On periodic solutions of 2periodic Lyness difference equations nosa - PowerPoint PPT Presentation

On periodic solutions of 2–periodic Lyness difference equations nosa 2 and Marc Rogalski 3 Guy Bastien 1 , V´ ıctor Ma˜ 1 Institut Math´ ematique de Jussieu, Universit´ e Paris 6 and CNRS, 2 DMA3-CoDALab, Universitat Polit` ecnica de Catalunya ∗ . 3 Laboratoire Paul Painlev´ e, Universit´ e de Lille 1; Universit´ e Paris 6 and CNRS, 18th International Conference on Difference Equations and Applications July 2012, Barcelona, Spain. ∗ Supported by MCYT’s grant DPI2011-25822 and SGR program. Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 1 / 13

We study the set of periods of the 2-periodic Lyness’ equations u n + 2 = a n + u n + 1 , (1) u n where � a for n = 2 ℓ + 1 , a n = (2) b for n = 2 ℓ, and being ( u 1 , u 2 ) ∈ Q + ; ℓ ∈ N and a > 0 , b > 0. This can be done using the composition map: � a + y � , a + bx + y F b , a ( x , y ) := ( F b ◦ F a )( x , y ) = , (3) x xy � � y , α + y where F a and F b are the Lyness maps: F α ( x , y ) = . Indeed: x F b F b F a F a F a ( u 1 , u 2 ) − → ( u 2 , u 3 ) − → ( u 3 , u 4 ) − → ( u 4 , u 5 ) − → ( u 5 , u 6 ) − → · · · Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 2 / 13

The map F b , a : • Is a QRT map whose first integral is (Quispel, Roberts, Thompson; 1989): V b , a ( x , y ) = ( bx + a )( ay + b )( ax + by + ab ) , xy see also (Janowski, Kulenovi´ c, Nurkanovi´ c; 2007) and (Feuer, Janowski, Ladas; 1996). • Has a unique fixed point ( x c , y c ) ∈ Q + , which is the unique global minimum of V b , a in Q + . • Setting h c := V b , a ( x c , y c ) , for h > h c the level sets { V b , a = h } ∩ Q + are the closed curves. h := { ( bx + a )( ay + b )( ax + by + ab ) − hxy = 0 } ∩ Q + for h > h c . C + The dynamics of F b , a restricted to C + h is conjugate to a rotation with associated rotation number θ b , a ( h ) . Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 3 / 13

Theorem A Consider the family F b , a with a , b > 0 . (i) If ( a , b ) � = ( 1 , 1 ) , then ∃ p 0 ( a , b ) ∈ N s.t. for any p > p 0 ( a , b ) , ∃ at least an oval C + h filled by p–periodic orbits. (ii) The set of periods arising in the family { F b , a , a > 0 , b > 0 } restricted to Q + contains all prime periods except 2 , 3 , 4 , 6 , 10 . Corollary. Consider the 2 –periodic Lyness’ recurrence for a , b > 0 and positive initial conditions u 1 and u 2 . (i) If ( a , b ) � = ( 1 , 1 ) , then ∃ p 0 ( a , b ) ∈ N , s.t. for any p > p 0 ( a , b ) ∃ continua of initial conditions giving 2 p–periodic sequences. (ii) The set of prime periods arising when ( a , b ) ∈ ( 0 , ∞ ) 2 and positive initial conditions are considered contains all the even numbers except 4 , 6 , 8 , 12 , 20 . If a � = b, then it does not appear any odd period, except 1 . The value p 0 ( a , b ) is computable for an open and dense set in the parameter space. Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 4 / 13

To compute the allowed periods, the main issues to take into account are: • The fact that the rotation number function θ b , a ( h ) is continuous in [ h c , + ∞ ) . • The fact that generically θ b , a ( h c ) � = h → + ∞ θ b , a ( h ) ⇒ ∃ I ( a , b ) , a rotation interval . lim Proposition B. � 1 � �� 1 1 2 lim θ b , a ( h ) = σ ( a , b ) := arccos − 2 + , and h → + ∞ θ b , a ( h ) = lim . h → h + 2 π 2 x c y c 5 c Corollary � � σ ( a , b ) , 2 Set I ( a , b ) := . 5 • If σ ( a , b ) � = 2 / 5 ∀ θ ∈ I ( a , b ) , ∃ an oval C + h s.t. F b , a ( C + h ) is conjugate to a rotation , with a rotation number θ b , a ( h ) = θ . • In particular, ∀ irreducible q / p ∈ I ( a , b ) , ∃ periodic orbits of F b , a of prime period p . Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 5 / 13

The periods of the family F b , a . Using the previous results with the family a = b 2 we found that: � 1 � � � � 3 , 1 I ( b 2 , b ) = ⊂ I ( a , b ) ⊂ Image ( θ b , a ( h c , + ∞ )) . 2 b > 0 a > 0 , b > 0 a > 0 , b > 0 Proposition. • For each θ in ( 1 / 3 , 1 / 2 ) ∃ a , b > 0 and an oval C + h , s.t. F b , a ( C + h ) is conjugate to a rotation with rotation number θ b , a ( h ) = θ. • In particular, ∀ irreducible q / p ∈ ( 1 / 3 , 1 / 2 ) , ∃ p-periodic orbits of F b , a We’ll know some periods of { F b , a , a , b > 0 } ⇔ We know which are the irreducible fractions in ( 1 / 3 , 1 / 2 ) Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 6 / 13

Lemma (Cima, Gasull, M; 2007) Given ( c , d ) ; Let p 1 = 2 , p 2 = 3 , p 3 , . . . , p n , . . . be all the prime numbers. • Let p m + 1 be the smallest prime number satisfying that p m + 1 > max ( 3 / ( d − c ) , 2 ) , • Given any prime number p n , 1 ≤ n ≤ m , let s n be the smallest natural number such that p sn > 4 / ( d − c ) . n • Set p 0 := p s 1 − 1 p s 2 − 1 · · · p sm − 1 . m 1 2 Then, for any p > p 0 ∃ an irreducible fraction q / p s.t. q / p ∈ ( c , d ) . Proof of Theorem A (ii): • We apply the above result to ( 1 / 3 , 1 / 2 ) . ∀ p ∈ N , s.t. p > p 0 p 0 := 2 4 · 3 3 · 5 · 7 · 11 · 13 · 17 = 12 252 240 , ∃ an irreducible fraction q / p ∈ ( 1 / 3 , 1 / 2 ) . • A finite checking determines which values of p ≤ p 0 s.t. q / p ∈ ( 1 / 3 , 1 / 2 ) , resulting that there appear irreducible fractions with all the denominators except 2 , 3 , 4 , 6 and 10. • Proposition C = ⇒ ∃ a , b > 0 s.t. ∃ an oval with rotation number θ b , a ( h ) = q / p , thus giving rise to p –periodic orbits of F b , a for all allowed p . • Still it must be proved that 2, 3, 4, 6 and 10 are forbidden, since � � I ( a , b ) ⊆ Image θ b , a ( h c , + ∞ ) Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 7 / 13

Continuity and asymptotic behavior of θ b , a ( h ) . The curves C h , in homogeneous coordinates [ x : y : t ] ∈ C P 2 , are � C h = { ( bx + at )( ay + bt )( ax + by + abt ) − hxyt = 0 } . The points H = [ 1 : 0 : 0 ]; V = [ 0 : 1 : 0 ]; D = [ b : − a : 0 ] are common to all curves Proposition If a > 0 and b > 0 , and for all h > h c , the curves � C h are elliptic. Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 8 / 13

� � F b , a extends to C P 2 as � ayt + y 2 : at 2 + bxt + yt : xy F b , a ([ x : y : t ]) = . Lemma. Relation between the dynamics of F b , a and the group structure of C h (*) For each h s.t. � C h is elliptic, � F b , a | � ( P ) = P + H C h Where + is the addition of the group law of � C h taking the infinite point V as the zero element. Observe that F n ( P ) = P + n H , so � C h is full of p -periodic orbits ⇔ pH = V i.e. H is a torsion point of � C h . (*) Birational maps preserving elliptic curves can be explained using its group structure (Jogia, Roberts, Vivaldi; 2006). Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 9 / 13

Instead of looking to a normal form for F we look for a normal form for � C h . � � � � ∼ � = E L , + , � � C h , + , V − → V � G | E L : P �→ P + � � F | � C h : P �→ P + H − → H Where � E L is the Weierstrass Normal Form which in the affine plane is: E L = { y 2 = 4 x 3 − g 2 x − g 3 } with g i := g i ( a , b , h ) . WHY? Because we can parameterize it using the Weierstrass ℘ function... 1 ...that gives an integral expression for the rotation number function . 2 � + ∞ d s � 4 s 3 − g 2 s − g 3 X ( L ) 2 Θ( L ) = � + ∞ where θ b , a ( h ) ∼ Θ( L ) d s � 4 s 3 − g 2 s − g 3 e 1 The asymptotics of this integral expression can be studied. 3 This scheme was used in (Bastien, Rogalski; 2004). Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 10 / 13

The Weierstrass normal form of C h is E L = { y 2 = 4 x 3 − g 2 x − g 3 } where     7 11 � � 1 1  L 8 +  − L 12 + p i ( α, β ) L i q i ( α, β ) L i   , g 2 = and g 3 = 192 13824 i = 4 i = 6 being p 7 ( a , b ) = − 4 ( α + β + 1 ) , � � 3 ( α − β ) 2 + 2 ( α + β ) + 3 p 6 ( a , b ) = 2 , � � α 2 − 4 βα + β 2 − 1 p 5 ( a , b ) = − 4 ( α + β − 1 ) , ( α + β − 1 ) 4 . p 4 ( a , b ) = and q 11 ( a , b ) = 6 ( α + β + 1 ) , � � − 5 α 2 + 2 αβ − 5 β 2 − 6 α − 6 β − 5 q 10 ( a , b ) = 3 � � 5 α 3 − 12 α 2 β − 12 αβ 2 + 5 β 3 + 3 α 2 − 3 αβ + 3 β 2 + 3 α + 3 β + 5 q 9 ( a , b ) = 4 � − 5 α 4 + 16 α 3 β − 30 α 2 β 2 + 16 αβ 3 − 5 β 4 + 4 α 3 q 8 ( a , b ) = 3 � − 12 α 2 β − 12 αβ 2 + 4 β 3 + 2 α 2 − 8 αβ + 2 β 2 + 4 α + 4 β − 5 � � α 2 − 4 αβ + β 2 − 1 ( α + β − 1 ) 3 q 7 ( a , b ) = 6 − ( α + β − 1 ) 6 q 6 ( a , b ) = where α = a / b 2 and b / a 2 and L → + ∞ ⇔ h → + ∞ . Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 11 / 13