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.
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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,
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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.
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