Deautonomising the Lyness mapping

Deautonomising the Lyness mapping B. Gramma ticos and A. Ramani Univ ersit ´ e P aris-Sacla y and Univ ersit ´ e de P aris-Cit´ e, CNRS/IN2P3, IJCLab, 91405 Orsa y , F rance R. Willox Graduate Sc ho ol of Mathematical Sciences, the Univ ersit y of T okyo, 3-8-1 Komaba, Meguro-ku, 153-8914 T okyo, Japan Abstract W e examine the Lyness mapping (an in tegrable N th-order discrete system whic h can b e generated from a one-dimensional reduction of the Hirota-Miwa equation) from the p oint of view of deautonomisation. W e sho w that only the N = 2 case can be deautonomised when one works with the standard form of the mapping. Ho w ev er it turns out that deautonomisation is p ossible for arbitrary N when one considers the deriv ative form of the Lyness mapping. The deautonomisation of the deriv ative of the N = 2 case leads to a result w e ha ve never met before: the secular dep endence in the co efficients of the mapping en ters through t w o differen t exp onen tial terms instead of just a single one. As a consequence, it turns out that a limit of this multiplicativ e dep endence tow ards an additive one is p ossible without mo difying the dep enden t v ariable. Finally , the analysis of the ‘late’ singularity confinement of the N = 2 case leads to a no v el realisation of the full-deautonomisation principle: the dynamical degree is not giv en (as is customary) simply by the solution of some linear or multiplicativ e equation, but is present in the growth of the non-linear (and non-in tegrable) late-confinemen t conditions. 1. Introduction While the beginning of the mo dern era of in tegrabilit y in differen tial systems can unarguably be asso ciated with the discov ery of the soliton b y Krusk al and Zabusky [1], this is less clearcut when it comes to difference systems. Still, it is indisputable that the deriv ation of what came to b e known as the Quisp el- Rob erts-Thompson (QR T) mapping [2] did pla y an important role in launc hing the activity in the domain of discrete systems that w e ha v e seen ov er the last 30 years. T o be sure, the fundamentals of the approac h, as V eselov points out in [3], can be traced back to Euler with the parametrisation of the bi-quadratic relation αx 2 y 2 + β xy ( x + y ) + γ ( x 2 + y 2 ) + xy + ζ ( x + y ) + µ = 0 , (1) in terms of elliptic functions. (The latter is actually given in Baxter’s b o ok [4] without any attribution or references, which suggests that b y then the result was already long part of the common heritage). Still, the deriv ation of the QR T mapping, of whic h (1) is the in v arian t, w as the crucial step establishing the link with integrabilit y studies. Moreo v er, soon after proposing a mapping that has (1) as in v arian t, for general v alues of the parameters, the authors of [2] presented an extension to what came to b e called the asymmetric case, asso ciated to the in v ariant αx 2 y 2 + β x 2 y + δ xy 2 + γ x 2 + κy 2 + xy + ζ x + λy + µ = 0 . (2) (It is in teresting that while the in tegration of (1) w as kno wn for o v er a cen tury , that of (2) had to w ait for more than a decade. In the end it turned out that it can b e parametrised in terms of elliptic functions [5,6], just as (1)). Be that as it may , the existence of whole families of integrable mappings pro vided a ric h playground for the prop osal and assay of v arious tec hniques, foremost of whic h w ere those designed to detect the 1 in tegrable character of a given mapping of the plane. While studying the singularities that may app ear sp on taneously in suc h systems, it w as observed that when the system was in tegrable b y sp ectral metho ds, an y singularity appearing due to some special choice of initial conditions did disapp ear after a certain n um b er of iteration steps. This prop ert y was dubbed ‘singularity confinemen t’ [7] and it was readily elev ated to a discrete in tegrabilit y criterion. Here tw o imp ortant cav eats are necessary . First, there exists a whole class of discrete systems, in tegrable through linearisation [8], which can possess unconfined singularities but this do es not compromise their integrable character. Second, a whole class of non- in tegrable mappings with only confined singularities also exists [9,10]. Do es this signal the inadequacy of singularit y confinement as an in tegrabilit y criterion? Nothing is further from the truth [11], as we shall see shortly , but for this we first need to introduce the notion of ‘deautonomisation’. A domain where singularity confinemen t excels is that of deautonomisation [12]. The latter consists in deriving non-autonomous extensions of initially autonomous, integrable, mappings, by assuming that the v arious parameters app earing in them can be functions of the indep endent v ariable. Using the singularit y confinemen t criterion in order to obtain the precise forms of said functions then necessarily results into an integrable non-autonomous system [12]. In fact it was the deautonomisation approac h that pro vided the key to resolving the paradox of non-integrable mappings with confined singularities. It was first noticed [13], and subsequently rigorously pro ven [14], that the deautonomisation based on singularit y confinemen t pro vides a definite wa y to rule on the integrabilit y of a giv en system, leading to what was termed the ‘full-deautonomisation’ metho d as a discrete in tegrabilit y criterion. The mechanism that underlies the confinement of singularities in the case of birational mappings (which constitute the v ast ma jority of those studied) is based on simplifications of the rational expressions for the iterates of the mapping, leading to the disappearance of an y indefiniteness in those iterates. These simplifications also hav e a consequence that gav e birth to another discrete in tegrabilit y criterion. Studying the degrees of the polynomials (in the v ariables in tro duced by the initial conditions) in the numerator and denominator of the iterates of the mapping, it w as observ ed that their degree increased slo wly (in fact, p olynomially) in the case of integrable mappings, while for non-integrable ones the growth was exp onen tial [10]. This gro wth can b e assessed quan titativ ely b y considering what is called the dynamical degree. The latter is obtained from the degree d n of the iterates, for some generic initial conditions, as λ = lim n →∞ d 1 /n n . (3) It is greater than 1 when the gro wth is exp onen tial and equal to 1 otherwise [10]. The full-deautonomisation approac h men tioned ab o v e originated in the observ ation that the conditions for deautonomisation of some autonomous mapping, be it integrable or not, yield direct information on the dynamical degree. When the non-autonomous mapping obtained from the deautonomisation is integrable this amounts to the trivial observ ation that the dynamical degree inferred from the deautonomisation conditions is equal to 1. It is in the non-integrable case ho wev er that things become truly interesting, as the confinement conditions lead to the exact v alue of the dynamical degree for b oth the non-autonomous map as w ell as for the original autonomous one. The main bulk of integrabilit y studies for one-dimensional discrete systems concerns second-order map- pings. Over the past decades an enormous effort has gone into developing metho ds for the study of these lo w-order systems. And th us, naturally , the question arose whether these methods can still be of help when v en turing into what is largely a terra incognita of higher-order systems. In [15] we offered some in- sigh t in to ho w one can adapt the existing methods in order to deal with systems of orders 3 or higher. W e sho w ed that the study of ‘singularities’ and of ‘growth’ is again an essential tool for the assessment of the in tegrable character of a given higher order mapping. How ever, b oth notions, unambiguously defined in the case of second order, need some clarification in the higher-order case. In all cases of order N studied, 2 it turned out that w e could tak e generic initial conditions for the dependent v ariables x 0 , x 1 , · · · , x N − 2 and compute the degree of the successive iterates in terms of x N − 1 . Similarly , in our singularity analysis, it was sufficient to restrict ourselves to a situation where x N − 1 w as the only initial condition allow ed to tak e sp ecial v alues leading to singularities. It is our empirical finding that this approach works and leads to the prop er answer; Still, it is clear that it is dev oid of a rigorous foundation, the results obtained in [10] b eing sp ecific to the second-order case. With this p oint clarified w e can now pro ceed to the study of the Lyness mapping. 2. The L yness mapping This pap er focuses on the Lyness mapping [16,17,18] and its non-autonomous extensions. The Lyness system is particularly interesting because it is a N th-order mapping, in tegrable at all orders in a sense whic h will b e made more precise b elow (see also [19] for a construction of sufficien tly man y first integrals and a conjecture that the Lyness map is in fact Liouville integrable at all orders). The autonomous form of the map is usually written as: x n + N x n = a + x n +1 + x n +2 + . . . + x n + N − 1 , (4) where N ≥ 2. As rep orted by Lyness in [16] and [17], when a = 1, this mapping is in fact p erio dic with p erio d 3 N − 1 when N = 2 or 3. Although this prop ert y seems to break down for all N ≥ 4 (see e.g. [18] and [20]), w e shall nev ertheless imp ose a 6 = 1 in (4) for all N ≥ 2. As we shall see it is also in teresting to study (4) in its deriv ative form, which is the follo wing N + 1st order mapping: x n + N (1 + x n ) = x n +1 (1 + x n + N +1 ) . (5) (When integrating (5) once, so as to obtain (4), the degree of freedom in the initial conditions for (5) that one loses compared to (4), is comp ensated by the parameter a which plays the role of an integration constan t.) In order to assess the integrabilit y of the Lyness mapping we start by considering its singularit y structure. The simplest case is that for N = 2, x n +2 x n = a + x n +1 , (6) whic h is a w ell-known integrable mapping b elonging to the QR T family [2]. W e shall not go into a detailed singularit y analysis of this mapping and therefore omit the (cyclic) singularity pattern that arises from x n +1 = ∞ (the reader is referred to [21] or [15] for more detail). The only singularity that is relev ant to our further discussion is that arising from x n +1 = − a (for generic x n ) for whic h we obtain the confined pattern {− a, 0 , − 1 , ∞ , ∞ , − 1 , 0 , − a } . Similarly , for the case N = 3, x n +3 x n = a + x n +1 + x n +2 , (7) for generic x n and x n +1 , the sole singularit y that is of interest to us arises at x n +2 = − a − x n +1 ≡ x ∗ . This singularity again confines, but one only reco v ers full (2 dimensional) freedom in terms of x n and x n +1 at the iterate x n +13 , which yields the confined singularity pattern { x ∗ , 0 , − 1 , • , ∞ , ∞ , • , − 1 , 0 , ∗ , ∗} , where • is used to indicate finite expressions in volving a and the free initial conditions (here x n and x n +1 ), and ∗ to indicate a succession of such functions that are not functionally indep endent as functions of the (free) initial conditions. A similar singularity analysis can b e easily carried out for se v eral higher v alues of N . In ev ery case we obtain for the singularit y at x n + N − 1 = x ∗ ≡ a − ( x n + · · · + x n + N − 2 ) a confined singularit y pattern of 3 the form: { x ∗ , 0 , − 1 , ( N − 2 finite v alues • ) , ∞ , ∞ , ( N − 2 finite v alues • ) , − 1 , 0 , ( N − 1 finite v alues ∗ ) } . W e consider it safe to surmise that this holds for all v alues of N , i.e. all higher-order Lyness mappings p ossess confined singularities of this t yp e. As explained in detail in [15], assuming this to be the case, the express metho d for asse ssing degree growths for second order mappings that we in tro duced in [22] then suggests that the dynamical degree of the Lyness mapping (4), for general N , should be given by the greatest real ro ot of the c haracteristic p olynomial 1 − λ − λ 2 N + λ 2 N +1 = ( λ − 1)( λ 2 N − 1) , (8) that can b e deduced from the abov e singularity pattern. This ro ot b eing equal to 1 suggests that the mapping should b e integrable for all N ≥ 2 and the fact that 1 is actually a double ro ot of the c haracteristic p olynomial (8) p oin ts to quadratic degree growth for the iterates of the map. Assuming of course that the express method can actually b e extended to such higher order mappings, whic h is still conjectural. In [21] some of the presen t authors, together with T. T amizhmani, studied the Lyness mapping in detail from the p oint of view of in tegrability . F or example, its gro wth prop erties w ere studied using the approac h in tro duced b y Halburd under the name of Diophantine in tegrability [23]. In this metho d one considers the iterates of the mapping for rational initial conditions and with rational v alues of the parameter and one studies the gro wth of their arithmetic heigh t H n = max( | p n | , | q n | ) (where x n = p n /q n , with gcd( p n , q n ) = 1 . The Diophantine integrabilit y criterion introduced by Halburd requires that log H n gro w no faster than a p olynomial in n . In all cases studied in [21] (up to N = 20) it was found that the limit (log H n ) /n 2 for n → ∞ conv erges to a finite n um b er, again indicating quadratic growth for the map and hence in tegrabilit y . Still, the fact that tw o, admittedly , stringent in tegrability criteria are satisfied does not constitute a pro of of the in tegrability of the Lyness mapping. It turns out ho w ev er that w e can offer suc h pro of with the help of the bilinear formalism. In this formalism the first step is to find an adequate ansatz linking the nonlinear v ariable x n to a set of τ -functions. As explained in [21], the deriv ed Lyness map (5) has a confining singularity that is similar to that for (4). The corresp onding (confined) singularity pattern easily yields a useful ansatz and it turns out that a single τ -function τ n suffices to cast the general mapping in a bilinear form. In fact, it suffices to enco de t he confined singularity pattern (for general N ≥ 2) in t wo differen t w a ys, one describing the sequence of app earance of zeros in the pattern, the other one describing the app earances of -1 and b oth yielding the same p ositions for infinities, in terms of zero v alues for τ n : x n = A n τ n − N τ n + N +1 τ n τ n +1 = − 1 − τ n − N +1 τ n + N τ n τ n +1 . (9) As explained in great detail in [21], it can b e seen that for this ansatz to satisfy the discrete deriv ative of the Lyness mapping it is necessary that A n b e a ( N − 1)-p erio dic function. The bilinear equation for τ n is then obtained b y simply equating the tw o rational expressions in (9), leading to A n τ n − N τ n + N +1 + τ n − N +1 τ n + N + τ n τ n +1 = 0 , (10) whic h is just a reduction of the Hirota-Miw a equation [21], whic h is the archet ypal integrable system in three discrete dimensions (discrete analogue of the KP equation) [24], [25]. The relation of Lyness to the Hirota-Miw a explains also the quadratic growth of the iterates. As shown in [26] by T. Mase, the degree of any mapping that can b e obtained as a direct reduction of the A-or B-type discrete KP equations, can only gro w as n ` where ` is 0 , 1 or 2. Hence, the degree growth of any non-linear mapping obtained from suc h a bilinear form through a rational transformations suc h as those in (9) can only hav e p olynomial gro wth as w ell and the Lyness mapping is integrable for all N . 4 3. Deautonomising the L yness mapping As a warm-up (and for reasons that will b ecome clear shortly) we start with a detailed study of the deautonomisation of the Lyness mapping (4) for the case N = 2. More precisely , w e assume that the parameter a en tering (4) is no w no longer a constant but dep ends on n , x n +1 x n − 1 = a n + x n , (11) and require that the s ingularit y pattern of the non-autonomous mapping (11) b e iden tical to that for the autonomous a n = a case. Again we start with a generic x 0 and set x 1 = − a 1 . Iterating, as before, w e repro duce the pattern obtained in the autonomous case, {− a 1 , 0 , − a 2 a 1 , ∞ , ∞ , − a 1 a 2 , 0 , − a 2 a 7 a 1 } , up to x 8 but find that for x 9 to dep end on x 0 a condition on a n is necessary: a n +7 a n = a n +1 a n +6 . (12) The solution of this condition is a n = κλ n times a p erio dic function of p erio d 6. (It is in teresting to remark that Cima and Zafar, who in [27] studied purely p erio dic deautonomisations of mapping (11) in whic h a n do es not con tain any non-p erio dic (secular) contributions, hav e rigorously shown that the only p erio dicities in a n that lead to an in tegrable mapping are 1, 2, 3 and 6, in p erfect agreemen t with our result). At this p oint it is useful to introduce tw o handy p erio dic functions (that we hav e b een regularly using in publications of ours). The first, φ m ( n ), of p erio d m , can be expressed in terms of the ro ots of unit y as φ m ( n ) = m − 1 X l =1 δ ( m ) l exp  2 iπ l n m  . (13) (Note that the constant term is absent, the sum starting at i = 1, inv olving m − 1 parameters). Since in this paper w e deal with multiplicativ e equations, w e also introduce the ‘m ultiplicativ e’ analogue of φ m ( n ), b y defining ϕ m ( n ) = exp( φ m ( n )). Note that while φ 2 ( n ) + φ 2 ( n + 1) = 0 we ha ve no w ϕ 2 ( n ) ϕ 2 ( n + 1) = 1 and analogous relations for the higher p erio ds. Similarly we introduce the p erio dic function χ 2 m ( n ) b y χ 2 m ( n ) = m X ` =1 η ( m ) ` exp  iπ (2 ` − 1) n m  . (14) whic h ob eys the relation χ 2 m ( n + m ) + χ 2 m ( n ) = 0. (Note that χ 2 m ( n ) inv olv es just m parameters). Again due to the present multiplicativ e setting we m ust introduce ψ 2 m = exp( χ 2 m ( n )), an m - perio dic function which satisfies the relation ψ 2 m ( n + m ) ψ 2 m ( n ) = 1. The identit y ϕ 2 N ( n ) = ϕ N ( n ) ψ 2 N ( n ), will b e useful in what follo ws. The solution of (12) can no w b e written as a n = κλ n ϕ 6 ( n ). How ever a ternary gauge freedom x n → ψ 6 x n is present in (11) and when this spurious freedom is remo ved the solution b ecomes finally a n = κλ n ϕ 3 ( n ) ϕ 2 ( n ). Equation (11) with this precise expression for a n is a well-kno wn q discrete P ainlev ´ e equation. Before pro ceeding to the study of higher N s, it is in teresting to inv estigate the p ossibility of late con- finemen ts for (11). It turns out that this is indeed p ossible. The first late confinement corresp onds to a singularity pattern {− a 1 , 0 , − a 2 a 1 , ∞ , ∞ , − a 1 a 2 , 0 , − a 2 a 7 a 1 , ∞ , ∞ , − a 1 a 2 a 7 , 0 , − a 2 a 7 a 12 a 1 } . The confinemen t condition needed for x 14 to dep end on x 0 instead of b ecoming infinite, is a n +12 a n = a n +11 a n +6 a n +1 . (15) The corresp onding c haracteristic p olynomial, k 12 − k 11 − k 6 − k + 1 = ( k 2 − k + 1)( k 10 − k 8 − k 7 + k 5 − k 3 − k 2 + 1) , (16) 5 is the pro duct of an inconsequen tial cyclotomic factor and a degree 10 p olynomial. F ollo wing the full- deautonomisation approac h w e claim that the largest zero of the latter, approximately 1.29348, should giv e the dynamical degree of the system. The Diophan tine method leads to an appro ximate v alue in p erfect agreemen t with the one obtained from this characteristic p olynomial. This is not the only p ossibility for late confinement for (11). In fact, infinitely many do exist. The next confined singularit y pattern can be represen ted as {− a 1 , 0 , • , ∞ , ∞ , • , 0 , • , ∞ , ∞ , • , 0 , • , ∞ , ∞ , • , 0 , • } with confinemen t condition a n +17 a n = a n +16 a n +11 a n +6 a n +1 , (17) leading to the c haracteristic p olynomial k 17 − k 16 − k 11 − k 6 − k + 1 = ( k 3 + 1)( k 14 − k 13 − k 11 + k 10 − k 7 + k 4 − k 3 − k + 1) . (18) The largest zero is now approximately 1.31820 and this estimate of the dynamical degree is again verified b y the Diophan tine approac h. Once the structure of the c haracteristic p olynomial b ecomes clear it is easy to guess the equation satisfied b y the dynamical degree for an infinitely postp oned confinemen t. (Details on how to deal with a infinitely- late confinemen t can b e found in [28]). T o make a long story short we find that the dynamical degree for an infinitely late confinemen t of the singularit y of (11) is the largest ro ot of the equation k 3 − k − 1 = 0 , (19) whic h we found to b e equal to  1 2 + q 23 108  1 / 3 +  1 2 − q 23 108  1 / 3 , approximately equal to 1.32472, a v alue again confirmed by the Diophantine approach for (11) with arbitrary a n . (This v alue of the dynamical degree is corrob orated by the rigorous results in [27] where dynamical degrees were calculated for iterates of (11) with the wrong p erio dicities in a n , i.e. not compatible with integrabilit y). W e now turn our atten tion to the case of higher N . W e studied the cases of several lo w v alues for N , starting from N = 3 and all of them exhibited the same behaviour. Starting from generic x 0 , x 1 , · · · , x N − 2 and choosing x N − 1 = x ∗ suc h that x N = 0, w e found that after the tw o expected infinities for x 2 N and x 2 N +1 , an extra infinity app eared in x 2 N +2 instead of a non-zero finite v alue as for the autonomous case. In fact, it turns out that the only w a y to make this infinity disapp ear is to require that a n b e constant. Hence, it do es not app ear to b e p ossible to deautonomise the Lyness mapping for any N higher than 2. Still, it is interesting to study the non-confining singularit y pattern produced b y this deautonomisation attempt. F or N = 3 w e found the pattern { x ∗ , 0 , • , • , ∞ , ∞ , ∞ , • , • , 0 , • , • , ∞ , ∞ , ∞ • , · · ·} in whic h the basic pattern { 0 , • , • , ∞ , ∞ , ∞ , • , • } rep eats indefinitely . F or N = 4 the basic rep eating pattern is { 0 , • , • , • , ∞ , ∞ , ∞ , ∞ , • , • , • } , while for N = 5 we find { 0 , • , • , • , • , ∞ , ∞ , ∞ , ∞ , ∞ , • , • , • , • } . W e therefore surmise that for general N the non-confined singularity pattern will b e given by the infinite rep etition of the basic pattern { 0 , ( N − 1 finite v alues • ) , ( N v alues ∞ ) , ( N − 1 finite v alues • ) } . Based on the structure of this rep eating pattern it is straightforw ard to obtain a characteristic equation that w ould giv e the v alue of the dynamical degree of a (non-in tegrable) non-autonomous mapping for N > 2. W e shall not go into the details of the deriv ation (whic h follo ws the metho d of Halburd [29], as w e explained in [22]) and just give the result. W e claim that the dynamical degree is giv en by the largest ro ot of the equation k 2 N − k 2 N − 1 − k N + 1 = 0 . (20) 6 F or N = 3 , 4 , 5 we find the (approximate) v alues 1.3247, 1.2964, and 1.2686, resp ectively . (Note that for N = 3 the v alue of this ro ot is exactly that obtained for the infinitely-late confinemen t for N = 2. In fact, the characteristic equation (20) con tains (19) as a factor b oth for N = 2 and N = 3). Using the Diophan tine approac h w e computed the dynamical degrees of a non-autonomous Lyness mapping (for an arbitrary function for a n ) for N = 3 , 4 and 5, and w e found that the dynamical degree obtained by the direct calculation, after 50 iterations, coincided with the largest root of the characteristic equation with a 10 − 4 precision. F rom the v alues obtained for N = 3 , 4 , 5 it is clear that the largest ro ot of (20) diminishes with increasing N . A rough estimate of the said ro ot giv es a b ehaviour 1 + log( N ) / N , with a b etter appro ximation b eing 1 + (log( N − 1) − log (log( N − 1))) / ( N − 1). 4. Deautonomising the deriv a tive L yness mapping While the study of the higher order non-autonomous Lyness mappings for N ≥ 3 led to some interesting results, it remains that the fact that these mappings cannot b e deautonomised while preserving their in tegrabilit y is somewhat frustrating. W e therefore turn to a different approach and inv estigate the p ossibilit y of deautonomising their deriv ative form (5) instead. W e thus introduce the mapping x n + N ( p n + x n ) = x n +1 ( q n + N +1 + x n + N +1 ) , (21) where p n and q n are functions the form of which will be determined by the application of the confine- men t criterion, i.e. by requiring that the nonautonomous mapping still leads to the confined singularit y pattern { 0 , ( N − 1 finite v alues • ) , ∞ , ∞ , ( N − 1 finite v alues • ) , 0 , ( N − 1 finite v alues ∗ ) } w e find in the autonomous case (here • is used to indicate finite and non-zero expressions in p n , q n and the free initial conditions, and ∗ to indicate a succession of suc h functions that are not functionally indep endent as functions of the (free) initial conditions). W e start by examining the 4th order mapping w e obtain in the N = 3 case. Starting from generic x 0 , x 1 and x 2 , and x 3 = 0, we find that subsequent iterates match the required singularity pattern, but that x 8 is also infinite unless q n = p n . W e apply this first confinemen t constraint and pursue the iterations. W e find that x 9 and x 10 are finite but that x 10 is not 0 unless p n satisfies the condition p n +7 p n = p n +6 p n +1 . (22) Implemen ting this condition as w ell we find that the resulting mapping does indeed p ossess the desired confined singularity pattern. Integrating (22) is straigh tforward and leads to p n = κλ n ϕ 6 ( n ). Ho w ev er, since equation (21) is in v ariant under a gauge transformation x n → ψ 2 N x n w e find, after remo ving a spurious factor ψ 6 ( n ), that the co efficien ts in (21) m ust take the form q n = p n = κλ n ϕ 3 ( n ). What happ ens if w e decide to take q n 6 = p n ? In that case the singularity pattern b ecomes unconfined { 0 , • , • , ∞ , ∞ , ∞ , • , • , • , ∞ , ∞ , ∞ , · · ·} , where the pattern {∞ , ∞ , ∞ , • , • , • } rep eats indefinitely . The re- sulting characteristic equation in this case is k 6 − k 3 − k 2 − k − 1 = 0 with largest root, appro ximately , 1.3803. If we decide to tak e q n = p n but ignore the second confinement condition, equation (22), we find a singularity pattern { 0 , • , • , ∞ , ∞ , • , • , • , ∞ , ∞ , · · ·} with the pattern {∞ , ∞ , • , • , • } rep eating indefinitely . The c haracteristic equation is now k 5 − k 2 − k − 1 = 0, which contains (19) as a factor (the v alue of the dynamical degree of b eing confirmed b y a Diophan tine calculation). The case N = 2 has also a similar deautonomisation. Starting with generic x 0 , x 1 and x 2 = 0, subsequent iterates matc h the desired pattern but x 6 is infinite unless q n = p n . Iterating further under this constrain t w e find that x 7 is finite but non-zero unless p n satisfies the condition p n +5 p n = p n +4 p n +1 . (23) 7 Under this condition the mapping then automatically p ossess the required confined singularit y pattern. The solution of (23) is p n = κλ n with, moreov er, a perio d-4 freedom, but after the remo v al of the freedom due to gauge w e ha v e finally q n = p n = κλ n ϕ 2 ( n ). W e hav e studied sev eral cases corresp onding to higher v alues of N . F or all of them a first confinemen t constrain t is q n = p n . Imp osing this constraint we obtain, after a certain n um b er of iterations, that p n m ust satisfy a second constrain t whic h, we surmise, has the form p n +2 N +1 p n = p n +2 N p n +1 . (24) Once gauge freedom has b een remo v ed, one finds that q n = p n = κλ n ϕ N ( n ). As in the case N = 3 ab ov e we can decide to tak e q n 6 = p n or, when q n = p n , to ignore the second confinemen t condition. The resulting singularity patterns are obtained by the infinite rep etitions of a simple basic pattern, which in the former case consists of N infinities follo w ed by N finite v alues, while in the latter case it starts with 2 successive infinities, follow ed b y N finite v alues, follo w ed b y N − 1 infinities and N finite v alues. It is straigh tforw ard to obtain the characteristic equation in b oth cases. In the former w e find k N +1 − k N − 1 = 0 , (25) and in the latter k 3 N +2 − k 3 N +1 − k 2 N +2 + k 2 N − k N + 1 = 0, whic h by dividing b y factor ( k N +1 − 1) can b e brough t to the form k 2 N +1 − k 2 N − k N +1 + k N − 1 = 0 . (26) F or (25) a rough estimate of the largest root is 1 + log( N ) / N , while for (26) the same reasoning giv es 1 + log(2 N ) / (2 N ). T aking q n = p n in (21) allo ws one to integrate once (and this, in fact, indep endently of the explicit expression for p n ). W e s tart from (21) rewritten as p n + N +1 + x n + N +1 p n + x n = x n + N x n +1 . (27) W e remark that in the left-hand side the indices of the n umerator and denominator are at a distance N + 1 while in the righ t-hand side the distance is N − 1. Next w e m ultiply the numerator and denominator of the left-hand side b y ( p n + N + x n + N )( p n + N − 1 + x n + N − 1 ) · · · ( p n +1 + x n +1 ) while for the righ t-hand side the corresp onding multiplier is x n + N − 1 x n + N − 2 · · · x n +2 . It is now straigh tforward to integrate the resulting equation. W e find ( p n + N + x n + N )( p n + N − 1 + x n + N − 1 ) · · · ( p n + x n ) = C x n + N − 1 x n + N − 2 · · · x n +1 (28) It is interesting at this p oin t to give the explicit expressions of these mappings for a few lo w-N cases. In order to write the equations in a more familiar form we introduce the v ariable X n = x n + p n . F or N = 2 w e ha v e X n +1 X n − 1 = C X n − p n X n (29) whic h, with p n = κλ n ϕ 2 ( n ), is a w ell-kno wn [30] discrete Painlev ´ e equation. F or N = 3 w e start by remarking that the distance of indices in b oth sides of (27) is ev en. (And this remark is v alid for all o dd N s). Thus it suffices to multiply b y only one out of the tw o factors we used ab ov e in order to b e able to in tegrate. The in tegration no w in troduces a p erio d-2 function in lieu of the in tegration constan t we had b efore, but given the parit y of the factors this is precisely the freedom that can be remo ved by the appropriate c hoice of the gauge. Finally , w e find X n +1 X n − 1 = C ( X n − p n ) , (30) 8 with p n = κλ n ϕ 6 ( n ), which is nothing but the discrete P ainlev´ e equation that w e obtained in Sec. 3 by deautonomising the N = 2 case of the Lyness mapping in its original form. F or N = 4 we find X n +2 X n +1 X n X n +1 X n +2 = C ( X n +1 − p n +1 )( X n − p n )( X n − 1 − p n − 1 ) , (31) while for N = 5 we again find a 4th-order mapping: X n +2 X n X n +2 = C ( X n +1 − p n +1 )( X n − 1 − p n − 1 ) . (32) W e claim that (with the appropriate p n ) equation (28) for N even, and integrated once more for N o dd, is a p ossible in tegrable deautonomisation of the higher N Lyness mapping, a deautonomisation that was imp ossible to obtain b y the direct deautonomisation of (4). 5. The mysterious aff air of the deriv a tive L yness mapping for N = 2 A t this point the careful reader has certainly noticed that w e started our presen tation in the previous section with the N = 3 case and, once the results were obtained for that case, we pro ceeded to deal with the N = 2 case. The reason for this will b ecome clear in this section. W e start from x n +2 ( p n + x n ) = x n +1 ( q n +3 + x n +3 ) , (33) and study its singularity pattern starting with generic x 0 , x 1 and x 2 = 0. W e obtain the sequence of v alues finite, infinite, infinite, finite, as exp ected but then x 7 is finite, instead of 0, thus leading to an infinite v alue for x 8 . In order to remedy this a constraint must b e imp osed on p n and q n : p n +1 p n − q n +5 p n − q n +1 p n + q n +1 q n +4 = 0 . (34a) Ho w ev er, iterating further w e find that x 9 b ecomes infinite unless a second constrain t is introduced p n +4 q n +1 − p n +1 p n + q n +1 p n − q n +1 q n +4 = 0 . (34b) Once the tw o constraints are satisfied the singularity b ecomes indeed confined with the exp ected pattern { 0 , • , ∞ , ∞ , • , 0 , ∗ , ∗} . Solving the confinemen t constraints is particularly interesting. T aking the sum of (34a) and (34b) w e find that p n and q n ob ey the simpler relation p n +4 q n +1 = q n +5 p n whic h means that p n = z n q n +1 with z n +4 = z n . Next we in troduce w n = p n +1 − q n +1 and from (34b), rewritten as p n w n = q n +1 w n +3 , we find that w n +3 = z n w n . Eliminating z n , w e obtain for w n the equation w n +7 w n = w n +3 w n +4 . (35) The solution of (35) is w n = κλ n m ultiplied b y t w o p erio dic terms with p erio ds 3 and 4. Rewriting the relation w n = p n +1 − q n +1 in terms of w n using z n = w n +3 /w n w e hav e w n +4 q n +2 − w n +1 q n +1 = w n w n +1 . Introducing Q n = w n +1 w n +2 w n +3 q n +1 w e find that Q n m ust ob ey the equation Q n +1 − Q n = w n w n +1 w n +2 w n +3 , (36) that can b e in tegrated b y a simple quadrature. 9 W e find q n +1 = ϕ 4 ( n )( σ ϕ 3 ( n ) λ n + µλ − 3 n ) and p n = λ 3 ϕ 4 ( n − 1)( σϕ 3 ( n ) λ n + µλ − 3 n ), where µ is a free constan t and σ is proportional to κ . This is an original non-autonomous form with t w o different exp onen tial terms in the same parameter, which is something that we hav e nev er encoun tered b efore. (Whether this is a general feature of third- and higher-order equations remains to b e seen). A consequence of the presence of tw o exp onentials in q n and p n is that we can turn the n dep endence in these parameters into an additive one, by taking the limit λ → 1, without c hanging the functional form of the equation. First, w e remark that σϕ 3 ( n ) b ecomes σ + φ 3 ( n ) by taking the appropriate limits on the parameters of the p erio dic function. Next w e in tro duce λ = 1 + δ and take δ → 0. W e assume that σ and µ diverge as 1 /δ , obeying the relations σ − 3 µ = α/δ and σ + µ = β . T aking the limit δ → 0 w e find for q n and p n the expressions q n +1 = ϕ 4 ( n )( αn + β + φ 3 ( n )) and p n = ϕ 4 ( n − 1)( αn + β + φ 3 ( n )). But what is ev en more interesting is the case of late confinemen t of (33). The expected pattern is now { 0 , • , ∞ , ∞ , • , • , ∞ , ∞ , • , 0 } , from whic h w e can infer the characteristic equation k 9 − k 7 − k 6 − k 3 − k 2 + 1 = 0 with largest ro ot approximately 1.42501. Obtaining the confinement constraints requires some extensive calculations. Performing them we find the tw o confinement conditions p n +8 p n +1 p n q n +1 − p n +8 p n q n +5 q n +1 − p n +8 p n q 2 n +1 + p n +8 q n +4 q 2 n +1 + p n +4 p n q n +9 q n +1 − p n +1 p 2 n q n +9 + p 2 n q n +9 q n +1 − p n q n +9 q n +4 q n +1 = 0 , (37) and p n +8 p n +1 p n q n +1 − p n +8 p n q n +5 q n +1 − p n +8 p n q 2 n +1 + p n +8 q n +4 q 2 n +1 + p n +5 p n +4 p n q n +1 − p n +5 p n +1 p 2 n + p n +5 p 2 n q n +1 − p n +5 p n q n +4 q n +1 + p n +4 p n +1 p n q n +1 − p n +4 p n q n +5 q n +1 − p n +4 p n q 2 n +1 + p n +4 q n +4 q 2 n +1 − p 2 n +1 p 2 n + p n +1 p 2 n q n +5 + 2 p n +1 p 2 n q n +1 − p n +1 p n q n +8 q n +1 − 2 p n +1 p n q n +4 q n +1 − p 2 n q n +5 q n +1 − p 2 n q 2 n +1 + p n q n +5 q n +8 q n +1 + p n q n +8 q 2 n +1 + p n q n +5 q n +4 q n +1 + 2 p n q n +4 q 2 n +1 − q n +8 q n +4 q 2 n +1 − q 2 n +4 q 2 n +1 = 0 . (38) The full-deautonomisation metho d stipulates that the confinement conditions lead directly to the v alue of the dynamical degree. In all cases studied up to no w the deautonomisation conditions w ere simple linear or multiplicativ e equations and the dynamical degree could b e obtained through the study of their c haracteristic equation. Here w e are in presence of a system that is non-linear and, in fact, non-integrable. Still, it can lead to the v alue of dynamical degree. T o this end we study the gro wth of the solutions of the system, introducing initial conditions where p i , q i , i = 0 , 6 are generic, and, for simplicit y , taking p 7 generic and q 7 = µ . W e compute the degrees in µ of the iterates of (37) and (38) and, starting at n = 8, find the sequence 1, 1, 1, 2, 3, 4, 6, 10, 13, 18, 27, 38, 54, 75, 111, 158, 225, 321, 457, 652, · · · . The ratio of 457/321 is 1.42368, the ratio 652/457 is 1.42670 and taking their geometric mean we obtain 1.42519, very close to the dynamical degree 1.42501 inferred ab o v e. As a further verification, w e studied the degree gro wth of (37), (38) through the Diophantine approac h and obtained a dynamical degree appro ximately equal to 1.4256, confirming the result of the c haracteristic equation and that of the direct calculation. Thus the dynamical degree is indeed given by the gro wth of the v ariables p n and q n when they ob ey the confinemen t conditions. 6. Conclusion The Lyness mapping is a simple integrable system, the deautonomisation of whic h ho w ev er turned out to b e particularly interesting. Its integrable character w as already established in [21] where we ha v e sho wn that it can b e obtained as a reduction of the Hirota-Miwa integrable lattice equation. The present w ork w as dev oted to the study of the deautonomisation of this mapping in t wo differen t forms. As we 10 ha v e already sho wn, the deautonomisation process allo ws to accen tuate the deep relationship betw een the singularit y structure of the solutions of an equation and its integrable c haracter, the latter b eing deduced from the v alue of the dynamical degree. Deautonomising the Lyness mapping under its customary form turned out to be possible only in the case N = 2. How ever, by considering the deriv ativ e form of the mapping, we were able to pro duce an (admittedly tailored to this sp ecial form) deautonomisation for ev ery v alue of N . In this wa y we presented a metho d to construct integrable equations of arbitrary (ev en) order whic h include w ell-kno wn q -discrete Painlev ´ e equations at N = 2 and 3. Studying cases of ‘late’ confinemen t for b oth the N = 2 and higher N cases (the latter through the deriv ative form), the deautonomisation allow ed once more to confirm the predictions of the ‘full- deautonomisation’ approach, successfully comparing the dynamical degree th us obtained with the v alue giv en b y a direct Diophantine calculation. The deautonomisation of the N = 2 case, how ev er, reserv ed some surprises. First, instead of the situation familiar in the case of q -Painlev ´ e equations where the n -dep endence in the parameters enters through a single exp onential, here we were in the presence of tw o exp onen tial terms in the same parameter. And as a consequence of this we found that, remark ably , a deautonomisation where n en ters linearly was also p ossible for the same functional form of the equation. But the most interesting result w as the one obtained by considering the late confinement in this case. W e obtained a system of t w o nonlinear, non-in tegrable equations, where the dynamical degree did not appear explicitly (as in all previously studied cases) as the largest ro ot of some c haracteristic equation. Here w e had to study the growth of the solutions of the nonlinear system, with resp ect to some initial condition. It turned out that the dynamical degree obtained from the gro wth of the solutions of the confinement conditions coincides with the dynamical degree of the solution of the mapping (obtained using Halburd’s metho d and confirmed b y the Diophan tine approac h). This result is one more v alidation of the full-deautonomisation approach as a reliable integrabilit y crite- rion. A t this point we cannot resist the temptation to quote the ‘prophetic’ statemen t of Gambier [31], who, while studying differen tial equations from the p oin t of view of the Painlev ´ e prop erty , remarked: Je rencontrais des syst` emes de conditions diff´ erentielles dont l’in t ´ egration ´ etait, quoiqu’au fond bien simple, assez difficile ` a apercevoir. Par un m´ ecanisme qui est g ´ en´ eral, mais qui ´ etait difficile ` a pr ´ evoir, la r´ esolution de ce premier probl ` eme, int ´ egration des conditions, est intimemen t li´ ee ` a l’in t ´ egration de l’ ´ equation diff ´ eren tielle elle-m ˆ eme . (I found m yself confron ted with systems of differen tial conditions whose integration, though in itself quite simple, was nev ertheless difficult to detect. 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