On two and three periodic Lyness difference equations

On wo and h ee pe iodic Lyness di e ence equa ions∗ Anna Cima(1), A mengol Gasull(1) and V´ıc o Ma˜nosa (2) (1) Dep . de Ma em`a iques, Facul a de Ci`encies, Uni e si a Au `onoma de Ba celona, 08193 Bella e a, Ba celona, Spain [email p o ec ed] , gasull@ma .uab.ca (2) Dep . de Ma em`a ica Aplicada III (MA3), Con ol, Dynamics and Applica ions G oup (CoDALab) Uni e si a Poli `ecnica de Ca alunya (UPC) Colom 1, 08222 Te assa, Spain ic o [email protected] Abs ac We desc ibe he sequences {xn}ngi en by he non-au onomous second o de Lyness di e ence equa ions xn+2 = (an+xn+1)/xn,whe e {an}nis ei he a 2-pe iodic o a 3- pe iodic sequence o posi i e alues and he ini ial condi ions x1, x2a e as well posi i e. We also show an in e es ing phenomenon o he disc e e dynamical sys ems associa ed o some o hese di e ence equa ions: he exis ence o one oscilla ion o hei associa ed o a ion numbe unc ions. This beha io does no appea o he au onomous Lyness di e ence equa ions. 2000 Ma hema ics Subjec Classi ica ion: 39A20, 39A11 Keywo ds: Di e ence equa ions wi h pe iodic coe icien s, ci cle maps, o a ion numbe . 1 In oduc ion and main esul This pape ully desc ibes he sequences gi en by he non-au onomous second o de Lyness di e ence equa ions xn+2 =an+xn+1 xn ,(1) whe e {an}nis a k-pe iodic sequence aking posi i e alues, k= 2,3,and he ini ial con- di ions x1, x2a e as well posi i e. This ques ion is p oposed in [4, Sec. 5.43]. Recall ∗Acknowledgemen s. GSD-UAB and CoDALab G oups a e suppo ed by he Go e nmen o Ca alonia h ough he SGR p og am. They a e also suppo ed by MCYT h ough g an s MTM2008-03437 ( i s and second au ho s) and DPI2008-06699-C02-02 ( hi d au ho ). 1 ha non-au onomous ecu ences appea o ins ance as popula ion models wi h a a iable s uc u e a ec ed by some seasonali y [10, 11], whe e kis he numbe o seasons. Some dynamical issues o simila ype o equa ions ha e been s udied in se e al ecen pape s [1, 8, 9, 12, 14, 16, 17]. Recall ha when k= 1, ha is an=a > 0, o all n∈N, hen (1) is he amous Lyness ecu ence which is well unde s ood, see o ins ance [2, 18]. The cases k= 2,3 ha e been al eady s udied and some pa ial esul s a e es ablished. Fo bo h cases i is known ha he solu ions a e pe sis en nea a gi en k-pe iodic solu ion, which is s able. This is p o ed by using some known in a ian s, see [14, 16, 17]. Recall ha in ou con ex i is said ha a solu ion {xn}nis pe sis en i he e exis wo eal posi i e cons an s cand C, which depend on he ini ial condi ions, such ha o all n≥1,0< c < xn< C < ∞.We p o e: Theo em 1. Le {xn}nbe any sequence de ined by (1) and k∈ {2,3}. Then i is pe sis en . Fu he mo e, ei he (a) he sequence {xn}nis pe iodic, wi h pe iod a mul iple o k; o (b) he sequence {xn}ndensely ills one o wo ( esp. one, wo o h ee) disjoin in e als o R+when {an}nis 2-pe iodic ( esp. 3-pe iodic). Mo eo e i is possible by algeb aic ools o dis inguish which is he si ua ion. Ou app oach o desc ibe he sequences {xn}nis based on he s udy o he na u al dynamical sys em associa ed o (1) and on he esul s o [6]. The main ool ha allows o dis inguish he numbe o in e als o he adhe ence o he sequences {xn}nis he compu a ion o se e al esul an s, see Sec ion 4. I is wo h o commen ha Theo em 1 is an ex ension o wha happens in he classical case k= 1. The e, he same esul holds bu in s a emen (b) only appea s one in e al. Ou second main esul will p o e ha he e a e o he mo e signi ica i e di e ences be ween he case k= 1 and he cases k= 2,3.These di e ences a e ela ed wi h he lack o mono onici y o ce ain o a ion numbe unc ions associa ed o he dynamical sys ems gi en by he Lyness ecu ences, see Theo em 3. The beha io s o hese o a ion numbe unc ions a e impo an o he unde s anding o he ecu ences, because hey gi e he possible pe iods o hem, see [2, 3, 18]. On he o he hand in [9, 17] i is p o ed ha , a leas o some alues o {an}n, he beha iou o {xn}n o he case k= 5 is o ally di e en . In pa icula unbounded posi i e solu ions appea . In he o hcoming pape [7] we explo e in mo e de ail he di e ences be ween he cases k= 1,2,3 and k≥4. This pape is o ganized as ollows: Sec ion 2 p esen s he di e ence equa ions ha we a e s udying as disc e e dynamical sys ems and we s a e ou main esul s on hem, see 2 Theo ems 2 and 3. Sec ion 3 is de o ed o he p oo o Theo em 2. By using i , in Sec ion 4, we p o e Theo em 1 and we gi e some examples o how o apply i o de e mine he numbe o closed in e als o he adhe ence o {xn}n. In Sec ion 5 we demons a e Theo em 3 and we also p esen some examples whe e we s udy in mo e de ail he o a ion numbe unc ion o he dynamical sys ems associa ed o (1). 2 Main esul s om he dynamical sys ems poin o iew In his sec ion we educe he s udy o he sequence {xn}n o he s udy o some disc e e dynamical sys ems and we s a e ou main esul s on hem. Fi s we in oduce some no a ions. When k= 2,se an=(a o n= 2ℓ+ 1, b o n= 2ℓ, (2) and when k= 3,se an=       a o n= 3ℓ+ 1, b o n= 3ℓ+ 2, c o n= 3ℓ, (3) whe e ℓ∈Nand a > 0, b > 0 and c > 0. We also conside he maps Fα(x, y), wi h α∈ {a, b, c},as Fα(x, y) = y, α+y x, de ined on he open in a ian se Q+:= {(x, y) : x > 0, y > 0} ⊂ R2. Conside o ins ance k= 2.The sequence gi en by (1), x1, x2, x3, x4, x5, x6, x7,..., (4) can be seen as (x1, x2)Fa −→ (x2, x3)Fb −→ (x3, x4)Fa −→ (x4, x5)Fb −→ (x5, x6)Fa −→ ··· . Hence he beha io o (4) can be ob ained om he s udy o he dynamical sys em de ined in Q+by he map: Fb,a(x, y) := Fb◦Fa(x, y) = a+y x,a+bx +y xy . Simila ly, o k= 3 we can conside he map: Fc,b,a(x, y) := Fc◦Fb◦Fa(x, y) = a+bx +y xy ,a+bx +y+cxy y(a+y). 3 No ice ha bo h maps ha e an only ixed poin in Q+,which depends on a, b (and c), ha o sho we deno e by p. I is easy o in e p e he in a ian s o (1) and k= 2,3,gi en in [12, 14], in e ms o i s in eg als o he abo e maps, see also Lemma 6. We ha e ha Vb,a(x, y) := ax2y+bxy2+bx2+ay2+ (b2+a)x+ (b+a2)y+ab xy , is a i s in eg al o Fb,a and Vc,b,a(x, y) := cx2y+axy2+bx2+by2+ (a+bc)x+ (c+ab)y+ac xy , is a i s in eg al o Fc,b,a. The opology o he le el se s o hese in eg als in Q+as well as he dynamics o he maps es ic ed o hem is desc ibed by he ollowing esul , ha will be p o ed in Sec ion 3. Theo em 2. (i) The le el se s o Vb,a ( esp. Vc,b,a) in Q+ {p}a e di eomo phic o ci cles su ounding p, which is he unique ixed poin o Fb,a ( esp. Fc,b,a). (ii) The ac ion o Fb,a ( esp. Fc,b,a) on each le el se o Vb,a ( esp. Vc,b,a) con ained in Q+ {p}is conjuga ed o a o a ion o he ci cle. Once a esul like he abo e one is es ablished he s udy o he possible pe iods o he sequences {xn}ngi en by (1) is qui e s anda d. I su ices, i s o ge he o a ion in e al, which is he open in e al o med by all he o a ion numbe s gi en by he abo e heo em, a ying he le el se s o he i s in eg als. A e wa ds, i su ices o ind which a e he denomina o s o all he i educible a ional numbe s ha belong o he co esponding in e al, see [3, 5, 18]. The s udy o he o a ion numbe o hese kind o a ional maps is no an easy ask, see again [2, 3, 5, 18]. In pa icula , in [2] was p o ed ha he o a ion numbe unc ion pa ame e ized by he ene gy le els o he Lyness map Fa, a 6= 1,is always mono onous, sol ing a conjec u e o Zeeman gi en in [18], see also [15]. As a as we know, in his pape we gi e he i s simple example o which his o a ion numbe unc ion is nei he cons an no mono onous. We p o e: Theo em 3. The e a e posi i e alues o aand b, such ha he o a ion numbe unc ion ρb,a(h)o Fb,a associa ed o he closed o als o {Vb,a =h} ⊂ Q+has a local maximum. Hence, apa om he known beha io s o he au onomous Lyness maps, ha is global pe iodici y o mono onici y o he o a ion numbe unc ion, which i ially holds o Fb,a, aking o ins ance a=b= 1 o a=b6= 1, espec i ely, he e appea mo e complica ed beha io s o he o a ion numbe unc ion. 4 Ou p oo o his esul elies on he s udy o lowe and uppe bounds o he o a ion numbe o Fb,a on a gi en o al o a le el se o Vb,a gi en o some (a, b)∈(Q+)2and {Vb,a(x, y) = Vb,a(x0, y0)}, o (x0, y0)∈(Q+)2. This can be done because he map on his o al is conjuga ed o a o a ion and i is possible o use an algeb aic manipula o o ollow and o o de a ini e numbe i e a es on i , which a e also gi en by poin s wi h a ional coo dina es. So, only exac a i hme ic is used. A simila s udy could be done o Fc,b,a. 3 P oo o Theo em 2 P oo o (i) o Theo em 2. The o bi s o Fb,a and Fc,b,a lie on he le el se s Vb,a =hand Vc,b,a =h espec i ely. These le el se s can be seen as he algeb aic cu es gi en by C2:= {c2(x, y) = ax2y+bxy2+bx2−hxy +ay2+ (b2+a)x+ (b+a2)y+ab = 0} and C3:= {c3(x, y) = cx2y+axy2+bx2−hxy +by2+ (a+bc)x+ (c+ab)y+ac = 0}, espec i ely. Taking homogeneous coo dina es on he p ojec i e plane PR2bo h cu es C2and C3 ha e he o m C:= {Sx2y+T xy2+Ux2z+V xyz +Wy2z+Lxz2+Myz2+Nz3= 0}. In o de o ind he b anches o hem ending o in ini y, we examine he di ec ions o app oach o in ini y (z= 0) in he local cha s de e mined by x= 1 and y= 1 espec i ely. In he local cha gi en by x= 1, he cu e Cw i es as Sy +Ty2+Uz +V yz +Wy2z+Lz2+Myz2+Nz3= 0 and i mee s he s aigh line a in ini y z= 0 when y(S+Ty) = 0.Since o bo h cu es C2and C3 he coe icien s Sand Ta e posi i e, he only in e sec ion poin ha could gi e poin s in Q+is (y, z) = (0,0).The algeb aic cu e Ca i es o (y, z) = (0,0) angen ially o he line Sy +Uz = 0.Since o bo h cu es, C2and C3, he coe icien s Sand Ua e also posi i e, we ha e ha he b anches o he le el se s ending o in ini y a e no included in Q+. An analogous s udy can be made in he cha gi en by y= 1,ob aining he same conclusions. Mo eo e , i can be easily checked ha in he a ine plane bo h cu es C2and C3do no in e sec he pa o he axes x= 0 and y= 0 which is in he bounda y o Q+. 5 In summa y, he e a e no b anches o he cu es C2and C3 ending o in ini y o c ossing he axes x= 0 and y= 0 in Q+, and he e o e he connec ed componen s o Ci∩Q+ o i= 2,3 a e bounded. No ice ha his esul in pa icula al eady implies he pe sis ence o he sequences gi en by (1). Conside k= 2. We claim he ollowing ac s: (a) In Q+, he se o ixed poin s o Fb,a and he se o singula poin s o C1coincide and hey only con ain he poin p= (¯x,¯y). (b) The unc ion Vb,a(x, y),has a local minimum a p. We ema k ha i em (b) is al eady known. We p esen a new simple p oo o he sake o comple eness. F om he abo e claims and he ac ha he connec ed componen s o he le el se s o Vb,a in Q+a e bounded i ollows ha he le el se s o Vb,a in Q+ {p}a e di eomo phic o ci cles. Le us p o e he abo e claims. The ixed poin s o Fb,a a e gi en by    x=a+y x, y=a+bx+y xy ,⇔   x2=a+y, x(y2−b) = a+y, and so x2=x(y2−b).Hence in Q+,we ha e ha x=y2−band he abo e sys em is equi alen o    x=y2−b, xy2−bx −y−a= 0,⇔   x=y2−b, P(y) := y4−2by2−y+b2−a= 0. I is no di icul o check ha he las sys em o equa ions is p ecisely he one ha gi es he c i ical poin s o he cu es Va,b =h. Mo eo e , om he i s equa ion i is necessa y ha x=y2−b > 0 and hence y > √b. Since P(y) has only one eal oo in (√b, ∞) he uniqueness o he c i ical poin holds. Le us p o e ha his c i ical poin co esponds wi h a local minimum o Vb,a.We will check he usual su icien condi ions gi en by he Hessian o Vb,a a p. Fi s ly, ∂2 ∂x2Vb,a(y2−b, y) = 2 (y+a) (ay +b) (y2−b)3y>0 o y > √b. Secondly, he de e minan o he Hessian ma ix a he poin s (y2−b, y) is h(y) = (y) (b−y2)4y4, 6 whe e (y) := (by2+a−b2)(−by6+ 3 a+b2y4+ 4 a2+by3+ 3b2a−b2y2+b2(b2−a). A edious compu a ion shows ha (y) = q(y)P(y) + (y),wi h (y) = 4a2b2+ 4a3+ 4b3+ 6aby3+18a2b+ 8b3a+ 3b2y2 +−4b3a2+ 4a3b−4b4+ 12ab2+ 3a2y−8b4a+ 5a2b2+ 3a3. Obse e ha i ¯yis he posi i e oo o P(y), hen sign(h(¯y)) = sign( (¯y)). Taking in o accoun ha P(¯y) = 0 implies ha a= ¯y4−2b¯y2−¯y+b2we ha e ha (¯y) = ¯y24 ¯y3−4b¯y−1b¯y+ 1 −¯y32b−¯y22. So, sign 4 ¯y3−4b¯y−1= sign (P′(¯y)) .Since P(√b) = −√b−a < 0,lim y→∞ P(y) = +∞ and, on his in e al, he e is only one c i ical poin o P(y),which is simple, we ge ha P′(¯y)>0 and so h(¯y)>0.Hence pis a local minimum o Vb,a(x, y), as we wan ed o p o e. The same kind o a gumen s wo k o end he p oo o he case k= 3,bu he compu- a ions a e ex emely mo e edious. We only make some commen s. The ixed poin s o Fc,b,a in Q+a e gi en by:      P(x) := x5+ax4−2x3−(2a+bc)x2+ (1 −b2−c2)x+a−bc = 0, y=Q(x) := (xb +a) (x−1)(x+ 1). I can be p o ed again ha hey coincide wi h he singula poin s o Vc,b,a in Q+.This ac ollows om he compu a ion o se e al sui able esul an s be ween ∂Vc,b,a/∂x, ∂Vc,b,a/∂y and Vc,b,a. The uniqueness o he ixed poin pin Q+can be shown as ollows: since Q(x)>0 implies ha x > 1, we only need o sea ch solu ions o P5(x) = 0 in (1,+∞). Wi h he new a iable z=x−1, e P(z) := P(z+ 1) := z5+ (c+ 5)z4+ (8 + 4c)z3+ (4 −ab + 4c)z2−(a+b)z−(a+b)2= 0, Since e P(0) <0; lim z→+∞e P(z) = +∞; and he Desca e’s ule, we know ha he e is only one posi i e solu ion, as we wan ed o see. Finally i can be p o ed ha pis a non-degene a ed local minimum o Vc,b,a. These compu a ions a e complica ed, and hey ha e been pe o med in a e y sma way in [14], so we skip hem and we e e he eade o his las e e ence. 7 3.1 P oo o (ii) o Theo em 2 In [6] i is p o ed a esul ha cha ac e izes he dynamics o in eg able di eomo phisms ha ing a Lie Symme y, ha is a ec o ield Xsuch ha X(F(p)) = (DF(p)) X(p). Nex heo em s a es i , pa icula ized o he case we a e in e es ed. Theo em 4 ([6]).Le U ⊂ R2be an open se and le Φ : U → U be a di eomo phism such ha : (a) I has a smoo h egula i s in eg al V:U → R,ha ing i s le el se s Γh:= {z= (x, y)∈ U :V(z) = h}as simple closed cu es. (b) The e exis s a smoo h unc ion µ:U → R+such ha o any z∈ U, µ(Φ(z)) = de (DΦ(z)) µ(z). Then he map Φ es ic ed o each Γhis conjuga ed o a o a ion wi h o a ion numbe τ(h)/T(h), whe e T(h)is he pe iod o Γhas a pe iodic o bi o he plana di e en ial equa ion ˙z=µ(z)−∂V (z) ∂y ,∂V (z) ∂x  and τ(h)is he ime needed by he low o his equa ion o going om any w∈Γh o Φ(w)∈Γh. Nex lemma is one o he key poin s o inding a Lie symme y o amilies o pe iodic maps, like he 2 and 3–pe iodic Lyness maps. Lemma 5. Le {Ga}a∈Abe a amily o di eomo phisms o U ⊂ R2. Suppose ha he e exis s a smoo h map µ:U → Rsuch ha o any a∈Aand any z∈ U, he equa ion µ(Ga(z)) = de (DGa(z)) µ(z)is sa is ied. Then, o e e y choice a1,...,ak∈A, we ha e µ(G[k](z)) = de (DG[k](z)) µ(z), whe e G[k]=Gak◦···◦Ga2◦Ga1. P oo . I is only necessa y o p o e he esul o k= 2 because he gene al case ollows easily by induc ion. Conside a1, a2∈A hen µ(Ga2,a1(z)) = µ(Ga2◦Ga1(z)) = de (DGa2(Ga1(z))) µ(Ga1(z)) = = de (DGa2(Ga1(z))) de (DGa1(z)) µ(z) = de (D(Ga2◦Ga1(z))) µ(z) = = de (DGa2,a1(z))µ(z), and he lemma ollows. 8 P oo o (ii) o Theo em 2. F om pa (i) o he heo em we know ha he le el se s o Vb,a and Vc,b,a in Q+ {p}a e di eomo phic o ci cles. Mo eo e hese unc ions a e i s in eg als o Fb,a and Fc,b,a, espec i ely. No ice also ha o any a, he Lyness map Fa(x, y) = (y, a+y x) sa is ies µ(Fa(x, y)) = de (DFa(x, y))µ(x, y), wi h µ(x, y) = xy. Hence, by Lemma 5, µ(Fb,a(x, y)) = de (DFb,a(x, y))µ(x, y) and µ(Fc,b,a(x, y)) = de (DFc,b,a(x, y))µ(x, y). Thus, om Theo em 4, he esul ollows. I is wo h o commen ha once pa (i) o he heo em is p o ed i is also possible o p o e ha he dynamics o Fb,a ( esp. Fc,b,a) es ic ed o he le el se s o Vb,a ( esp. Vc,b,a) is conjuga ed o a o a ion by using ha hey a e gi en by cubic cu es and ha he map is bi a ional, see [13]. We p e e ou app oach because i p o ides a dynamical in e p e a ion o he o a ion numbe oge he wi h i s analy ic cha ac e iza ion. 4 P oo o Theo em 1 In o de o p o e Theo em 1 we need a p elimina y esul . Conside he maps Fb,a and Fa,b,join ly wi h hei co esponding i s in eg als Vb,a and Va,b.In a simila way conside Fc,b,a , Fa,c,b and Fb,a,c wi h Vc,b,a , Va,c,b and Vb,a,c.Some simple compu a ions p o e he ollowing elemen a y bu use ul lemma. No ice ha i can be in e p e ed as he ela ion be ween he i s in eg als and he non-au onomous in a ian s. Lemma 6. Wi h he abo e no a ions: (i) Vb,a(x, y) = Va,b(Fa(x, y)). (ii) Vc,b,a(x, y) = Va,c,b(Fa(x, y)) = Vb,a,c(Fb(Fa(x, y))). P oo o Theo em 1. We spli he p oo in wo s eps. Fo k= 2,3 we i s p o e ha he e a e only wo ypes o beha io s o {xn}n, ei he his se o poin s is o med by kp poin s o some posi i e in ege p, o i has in ini ely many poin s whose adhe ence is gi en by a mos kin e als. Secondly, in his la e case, we p o ide an algeb aic way o s udying he ac ual numbe o in e als. Fi s s ep: We s a wi h he case k= 2.Wi h he no a ion in oduced in (2), i holds ha Fb,a(x2n−1, x2n) = (x2n+1, x2n+2), Fa,b(x2n, x2n+1) = (x2n+2, x2n+3), 9 In ac when we say ha ρ2,3(34) ∈(ρlow, ρupp), he alue ρlow is he uppe lowe bound ob ained by ollowing all he conside ed poin s o he o bi , and ρupp is he lowes uppe bound. No ice ha aking 1000 o 3000 poin s we ha e ob ained he same lowe bound o ρ2,3(34). Le us p o e Theo em 3 by using he abo e app oach. P oo o Theo em 3. Conside a= 1/2, b = 3/2 and he h ee poin s p1=149 100,173 100,p2=3 40,173 100,p3=1 1000,173 100. No ice ha h1:= V3/2,1/2(p1) = 10655559 1288850 ≃8.27, h2:= V3/2,1/2(p2) = 9328327 207600 ≃44.93, h3:= V3/2,1/2(p3) = 1056238343 346000 ≃3052.71. Hence hc< h1< h2< h3.By applying he algo i hm desc ibed abo e, using 100 poin s o each o bi s a ing a each pj, j = 1,2,3,we ob ain ha ρ3/2,1/2(h1), ρ3/2,1/2(h3)∈3 5,59 98and ρ3/2,1/2(h2)∈56 93,53 88. Since 59/98 <56/93 we ha e p o ed ha he unc ion ρ3/2,1/2(h) has a leas a local maximum in (h1, h3).F om he con inui y o he o a ion numbe unc ion, wi h espec a, b and h, we no ice ha his esul also holds o all alues o aand bin a neighbo hood o a= 1/2, b = 3/2. We belie e ha wi h he same me hod i can be p o ed ha a simila esul o he one gi en in Theo em 3 holds o some maps Fc,b,a,bu we ha e decided do no pe o m his s udy. 5.1 Some nume ical explo a ions o k= 2. We s a by s udying wi h mo e de ail he o a ion numbe unc ion ρ3/2,1/2(h), ha we ha e conside ed o p o e Theo em 3. In his case he ixed poin is p≃(1.493363282,1.730133891) and hc=Vb,a(p) = 8.267483381.Mo eo e ρb,a(hc)≃0.6006847931. By applying ou al- go i hm o app oxima ing he o a ion numbe , wi h 5000 poin s on each o bi , we ob ain he esul s p esen ed in Table 1. In Figu e 3 we also plo he uppe and lowe bounds o ρ3/2,1/2(h) ha we ha e ob ained by using a wide ange o alues o h. 16 Ini . cond. (x, ¯y) Ene gy le el h ρlow(h)ρupp(h) ¯x hc≃8.2675 ≃0.6006848 ≃0.6006848 1.3 8.3068 173 288 ≃0.6006944 2938 4891 ≃0.6006951 0.75 9.2747 1435 2388 ≃0.6009213 2087 3473 ≃0.6009214 0.3 14.7566 1548 2573 ≃0.6016323 2285 3798 ≃0.6016324 0.075 44.9347 657 1091 ≃0.6021998 2354 3909 ≃0.6022001 0.001 3052.75 2927 4867 ≃0.6013972 86 143 ≃0.6013986 5·10−6609716.07 1832 3049 ≃0.6008527 1409 2345 ≃0.6008529 5·10−256 6.097 ·10255 3 5= 0.62999 4998 ≃0.6000400 Table 1: Lowe and uppe bounds o he o a ion numbe ρ3/2,1/2(h), o some o bi s o F3/2,1/2s a ing a (x, ¯y),whe e p= (¯x, ¯y). Figu e 3: Lowe and uppe bounds o ρ3/2,1/2(h). On he ho izon al axis we ep esen −log10(h) and, on he e ical axis, he alue o he o a ion numbe . No ice ha o alues o −log10(h) smalle ha 70 bo h alues a e indis inguishable in he Figu e. Fo o he alues o aand bwe ob ain di e en beha io s. All he expe imen s a e pe o med by s a ing a he ixed poin p= (¯x, ¯y),and inc easing he ene gy le el by aking ini ial condi ions o he o m (x, y), by dec easing x o 0. Wi h his p ocess we ake o bi s app oaching o he bounda y o Q+, ha is lying on le el se s o Vb,a wi h inc easing ene gy. The s ep in he dec ease o x(and he e o e in he inc ease o h) is no uni o m, 17 and i has been manually uned making i smalle in hose egions whe e a possible non mono onous beha io could appea . Conside he se o pa ame e s Γ = {(a, b),∈[0,∞)2}, whe e no ice ha we also conside he bounda ies a= 0 o b= 0,whe e he map Fb,a is well de ined. We al eady know ha he o a ion numbe unc ion beha es equal a (a, b) and (b, a).Mo eo e we know pe ec ly i s beha io on he diagonal (a, a) (when a < 1 i is mono onous dec easing and when a > 1 i is mono onous inc easing) and ha ρ1,1(h)≡4/5 and ρ0,0(h)≡2/3.Hence a good s a egy o an nume ical explo a ion can be o p oduce sequences o expe imen s using ou algo i hm by ixing some a≥0 and a ying b. Fo ins ance we ob ain: •Case a= 1/2.Fo all he alues o b > 0 conside ed, he o a ion numbe unc ion seems o end o 3/5 when hgoes o in ini y. Mo eo e i seems –mono one dec easing o b∈ {1/4,1}; – o ha e a unique maximum when b∈ {7/5,3/2}; –mono one inc easing o b∈ {2,3}. •Case a= 0.Fo all he alues o b > 0 conside ed, he o a ion numbe unc ion seems o end o 5/8 when hgoes o in ini y. Mo eo e i seems –mono one dec easing o b∈ {1/10,3/10,1/2}; – o ha e a unique maximum when b∈ {7/10,3/4}; –mono one dec easing o b∈ {1,5}. The abo e esul s, oge he wi h some o he expe imen s o o he alues o aand b, no de ailed in his pape , indica e he exis ence o a subse o posi i e measu e in Γ whe e he co esponding o a ion numbe unc ions seem o p esen an unique maximum. This subse p obably sepa a es wo o he subse s o Γ, one whe e ρb,a(h) is mono onically dec easing o 3/5 , and ano he one whe e ρb,a(h) inc eases mono onically o he same alue. The “oscilla o y subse ” seems o sh ink o (a, b) = (1,1) when i app oaches o he line a=b and seems o inish in one in e al on each o he bo de s {a= 0}and {b= 0}. Fu he analysis mus be done in his di ec ion in o de o ha e a mo e accu a e knowledge o he bi u ca ion diag am associa ed o he beha io o ρb,a on Γ. Re e ences [1] W.J. Beyn, T. H¨uls, M.Ch. Sam enschniede . On –pe iodic o bi s o k–pe iodic maps, J. Di e ence Equ. Appl. 14 (2008), 865–887. 18 [2] G. Bas ien, M. Rogalski. Global beha io o he solu ions o Lyness’ di e ence equa ion un+2un=un+1 +a, J. Di e ence Equ. 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