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 Universitat Polit` ecnica de Catalunya, CoDALab ∗ . 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 / 20
1. INTRODUCTION 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 a F b F a F b 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 / 20
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 / 20
Theorem A Consider the family F b , a with a , b > 0 . (i) If ( a , b ) � = ( 1 , 1 ) , then ∃ p 0 ( a , b ) ∈ N , generically computable, 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 , generically computable, 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 . Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 4 / 20
Digression: why to focus on the 2-periodic case? Because of computational issues, and because is one of the few integrable ones. For each k , the composition maps are F [ k ] := F a k ,..., a 2 , a 1 = F a k ◦ · · · ◦ F a 2 ◦ F a 1 (4) where � � y , a i + y F a i ( x , y ) = and a 1 , a 2 , . . . , a k are a k-cycle. x The figure summarizes the situation. k = ˙ All k 5 k �∈ { 1 , 2 , 3 , 5 , 6 , 10 } (*) The cases 1,2,3 and 6 have first integrals given by V ( x , y ) = P 3 ( x , y ) (Cima, Gasull, M; 2012b) . xy (*) This phase portraits are the ones of the DDS associated with the recurrence obtained after the change z n = log ( u n ) . Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 5 / 20
2. THE STRATEGY: analysis of the asymptotic behavior of θ b , a ( h ) . The main issues that allow us to compute the allowed periods are: 1 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 ) � = → + ∞ θ b , a ( h ) = lim ⇒ ∃ I ( a , b ) , a rotation interval . 2 h − ∀ θ ∈ I ( a , b ) , ∃ at least an oval C + h s.t. F b , a restricted to the this oval is conjugate to a rotation, with a rotation number θ b , a ( h ) = θ In particular, for all the 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 6 / 20
Proof of Theorem A (i). Proposition B. h → + ∞ θ b , a ( h ) = 2 lim 5 � 1 � �� θ b , a ( h ) = σ ( a , b ) = 1 1 lim 2 π arccos − 2 + . 2 x c y c h → h c Theorem C. � � σ ( a , b ) , 2 Set I ( a , b ) := . 5 If σ ( a , b ) � = 2 / 5 , for any fixed a , b > 0 , and any θ ∈ I ( a , b ) , ∃ at least 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 . Which are the periods of a particular F b , a ? ⇔ Which are the irreducible fractions in I ( a , b ) ? • If σ ( a , b ) � = 2 / 5, it is possible to obtain constructively a value p 0 s.t. for any r > p 0 ∃ an irreducible fraction q / r ∈ I ( a , b ) . • A finite checking determines which values of p ≤ p 0 are s.t. ∃ q / p ∈ I ( a , b ) . � � • Still the forbidden periods must be detected. Since I ( a , b ) ⊆ Image θ b , a ( h c , + ∞ ) . Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 7 / 20
Generically? Set P := { ( a , b ) , a , b > 0 } : The curve σ ( a , b ) = 2 / 5 for a , b > 0 is given by � � t 3 − φ 2 � � , φ 4 − t 3 2 4 3 , φ 3 ) Γ := { σ ( a , b ) = 2 / 5 , a , b > 0 } = ( a , b ) = , t ∈ ( φ ⊂ P . t 2 t Of course P \ Γ is open and dense in P Bastien, Ma˜ nosa & Rogalski (Paris 6-UPC) 2-periodic Lyness’ equations 18th ICDEA 8 / 20
3. The periods of the family F b , a . Proof of Theorem A (ii) 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 D. For each θ in ( 1 / 3 , 1 / 2 ) ∃ a , b > 0 and at least 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 ) , ∃ periodic orbits of F b , a of prime period p . 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 9 / 20
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 ∈ ( 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 10 / 20
4. Back to the rotation number: an algebraic-geometric approach. 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 11 / 20
� � 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 + nH , so � C h is full of p -periodic orbits iff 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 12 / 20
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