1.2) are, if a +b>0andb + c+ d>0, bounded and persistent in R + ∗ 2 ,anddivergeif(u 1 ,u 0 ) = (,), the fixed point of F, and prove that this point is locally stable. In Section 4, we show that the case where u n+2 u n is a homographic function of u n+1 , studied in [5], comes down to our general model (1.2). This gives again, in a simpler way, results of [5], and improvements of them. In Section 5,westudythecasea = 0, where the quartic passes through the point (0,0). This case is easy, because a simple birational map transforms every quartic Q(K)intoa cubic curve studied in [4]. So we can apply the results of [4] without more work. In Section 6, we prove general results in the case a>0, which lead to the fact that the restriction of the map F to each curve Q 0 (K) is conjugated to a rotation onto the circle (see Theorem 6.11). We study also in Sections 6 and 7 whether the chaotic behavior de- fined in Section 2 holds in the general case of (1.2), w ith a general property of dichotomy (see Theorem 6.18), and what happens in some particular cases (Section 7) and in the general one (Section 8). In Section 9, we determine the possible periods of solutions of (1.2). G. Bastien and M. Rogalski 229 2. A topological tool for difference equations with an invariant In this section, we give an abstract and more or less classical general result which will be useful for the study of difference equations. This assertion extends [4, Proposition 1]. Proposition 2.1. Let X be a topological Hausdorff space. Let F : X →X and G : X →R be two maps. Suppose first that the following conditions hold: (a) F is continuous on X; (b) G is continuous and has a strict minimum K m at a point L; (c) ∀x ∈X, G◦F(x) = G(x) (the invariance property); (d) F has at most one fixed point. If K ≥ K m , the level sets (if nonempty) of G are defined by Ꮿ K ={x ∈ X | G(x) = K}. Then the following three results hold: (1) every point x ∈X lies in exactly one set Ꮿ K ; (2) the point L is the (unique) fixed point of F; (3) if M 0 ∈ X let M n+1 = F(M n ) be the points of the orbit of M 0 under F; then M n ∈ Ꮿ G(M 0 ) ,andifM 0 = L,thenthesequence(M n ) does not converge. Now suppose additional hypotheses: (e) X is connected and locally compact; (f) K ∞ := lim x→∞ G(x) ≤ +∞ exists, and G<K ∞ ; then (4) each Ꮿ K is compact and nonempty for K m ≤ K<K ∞ (w ith Ꮿ K m ={L}), and the equilibrium point L is locally stable. Suppose at last the additional hypothesis: (g) G has only one local minimum (its global one at L); then (5) for K>K m the set Ꮿ K is the boundary of the open set U K ={G<K} whichisa connected relatively compact s et. Proof. Assertions (1) and (2) are obvious. If M n+1 = F(M n ), then M n ∈ Ꮿ G(M 0 ) .Suppose that M n converges to a point N.ThenG(N) =G(M 0 )andF(N) = N, so by (d) and (1) N = L.ButG(M n ) =G(N) =G(L) =K m ,andby(b)M n = N for all n. Thus, if M 0 = L, then M n does not converge. If (e) and (f) hold, it is easy to see that Ꮿ K is nonempty and compact for every K ≥K m ; in particular, sequences (M n ) are bounded (i.e., relatively compact). We prove now that the sets U K ={G<K} form a basis of neighborhoods of L.LetV be an open neighborhood of L. The sets {G ≤ K},forK>K m , are compact, and their intersection is {L}; so there is a K>K m such that {G ≤ K}⊂V, and thus U K ⊂ V. We can now prove easily that L is locally stable: if V is a neighborhood of L, there exists K>K m such that U K ⊂ V.IfM 0 ∈ U K ,then,foreveryn, M n ∈ U K by (c), and M n ∈V. We prove now assertion (5), if (g) also holds. We have U K ⊂{G ≤K},andU K \U K ⊂ { G ≤K}\{G<K}=Ꮿ K .Thus,∂U K ⊂ Ꮿ K .Now,ifᏯ K ⊂ ∂U K , there exists x ∈ Ꮿ K , x/∈ ∂U K , thus there exists a neighborhood V of x such that V ∩U K =∅.Thus,G ≥ K on V,andG(x) = K;thusx is a local minimum of G,andx = L because K>K m : this is impossible, and ∂U K = Ꮿ K . Finally, we prove that U K is connected. If U K is the union of two disjoint nonempty open sets A and B (which are relatively compact), put α =inf A G and β =inf B G;wehave α,β<K.Ifα =G(u)withu ∈A and β =G(v)withv ∈ B,thenu and v are two distinct 230 Difference equations related to elliptic quartics local minima of G, which is impossible. Thus we can suppose that α = G(u)withu ∈ A \A.ButA ⊂ X \B (because U K is open), thus u/∈ B,andu ∈U K \U K = ∂U K = Ꮿ K .So we hav e G(u) = K>α, which is a contradiction.  We w ill use Proposition 2.1 when X is an open subset of R 2 ; in this context, F and G are given by (2) and (3). In the general case, we can ask the question whether a form of chaotic behavior may be described for the map F (we wil l study in this paper if it is the case with F and G given by (2) and (3)). Precisely, one may ask the question w hether F has an “invariant pointwise chaotic behavior,” denoted IPCB. Property of IPCB. We suppose t hat X, F,andG satisfy properties (a),(b), ,(g) of Proposition 2.1, and that X is a metric space, with distance d. We say that the dynam- ical system (X,F) with invariant G has IPCB if we have the following three properties. (a) There exists a partition of X\{L} into two dense subsets A and B which both are union of “curves” Ꮿ K , and then invariant under F : A is the set of initial periodic points M 0 , B is the set of initial points M 0 whose orbit is dense in the curve Ꮿ K which passes through M 0 (that is Ꮿ G(M 0 ) ). (b) Every point M 0 ∈ X\{L} has sensitivity to initial conditions, that is, there exists δ(M 0 ) > 0 (this dependance on M 0 explains the term “pointwise”) such that every neighborhood of M 0 contains a point M  0 whose iterates M  n satisfy d(M n ,M  n ) ≥ δ(M 0 ) for infinitely many integers n. (c) There exists an integer N such that every integer n ≥ N is the minimal period of some periodic orbit of F. IPCB is the essential result of [3] about the behavior of Lyness’ difference equation u n+2 u n = k + u n+1 if 0 <k= 1 (if k = 1, 5 is a common minimal period to all nonconstant solutions). In [4], we prove also that IPCB holds for the solutions of difference equations in R + ∗ 2 u n+2 u n = a + bu n+1 + u 2 n+1 , u n+2 u n = a +bu n+1 + cu 2 n+1 c + u n+1 . (2.1) An important tool to study the dynamical system linked to (1.2) may be an eventual propertyintheabstractcaseofProposition 2.1:foreveryK ∈ ]K m ,K ∞ [, is the dynamical system F |Ꮿ K conjugated to a rotation on the circle with angle 2πθ(K) ∈ ]0, π[? This even- tual property supposes that each set Ꮿ K is homeomorphic to a circle. Then the s tudy of the properties of function θ would be essential: continuity (analyticit y if X is an open set of R 2 ), limits when K → K m and K → K ∞ . 3. First general results of divergence and stability We begin by identifying the fixed point. Lemma 3.1. If a = b =0,thensequence(1.2)tendsto0.Ifa + b>0, then the fixed point of the dynamical system (1.3) is the unique positive root  of the equation Y 4 + dY 3 −bY −a = 0, (3.1) and it is the unique possible limit for sequence (1.2). G. Bastien and M. Rogalski 231 Figure 3.1 Proof. It is obvious that a fixed point of F has the form (Y,Y )whereY satisfies (3.1), and that (3.1)hasauniquepositiveroot such that (,)isinvariantbyF. For the limit of the sequence (u n )of(1.2), we must be more careful if a =0, because in this case Y = 0issolutionof(3.1). But then Q(K) passes through (0,0) and has as tangent at this point the line x + y = 0ifb>0, and so the point M n = (u n+1 ,u n ), which lies on Q(K) ∩R + ∗ 2 , cannot tend to (0, 0). If a = b = 0, the fixed point is solution of Y 4 + dY 3 = 0, which has no solution in R + ∗ . In this case, we have u n+2 /u n+1 = (u n+1 /u n )(c/(c + du n+1 + u 2 n+1 )), with c>0and d ≥ 0. Thus ρ n = u n+1 /u n is decreasing and tends to a limit λ.Ifλ<1, then u n → 0. If λ would satisfy λ ≥1, then u n would be increasing, and would tend to infinity. But then c/(c + du n+1 + u 2 n+1 ) ≤ 1/2forbign, and thus we would have λ = 0, which is a contradic- tion.  With the objective of using Proposition 2.1, it is necessary to study the function G.The first question is to know if G(x, y) →+∞ if (x, y) tends to the infinite point of the locally compact space R + ∗ 2 . It appears that this condition fails in the general case. Indeed, we look for a condition for the sets A K :={G ≤K}∩R + ∗ 2 to be compact. The hypothesis is xy+ d(x + y)+c  x y + y x  + b  1 x + 1 y  + a xy ≤ K. (3.2) Thus we have xy+ a/xy ≤K, d(x + y) ≤K, c(x/y + y/x) ≤ K, b/x ≤K,andb/y ≤K. If b>0, then x ≥b/K and y ≥ b/K, and thus, with the condition xy≤ K, the set A K is compact. If b =0, we will suppose a>0(thecasea =b = 0istrivialbyLemma 3.1), and the condition xy+ a/xy ≤ K implies that 0 <r 1 ≤ xy ≤r 2 : the point (x, y)isbetweentwo hyperbolas. But then if c or d is positive, we have x/y + y/x ≤K/c and thus 0 <s 1 ≤ y/x ≤ s 2 ,orx + y ≤ K/b. In the two cases, A K is compact; see Figure 3.1. So the good condition in (1.2), which we suppose in all the sequel, is a +b>0, b + c + d>0. (3.3) It is to be noticed that if b = c = d = 0, the function G does not tend to +∞ at the point at infinity of R + ∗ 2 , and then the solutions of the difference equation (1.2)maybe unbounded or not persistent. In fact, it is always the case. 232 Difference equations related to elliptic quartics Indeed, consider the difference equation u n+2 u n = a/u 2 n+1 ,witha>0. Its solutions are the sequences u n =a 1/4 exp[(−1) n (A +Bn)], with A=ln(u 0 a −1/4 )andB =−ln(u 0 u 1 a −1/2 ), which are neither bounded nor persistent if u 0 u 1 = √ a. Then, we must identify the minimum of G. The equations of critical points are x 2 y 2 + dx 2 y + c(x 2 − y 2 ) −by−a = 0andx 2 y 2 + dy 2 x + c(y 2 −x 2 ) −bx−a = 0. The difference of these two equations gives (x − y)(dxy +2c(x + y)+b) = 0. But if (x, y) ∈R + ∗ 2 ,theonly solution is x = y, so the previous equations give x 4 + dx 3 −bx −a =0, and thus we have x = y = : G has a unique critical point at the equilibrium L =(,), the minimum of G is achieved only at this point, and the value of the minimum is K m = d +2c + 3b  + 2a  2 . (3.4) If K>K m ,thenQ 0 (K) =Q(K) ∩R + ∗ 2 ={(x, y) ∈R + ∗ 2 | G(x, y) =K}is a nonempty com- pact component of Q(K), and through every point M ∈R + ∗ 2 passes a unique curve Q 0 (K). We can thus apply Proposition 2.1 and obtain the following theorem. Theorem 3.2. If a ≥0, b ≥0, c ≥ 0, d ≥ 0, a + b>0,andb + c + d>0,everysolutionof the difference equation (1.2) u n+2 u n = a +bu n+1 + cu 2 n+1 c + du n+1 + u 2 n+1 (3.5) is bounded and persistent in R + ∗ 2 .If(u 1 ,u 0 ) = (,), then (u n ) diverges, the point M n = (u n+1 ,u n ) moves on the curve Q 0 (K) which passes through M 0 ,andK>K m .Moreoverthe equilibrium L is locally stable. 4. The homographic case In [5], the authors study the difference equation u n+2 = αu n+1 + β u n  γu n+1 + δ  ,withα,β,γ,δ ≥ 0, α +β>0, γ +δ>0. (4.1) If γ = α =0, we find the classical sequence u n+2 = (β/δ)/u n which is always 4-periodic. If γ =0, α = 0, the sequence v n = (δ/α)u n satisfies v n+2 v n = v n+1 + βδ/α 2 : it is a Lyness sequence, and its behavior is known and given in [3]. So, we suppose γ>0, and thus we can suppose γ =1. Under this hypothesis, if we suppose that the two quadratic polynomials of (1.2)have a common root x =−p<0, then (4.1) is a particular case of (1.2). To see this fact, we examine some cases. (i) If δ =0, easy calculation shows that with a = αβ δ , b = α 2 δ + β, c = α, d = α δ + δ, (4.2) (1.2)isexactly(4.1)withγ = 1. (ii) If δ = α =0, (4.1)becomesu n+2 u n = β/u n+1 , which is a classical 3-periodic se- quence. G. Bastien and M. Rogalski 233 (iii) If δ = 0, α>0, β = 0, u n+2 = α/u n is another case of the previous 4-periodic se- quence. (iv) If δ =0, α>0, β>0, we put u n = β/v n , and obtain v n+2 v n = β 2 /(v n+1 + α), which has the form (α  v n+1 + β  )/(v n+1 + δ  ). Thus, (1.2)for(v n ) with the values a =c = 0, b = β 2 , d = α,isexactly(4.1)foru n = β/v n . In any cases, (4.1)comesdownto(1.2) or to a known sequence (Lyness’ one) or to elementary sequences (3- or 4-periodic). Thus, with the aid of elementary results on Ly- ness’ equation (see [3]), we deduce again from Section 2 the result of [5]about(4.1), but almost without calculation, and we can improve it. Proposition 4.1. The solutions of (4.1) are bounded and persistent, and diverge if (u 1 ,u 0 ) is different than the fixed point. Moreover the equilibrium point is locally stable. Of course, other properties of the solutions of (4.1)willfollowfromthegeneralprop- erty of solutions of (1.2) that we wil l prove in the following parts, see corollaries of Theorems 5.1 and 7.1,whereexamplesof(4.1) which have IPCB are given. 5. The case a =0 In this part, we solve the case when a =0, which is simple, because an easy birational map reduces the associated quartic curves to cubic ones which give a previous case already solved (see [4]). So we obtain the following general result. Theorem 5.1. Let the difference equation in R + ∗ be u n+2 u n = bu n+1 + cu 2 n+1 c + du n+1 + u 2 n+1 with b>0, c ≥ 0, d ≥0, c + d>0, (5.1) whose solutions diverge if (u 1 ,u 0 ) =(,).LetL =(,) be the equilibrium, with  positive solution of the equation Y 3 + dY 2 −b = 0.LetF(x, y) = ((bx + cx 2 )/y(c + dx + x 2 ),x) be the homeomorphism of R + ∗ 2 associated to (5.1): M n := (u n+1 ,u n ) = F n (M 0 ).LetQ b,c,d (K) be the quartic curve with equation x 2 y 2 + dxy(x + y)+c  x 2 + y 2  + b(x + y) −Kxy= 0 (5.2) whichpassesthroughM 0 = (u 1 ,u 0 ),andQ 0 b,c,d (K) its positive component, globally invariant under the action of F. (a) Thereexistsawell-definednumberθ b,c,d (K) ∈]0,1/2[ such that the restriction of F to Q 0 b,c,d (K) is conjugated to a rotation on the circle, of angle 2πθ b,c,d (K) ∈]0,π[. (b) For every b, c, d satisfying the conditions of (5.1)andb 2 = c 3 or bd = 2c 2 ,thediffer- ence equation (5.1) has IPCB. (c) Every integer n ≥ 4 is the minimal period of some solution of (5.1)forsomeb, c, d, and some initial point M 0 . One has b 2 = c 3 and bd = 2c 2 if and only if every solution of (5.1)is5-periodic. Proof. If a = 0, then the quartic curve (1.6)reducesto(5.2), and then it passes through (0,0). 234 Difference equations related to elliptic quartics (1) Case c = 0andd>0. We define the birational map X =  b d 1 x , Y =  b d 1 y , (5.3) that is the transformation on the solutions (u n )ofdifference equation (5.1)bythefor- mula v n =  b d 1 u n . (5.4) Under map (5.3) the quartic (5.2) becomes the cubic of paper [4]: Γ α (K  )withα =  b d 3 , K  = K √ bd , (5.5) associated to the difference equation v n+2 v n+1 v n = α + v n+1 (5.6) whose solutions are studied in [4]. Then results of Theorem 5.1 are nothing else but [4, Proposition 8 and Theorem 4]. (2) Case c>0. We define now the birational map X = b c 1 x , Y = b c 1 y , (5.7) that is the transformation on the solutions (u n )ofdifference equation (5.1)bythefor- mula v n = b c 1 u n . (5.8) Under map (5.7) the quartic (5.3) becomes the cubic of (see [4]) Γ α,β (K  )withα = b 2 c 3 , β = bd c 2 , K  = K c , (5.9) associated to the difference equation v n+2 v n = α +βv n+1 + v 2 n+1 v n+1 +1 (5.10) whose solutions are studied in [4]. Then the results in Theorem 5.1 are nothing but [4, Proposition 11 and Theorem 6]. The case of 5-periodicity in [4] corresponds to the values α = 1andβ = 2 (see [4, Lemma 8]), which gives the end of assertion (d) of the theorem. G. Bastien and M. Rogalski 235 Point (3) of [4,Theorem6]hastobemodified,becauseherewehaveonlyα>0and β ≥0 (arbitrary), but in [4]wehaveα ≥ 0andβ>−2 √ α.So,in[4, Lemma 10], we must replace the domain D by  D = R + 2 and the function f () =(1/π)cos −1 (1/2)(1 −1/ √  +1) by the function  f () =(1/π)cos −1 (1/2)  1+3/( + 1). Then it is easy to show that we have only o ψ(  D) =]0,1/3[. Thus every integer n ≥4 is actually a period.  Corollary 5.2. The solutions of (4.1)studiedin[5],whereαβγ > 0, δ = 0,satisfy Theorem 5.1. Remark 5.3. (1) If c>0, then solutions (u n )of(5.1) are rational if and only if the v n ’s are rational, when b, c, d are rational. Then, in this case, a rational periodic solution of (5.1) may have only periods which belong to the set {3, 4,5,6,7,8,9,10, 12} (see [4]). But if c = 0, the map (5.3) does not preserve rationality of real numbers, except if b/d =q 2 with q ∈ Q + ∗ . In this case, and with b rational, the periodic rational solutions of (5.1) may have only periods 7 or 10 (see [4, corollary of Proposition 7]). (2) The 5-periodic case b 2 = c 3 and bd = 2c 2 corresponds to initial Lyness’ sequence: v n = √ c/u n satisfies v n+2 v n = 1+v n+1 . We give now two easy cases with a =0, which are not covered by Theorem 5.1. First, the case a = b =0isgivenbyLemma 3.1: the sequence tends to 0. Second, we have the following classical result. Lemma 5.4 (case a = c = d = 0, b>0). The positive solutions of the difference equation u n+2 u n+1 u n = b are 3-periodic. 6. General results in the case a>0 It is easy to see that if a>0wecansupposethata = 1(putu n = v n 4 √ a). We make this hypothesis from now on. 6.1. Points on the diagonal and the birational transformation of the quartic. We k now that the quar tic curve has two double points at infinity in vertical and horizontal di- rection, which are ordinary if d 2 −4c = 0, the asymptotes being then the lines x = r 1 , x = r 2 , y = r 1 ,andy = r 2 ,wherether i are the roots (real or complex) of the equa- tion s 2 + ds + c = 0. Moreover, if K>K m , the quartics Q(K) have no singular point in R + ∗ 2 . Indeed, if the equation of Q(K)isp(x, y) −Kxy = 0, singular points are given by p  x −Ky= 0, p  y −Kx = 0, and p −Kxy =0. These relations give xp  x = p and yp  y = p, and these last relations are the equations whose solutions are the critical points of the function G(x, y) = p(x, y)/xy. But we have seen that G has no critical point in R + ∗ 2 except for L =(,), and so the only finite singular p oint of Q(K)inR + ∗ 2 would be L, but this point is not on Q(K)ifK>K m . So we can hope that the quartic curve Q(K)isanellipticone,andthusthatitcan be transformed in a regular cubic curve by a birational transformation. To make such a transformation, some point of Q(K) should disappear, and to preserve the symmetry of the curve with respect to the diagonal δ : x = y, we choose this point on this diagonal. So the fundamental technical result will be the behavior of the points of Q(K)onthe diagonal. 236 Difference equations related to elliptic quartics f 1 f 0 −λ 0 −λ f 2 f 3 t K m K G(t,t) Figure 6.1 Lemma 6.1. For K>K m , the coordinates of the intersection points of Q(K) with the diagonal δ are solutions of the equation t 4 +2dt 3 +(2c −K)t 2 +2bt +1= 0. (6.1) These coordinates are real numbers f 1 , −λ, f 2 , f 3 which satisfy f 1 < −<−λ<0 <λ< f 2 << f 3 if d + b>0, f 1 =− 1 λ < −1 < −λ<0 <λ= f 2 < 1 =<f 3 = 1 λ if d = b = 0. (6.2) Moreover, numbers f i and λ are continuous functions of K on ]K m ,+∞[, whose limits when K → +∞ and K → K m are lim K→+∞ λ = lim K→+∞ f 2 = 0, lim K→+∞ f 1 =− lim K→+∞ f 3 =−∞, (6.3) lim K→K m f 2 = lim K→K m f 3 = , λ 0 := lim K→K m λ = (d + ) −  (d + ) 2 − 1  2 , lim K→K m f 1 := f 0 =−(d + ) −  (d + ) 2 − 1  2 . (6.4) Proof. Formula (6.1)isobvious.Leth K (t) = t 4 +2dt 3 +(2c −K)t 2 +2bt +1= 0. By (3.1) and (3.4)wehave h K () =d 3 +3b +2+(2c −K) 2 <d 3 +3b +2− 2  d + 3b  + 2  2  = 0, (6.5) h K (0) =1, and so h K has two roots f 2 and f 3 which satisfy 0 <f 2 << f 3 .Wehavealso h K (−) = h K () − 4d 3 −4b < 0, and thus we have two other roots f 1 and −λ which satisfy f 1 < −<−λ<0. At last, h K (λ) = h K (−λ)+4dλ 3 +4bλ = 4dλ 3 +4bλ ≥ 0. If b + d> 0, thus we have 0 <λ< f 2 .Ifb = d = 0, the roots of h K are f 1 , −λ, λ,and−f 1 , whose product is 1. This gives (6.2). Then we remark that the equation h(t) = 0isequivalenttotherelationG(t,t) =K.But the graph of the function t → G(t,t) is easy to determine, see Figure 6.1. It is immediate from this graph that the roots are continuous functions of K. Their limits when K → +∞ [...]... has the factor U + 1/ p(K )) So, it is also the case of Γ(K): it is the union of the real line J with equation x + y = 1/ p(K) and a conic B But Γ(K) is symmetric with respect to the diagonal, contains the points P = (−1/ , 0 ), P = ( 0, −1/ ), and the three distinct points Fi = (1/( fi − ), 1/( fi − )) , i = 1,2 , 3, except if b = d = 0, when F2 is the point at in nity in direction x = y (see Lemma 6. 2). .. Gauthier-Villars, Paris, e 1897 (French) E Barbeau, B Gelbord, and S Tanny, Periodicities of solutions of the generalized Lyness recursion, J Differ Equations Appl 1 (199 5 ), no 3, 291–306 G Bastien and M Rogalski, Global behavior of the solutions of Lyness’ difference equation un+ 2 un = un+ 1 + a, J Differ Equations Appl 10 (200 4 ), no 1 1, 977–1003 , On some algebraic difference equations un+ 2 un = ψ (un+ 1 ) in R+ , related. .. in point ( 4) of Theorem 7. 1, with the aid of form (7. 2) of H(d,c,d )) Theorem 7.6 Let the difference equation un+ 2 un+ 1 un = 1 + bun+1 , d + un+ 1 b,d > 0 (7.1 1) If bd = 1, then (7.1 1) has IPCB If bd = 1, then all the nonconstant solutions of (7.1 1) have the common period 3 Conversely, if there exists (u1 ,u0 ) = ( , ) which is 3-periodic, then bd = 1 and all the nonconstant solutions of (7.1 1) have the. .. equation (4. 1): un+ 2 un = 1 + cun+1 , c + un+ 1 0 < c = 1, (7. 5) has IPCB Remark 7.3 It is easy to see that there is another case of difference equation (4. 1) which has IPCB: un+ 2 un = un+ 1 + α3 /δ 3 un+ 1 + δ (7. 6) 7.2 The difference equations un+ 2 un+ 1 un = b + 1 /un+ 1 and un+ 2 un+ 1 un = 1/(d + un+ 1 ) These cases are c = d = 0 and b = c = 0 First, we remark that by setting un = 1/vn and inverting b and... of solutions of (1. 2 ), for each set of parameters of the previous table, are given in Table 9.1 Proof In fact, the only difficult case is the first row of the table; rows ( 2 ), ( 4 ), and ( 6) are easy, and the case of row ( 3) will be deduced from row ( 1) The case of rows ( 5) and ( 7) will follow from rows ( 2) and ( 3) The essential tool is the determination of the range of the function , or at least a set J... of (7. 1 ), then (1 /un ) is also a solution ( 2) The following formula holds H(d,c,d) = d+2 2c + d (7. 2) ( 3) If 0 < c = 1, the difference equation (7. 1) has IPCB If c = 1, all the nonconstant solutions of (7. 1) have the common minimal period 4 Conversely, if there exists (u1 ,u0 ) = ( , ) with period 4, then c = 1 and all nonconstant solutions of (7. 1) have minimal period 4 in common ( 4) If c = 0, the difference... p(K)U (6.2 2) or or, in homogeneous coordinates, by U =u , V =v , W = w − p(K)u , (6.2 3) u =U , v =V , w = W + p(K)U (6.2 4) or We obtain a new cubic curve E(K) Remark that if for some K one has p(K) = 0, then ψK = Id If p(K) = 0, then the line with equation U = −1/ p(K) is an asymptote of in exion of Γ(K) which is sent to the line at in nity by ψK ; this line is a tangent of in exion to the cubic E(K)... common period 3 Proof Of course, if bd = 1, then un+ 2 un+ 1 un = 1/d, and so un is 3-periodic If there exists (u1 ,u0 ) = ( , ) which is 3-periodic, un is nonconstant, and un+ 2 un+ 1 un is constant, and thus the function t → (1 + bt)/(d + t) has the same value for at least two distinct values of the sequence (un ); so this homographic function is constant, that is bd = 1 If bd = 1, then function θ is nonconstant... distinct values of (un ) So the homographic function t → (1 + dt)/(d + t) is constant, that is d = 1; and thus all the solutions of (7. 3) are 3-periodic If 0 < c = 1 and d = c + 1, point ( 3) of Theorem 7.1 gives the behavior of a particular case of the difference equation of [5 ], because the factor 1 + un+ 1 appears both in the numerator and the denominator of (7. 1) Corollary 7.2 The particular case of difference... compact, and φK is a homeomorphism of Q0 (K) onto Γ0 (K) If b = d = 0, then Γ0 (K) is unbounded and has a point at in nity in direction x = y, which is the image by φK of the point ( , ) of Q0 (K ), and φK is a homeomorphism of Q0 (K) \ {( , )} on Γ0 (K) Proof ( 1) We work here in R+ 2 , and begin in the case when b + d > 0 Suppose that there ∗ is a point (X,Y ) in the set {G ≤ K }, lying on the hyperbola . is  Γ(K) splits into its asymptote U = 1/p(K) and a conic (besides one see easily that the left hand of (6.17)hasthefactorU +1/p(K )) . So, it is also the case of Γ(K): it is the union of the real line. +∞.Sowewilltakeλ as the variable, and find asymptotic expansion of the other parameters: K, , , p, f i , e i , and finally the parameters of integr als in (6.4 0): , . We have first from (6.1)fort = λ K. c, d are rational. Then, in this case, a rational periodic solution of (5. 1) may have only periods which belong to the set { 3, 4,5 , 6,7 , 8,9 ,1 0, 12} (see [4 ]). But if c = 0, the map (5. 3) does not