Singularity confinement in delay-differential Painlevé equations
aa r X i v : . [ n li n . S I] J un Singularity confinement in delay-differentialPainlev´e equations
Alexander Stokes
Department of Mathematics, University College London,Gower Street, London, WC1E 6BT, UK
Abstract
We study singularity confinement phenomena in examples of delay-differential Painlev´eequations, which involve shifts and derivatives with respect to a single independent variable.We propose a geometric interpretation of our results in terms of mappings between jet spaces,defining certain singularities analogous to those of interest in the singularity analysis of discretesystems, and what it means for them to be confined. For three previously studied examplesof delay-differential Painlev´e equations, we describe all such singularities and show they areconfined in the sense of our geometric description.
Singularity confinement is a phenomenon first proposed as an integrability criterion for discretesystems [GRP91], and has been used to great effect to obtain discrete analogues of the Painlev´edifferential equations [GRWS20, RGH91, GR93]. Its geometric interpretation has led to novel con-nections between discrete integrable systems and birational algebraic geometry, most notably Sakai’sgeometric framework and classification scheme for discrete Painlev´e equations [Sak01]. We studydelay-differential equations, for which a kind of singularity confinement test has been used to isolateintegrability candidates and obtain delay-differential equations of Painlev´e-type [GRM93, RGT93].These so-called delay Painlev´e equations possess analogues of many integrability properties of theirdiscrete and differential counterparts, and it is natural to ask whether a geometric theory may bedeveloped for them.Compared to the discrete case, the understanding of singularity confinement in this class ofequations is in its infancy. In particular, we do not have available to us the definition of singularityconfinement in second-order discrete systems as the iteration mappings of the systems lifting toisomorphisms between rational surfaces. Further, even for heuristic observations in the absenceof a proper definition of confinement, the presence of derivatives leads to challenges, as differentmultiplicities with which solutions take singular values lead to infinitely many behaviours to bechecked. We consider the following three examples of delay Painlev´e equations u (¯ u − ¯ u ) = au − bu ′ , (1.1) v (¯ v − ¯ v ) = pv + qv ′ , (1.2)¯ ww = ¯ w ( λzw + αw ′ ) , (1.3)where u, v and w are functions of the complex independent variable z , we take p, q, a, b, λ, α to becomplex parameters, and we denote up- and down-shifts by ¯ u ( z ) = u ( z + 1) , ¯ u ( z ) = u ( z −
1) etc.For the purpose of isolating integrability candidates in the class of delay-differential equations, it1eems to have been sufficient to require only that the simplest singularities exhibit confinement-type behaviour, and all three of the examples above may be obtained by such means. However,if singularity confinement is to lead to a geometric theory in this case, a more detailed analysis isrequired. It is the first steps in this direction that we take in this paper, by extending previousobservations to account for different multiplicities with which solutions take singular values, as wellas giving a geometric description of singularities that may arise in delay-differential equations andwhat it means for them to be confined.The equation (1.1) was obtained by Quispel, Capel and Sahadevan [QCS92] as a similarityreduction of the Kac-van Moerbeke differential-difference equation, also known as the Manakovequation or Volterra lattice. They also showed that it has a continuum limit to the first differentialPainlev´e equation and that it exhibits some singularity confinement-type behaviour. The equation(1.2) is a symmetry reduction of a known integrable differential-difference modified Korteveg-deVries equation, and extensions of it have been studied by Halburd and Korhonen from the point ofview of Nevanlinna theory [HK17]. Further, it has a continuum limit to the first Painlev´e equationand may be obtained from B¨acklund transformations of the third Painlev´e equation [Ber17], oralternatively using singularity confinement tests adapted from those in [TRGO99]. The third equa-tion (1.3) was isolated as an integrability candidate by Ramani, Grammaticos and Moreira [GRM93]using a kind of singularity confinement test (which also recovered equation (1.1)), and has a con-tinuum limit to the first Painlev´e equation. We also point out that other integrability propertiesanalogous to those of differential and discrete Painlev´e equations have been studied in equations(1.1),(1.2),(1.3), for example the fact that they may be rewritten in bilinear forms [Car11] and thatdegenerate cases admit elliptic function solutions [Ber17], in parallel with the discrete case whereautonomous degenerations of discrete Painlev´e equations are Quispel-Roberts-Thompson (QRT)mappings [QRT88, QRT89], solved by elliptic functions.We also remark that we are considering examples of so-called three-point delay differentialequations, which are of the form ¯ u = f ( u, u ′ , ... ) + f ( u, u ′ , ... )¯ uf ( u, u ′ , ... ) + f ( u, u ′ , ... )¯ u , (1.4)where f i are polynomials in u and its derivatives. There are known integrable delay-differentialequations of other forms, for example the so-called bi-Riccati equations [GRM93, Ber18], but studiesof singularity confinement in these more closely resembles classical Painlev´e analysis than birationalgeometry, and will not be discussed in this paper. The class of three-point equations is the oneconsidered by Halburd and Korhonen through the Nevanlinna theoretic approach [HK17], and fitsinto the family for which Viallet defined algebraic entropy in the delay-differential setting [Via14]. The differential Painlev´e equations P I -P VI are six nonlinear second-order ordinary differential equa-tions (ODEs), the study of which has become one of the cornerstones of the field of integrablesystems. Painlev´e, Gambier, Fuchs and their collaborators considered a large class of second-orderODEs, and isolated those for which all solutions are single-valued about any movable singulari-ties (those whose locations depend on the initial conditions). This condition is now known as thePainlev´e property, and of all the equivalence classes of equations obtained, the six Painlev´e equa-tions arose as representatives whose general solutions could not be expressed in terms of known2unctions. These new special functions, known as the Painlev´e transcendents, play a central role inmodern nonlinear physics, see e.g. [Cla06, FIN +
06] and numerous references within.The differential Painlev´e equations admit a geometric description in terms of rational surfacesobtained by blowing up certain singularities of the equations. Discovered by K. Okamoto [Oka79],for each equation this comes in the form of a bundle over the independent variable space whosefibres are rational surfaces with certain curves removed. The bundle, known as Okamoto’s space,admits a foliation by solution curves of the ODE system transverse to the fibres, and each fibrecan be regarded as a space of initial conditions for the system. Further, the curves which wereremoved from each fibre (the inaccessible divisors) have irreducible components whose intersectionconfiguration is encoded in a Dynkin diagram of affine type, also known as an extended Dynkindiagram. It was also shown that Okamoto’s space for each P I − P VI essentially determines thedifferential equation [MMT99, Mat97, ST97], and can be used to explain many of their properties(see [KNY17] and references within).Beginning in the 1990’s, important steps were made towards defining and understanding discreteanalogues of the Painlev´e equations, through the proposal by Ramani and Grammaticos, togetherwith Papageorgiou, of singularity confinement [GRP91] as the discrete counterpart to the Painlev´eproperty. We will illustrate the singularity confinement phenomenon in the second order differenceequation f n +1 = ( f n − k )( f n + k ) f n − k − f n + 2 tf n f n − , (1.5)with parameters k = 0 , ± t = 0. The initial value problem for this equation requires twovalues of the solution, say f , f , which in almost all cases will allow the values f , f and so onto be determined recursively. The system (1.5) has singular values f n = ± k , in the sense that ifwhile iterating the solution takes one of these values, f n +1 is zero independent of the value of f n − (provided f n − = 0). This is usually referred to as a loss of a degree of freedom occurring whileiterating the system. For generic (non-integrable) discrete systems, the singularity propagates, inthe sense that the subsequent values f n +2 , f n +3 , ... will all be determined independently of f n − and the lost degree of freedom is never recovered. In our case, we may compute the next iterate f n +2 = ∓ k , but then, importantly, arrive at an indeterminacy of the rational function giving f n +3 ,namely at ( f n +1 , f n +2 ) = (0 , ∓ k ). If, however, we consider a perturbation of the singular value f n = ± k by introducing a small parameter ε , we may compute the following in the small ε limit: f n − = 0 , f n = ± k + O ( ε ) , f n +1 = O ( ε ) , f n +2 = ∓ k + O ( ε ) , f n +3 = f n − + O ( ε ) . If we define the values of the iterates as the limits of the above sequence as ε →
0, the lost degreeof freedom is said to be recovered in the value of f n +3 , and the singularity at f n = ± k is said tobe confined. The singularity confinement property for second order discrete systems can be under-stood as the existence of a space of initial conditions for the system: a family of rational surfacesto which the birational iteration mappings lift to isomorphisms. In fact, defining the values of thesolution by iterating and taking limits as above implicitly lifts the system under certain blow-ups.The example (1.5) is in fact an example from the family of QRT mappings [QRT88, QRT89], thedefinition of which ensures they have a space of initial conditions given by a rational elliptic surface.The equation (1.5) can be considered as a birational mapping ϕ : P × P → P × P . Letting f n − = y , f n = x = ¯ y , f n +1 = ¯ x , the iteration ( f n , f n − ) ( f n +1 , f n ) gives a birational map3 x, y ) (¯ x, ¯ y ). We consider this on P × P via the usual charts. That is, we use x, y as affinecoordinates in the P factors, and introduce X = 1 /x, Y = 1 /y , so P × P is covered by the fourcharts ( x, y ) , ( X, y ) , ( x, Y ) , ( X, Y ). This mapping ϕ : P × P → P × P ( x, y ) (¯ x, ¯ y ) = (cid:18) ( x − k )( x + k ) yk − x + 2 txy , x (cid:19) (1.6)preserves each member of a pencil of elliptic curves on P × P , and the space of initial conditions isobtained from P × P by resolving its basepoints through a number of blow-ups. This is ensured bythe definition of the QRT map in terms of this pencil, which we outline now. Consider the matrices A = − t − t k t , B = , (1.7)where again k = 0 , ± t = 0, which define a pencil of biquadratic curves (cid:8) Γ [ α : β ] : [ α : β ] ∈ P (cid:9) in P × P , written in the affine coordinates ( x, y ) asΓ [ α : β ] : α x T Ay + β x T By = α t ( k − x − y + 2 txy ) + βx y = 0 , (1.8)where x T = (cid:0) x x (cid:1) , y T = (cid:0) y y (cid:1) . The QRT mapping is defined as follows. A genericpoint, say given by ( x, y ), lies on exactly one curve Γ [ α : β ] in the pencil. There is then exactly oneother point (¯ x, y ) on Γ [ α : β ] with the same y -coordinate, from which we can define the involution r x : ( x, y ) (¯ x, y ). Similarly we have another involution r y : ( x, y ) ( x, ¯ y ), and their composition r x ◦ r y is the QRT mapping. Following [CDT17] we introduce the involution σ xy : ( x, y ) ( y, x )and work with the map ϕ = σ xy · r y , which for the pencil (1.8) is precisely (1.6), and can be thoughtof as a ‘half QRT mapping’ due to the fact that ϕ = r x ◦ r y . The pencil (1.8) has four basepoints,given in coordinates by p : ( x, y ) = ( k, , p : ( x, y ) = ( − k, , p : ( x, y ) = (0 , k ) , p : ( x, y ) = (0 , − k ) . (1.9)Blowing these up, we denote the blow-up projection by π : Bl p ,p ,p ,p ( P × P ) → P × P , and denote the exceptional curves by π − ( p i ) = E i for i = 1 , , ,
4. The proper transform of thepencil under π still has four basepoints p ∈ E , p ∈ E , p ∈ E , p ∈ E , after the blow-ups ofwhich the proper transform of the pencil is basepoint-free and we obtain a rational elliptic surface X . Denote the projection under the second four blow-ups by π : X → Bl p ,p ,p ,p ( P × P ) , and the exceptional curves by π − ( p i ) = E i for i = 5 , , ,
8. Composing the projections we obtain π = π ◦ π : X → P × P , = 0 E − E E − E E E πp p x = 0 E − E E − E E E p p π Figure 1: Configuration of curves on X arising from the blow-ups of the basepointsand X is a rational surface fibred by the proper transform of the pencil. Under π , we have thepreimage of each basepoint p , . . . , p given by the union of two irreducible curves: π − ( p ) = ( E − E ) ∪ E , π − ( p ) = ( E − E ) ∪ E ,π − ( p ) = ( E − E ) ∪ E , π − ( p ) = ( E − E ) ∪ E , where we have used the usual notation for divisors to denote by E − E the proper transform of E under π , and so on, which we illustrate in Figure 1. The iteration mapping (1.6) lifts uniquelyunder the blow-ups to give a birational map˜ ϕ : X → X , which is in fact a true isomorphism, and the singularity confinement observed earlier can be under-stood in terms of this space of initial conditions as follows. Lifted under the blow-ups, the initialdata f n − = 0 , f n = k correspond to a point on the proper transform H x − E of the line x = k on P × P , while the pairs ( f n , f n +1 ) = ( k, f n +1 , f n +2 ) = (0 , − k ) correspond to the basepoints p , p respectively. Further, the recovery of the degree of freedom ( f n +2 , f n +3 ) = ( − k, f n − ) corre-sponds to a one-to-one correspondence between H x − E and H y − E under the iterated mapping˜ ϕ , as we illustrate in Figure 2. X H x − E E E H y − E x = k P × P y = − kp p ˜ ϕ ˜ ϕ ˜ ϕπ π Figure 2: Confined singularity pattern as isomorphisms between exceptional curves5he loss of a degree of freedom when f n = ± k can now be understood in terms of curves on P × P being blown down to points under the mapping ϕ : a codimension one subvariety beingblown down to one of codimension two. The recovery of the lost degree of freedom occurs preciselywhen, while iterating after a blow-down, we arrive at an indeterminacy of the forward iterationmap ϕ (in the case of the singularity f n = k , this is p ), so the point is blown back up to acurve. As remarked before, for a generic (non-integrable) system, after a blow-down we will notarrive at an indeterminacy of the forward mapping and the lost degree of freedom will never berecovered. In other words, we cannot lift the mapping to an isomorphism through a finite numberof blow-ups. This description of singularity confinement in terms of codimension increasing un-der the mapping, followed by a return to the same as the generic case, is the main reference pointfor our geometric formulation of singularity confinement for delay-differential equations in section 3.Ramani, Grammaticos and collaborators have obtained a plethora of discrete Painlev´e equationsvia the process of ‘deautonomisation by singularity confinement’ applied to members of the QRTfamily. This involves considering non-autonomous generalisations of a given QRT map by intro-ducing n -dependence into the coefficients of the mapping, then isolating examples for which thesingularity confinement behaviour persists. The definitive framework for discrete Painlev´e equa-tions was provided in a seminal paper by H. Sakai [Sak01]. Sakai defined a class of rational surfacesgeneralising both those associated with differential Painlev´e equations via Okamoto’s space andthe rational elliptic surfaces giving spaces of initial conditions for QRT mappings. Certain surfacesfrom this class come in families that admit actions of extended affine Weyl groups by birationaltransformations, with translation elements defining discrete Painlev´e equations. The theory ofSakai has had a huge impact on both the general theory of discrete integrable systems, as wellas on the applications in which they arise. While this theory provides a classification scheme fordiscrete Painlev´e equations in terms of the surfaces they are associated with, it has also led to asuite of geometric tools for their analysis (see [KNY17] and numerous references within), which areinvaluable in cases where a discrete system from an applied problem fits into the discrete Painlev´eframework [DFS19]. Sakai’s construction recovers many of the examples obtained by singularityconfinement methods, but we make an important remark here that lifting to isomorphisms under afinite number of blow-ups is not sufficient for integrability, and the geometry of the space of initialconditions plays a defining role. In particular, an example given by Hietarinta and Viallet [HV98]admits a space of initial conditions but exhibits exponential degree growth, which was explained interms of its geometry by Takenawa [Tak01]. It has since been shown [Mas18] that if a second-orderdiscrete system with the singularity confinement property (in the sense that it admits a space ofinitial conditions) is nontrivially integrable (in the sense of quadratic degree growth), then it mustarise from the surfaces defined by Sakai.As mentioned previously, the theory of delay-differential Painlev´e equations is in its infancycompared to the differential and discrete cases, but there is already a body of evidence showingits promise, which we hope to add to with this work. Delay-differential equations of the kind weconsider arise in a range of fields of applied mathematics, most notably in mathematical biology,for example as equations for steady states of systems of partial differential equations with a spatialdelay [FBM19]. Thus the possibility of a geometric framework for Painlev´e equations in the delay-differential class is an exciting prospect not only for the theory of Painlev´e equations itself, but forwidening the range of equations whose integrability can be exploited in applications.6 .2 Outline of the paper We will begin our analysis working on the level of equations, without invoking geometric language.In section 2 we recall previous observations of singularity confinement behaviour in the three equa-tions, and extend them to include infinite families of confined singularity patterns in each case. Theproofs of these are deferred to the appendix. In section 3 we shift to the geometric setting, firstrecasting our equations as mappings between jet spaces and defining ‘blow-down type’ singularities,and propose a notion of confinement for them. Rephrased in these geometric terms, we use theresults of Section 2 to show that in the three examples, all such singularities are, in the sense ofour definition, confined. We conclude with a discussion of how the geometric framework and thetechniques developed for proving the singularity confinement property may be utilised and builtupon in the study of other examples, as well as some open questions that arise from our work.
We begin by recalling previous observations of singularity confinement phenomena in the threeexamples we consider. Beginning with equation (1.2), the forward iteration, which gives ¯ v in termsof v, v ′ and ¯ v is given by ¯ v = ¯ v + p v + q v ′ v , (2.1)so if we take, as initial data, a pair of Laurent series expansions of v, ¯ v about z = z , then (2.1) andits upshifts determine all subsequent iterates ¯ v, ¯¯ v, . . . as Laurent series about z . If we only wishto iterate a finite number of steps forward from generic initial data, we need only finitely manycoefficients. For example, we could begin by giving initial ¯ v, v as Taylor expansions in ζ = z − z about some z = z : ¯ v = ¯ a + ¯ a ζ + ¯ a ζ + . . . , (2.2a) v = a + a ζ + a ζ + . . . . (2.2b)If we assume that the iterates ¯ v ( z ) = u ( z + 1) , ¯¯ v ( z ) = v ( z + 2) , . . . , v ( k ) ( z ) = v ( z + k ) are all regularand nonzero at z , it is clear from the form of the equation (2.1) that the value v ( z + k ) dependsonly on the following coefficients from the expansions (2.2a), (2.2b): (cid:18) ¯ a ¯ a . . . ¯ a k − a a . . . a k − a k (cid:19) . (2.3)We will be iterating systems arbitrarily many times forward, so we will use this kind of notation forthe iterates, i.e. v ( k ) ( z ) = v ( z + k ), throughout the remainder of the paper. Further, the form ofthe right-hand side of the forward iteration (2.1) ensures that if we start from ¯ v, v given by Taylorseries, the only way that a pole may develop is through some iterate having a zero first. If whileiterating, some iterate v develops a zero of order one, say at ζ = z − z = 0, with¯ v = ¯ a + ¯ a ζ + . . . , (2.4a) v = a ζ + a ζ + . . . , (2.4b)7here a = 0, then we have by direct calculation that¯ v = − qa ζ − + O ( ζ − ) , (2.5a)¯¯ v = − a ζ + O ( ζ ) , (2.5b)¯¯¯ v = (cid:18) a + 7 p qa + 2 pa a − qa a + 6 qa a (cid:19) + O ( ζ ) . (2.5c)We summarise the observations above by saying that the equation (1.2) admits the singularitypattern (cid:0) rg , , ∞ , , rg (cid:1) , where rg indicates a regular iterate with generic coefficients. We note here that this behaviour isexceptional for the following reason. In the computation of ¯¯ v here, it is natural to expect a zeroof order one, as this is what happens generically when v and ¯ v are of order ζ, ζ − respectively.However, while ¯ v, ¯¯ v having orders ζ − , ζ respectively would generically lead to ¯¯¯ v having anotherpole of order 2, in this singularity pattern we note that two terms have vanished as ¯¯¯ v regains reg-ularity. In the language of previous studies of singularity confinement behaviour, the informationlost when entering the singularity is recovered in the iterate ¯¯¯ v , in the form of the coefficient ¯ a from the initial data. Though this behaviour has not, to our knowledge, been reported explicitly,we note that the equation (1.2) may be obtained by singularity confinement tests along the lines of[GRM93, TRGO99].We next consider equation (1.1), which was first observed in [QCS92] to exhibit the followingsingularity confinement behaviour. The forward iteration is given by¯ u = ¯ u + a − b u ′ u , (2.6)so again it is clear that the only way that a pole may develop while iterating from formal Taylorseries is following a zero. Suppose that while iterating, the solution u develops a zero of order oneat ζ = z − z = 0, so ¯ u = ¯ c + ¯ c ζ + . . . , (2.7a) u = c ζ + c ζ + . . . , (2.7b)where c = 0. Then direct calculation shows that u (1) = − bζ + (cid:18) a + ¯ c − b c c (cid:19) + O ( ζ ) , (2.8a) u (2) = bζ + (cid:18) a + ¯ c − b c c (cid:19) + O ( ζ ) , (2.8b) u (3) = a b − ¯ c b + 2 c ¯ c c − c + 2 b (cid:0) c c − c (cid:1) c − c ! ζ + O ( ζ ) , (2.8c) u (4) = F (¯ c , ¯ c , ¯ c , c , c , c , c ) + O ( ζ ) , (2.8d)where F is a known rational function of the generic initial data, which we omit for conciseness.Again, this behaviour is exceptional as u (1) , u (2) both having simple poles would generically lead to8 (3) also having a simple pole, but here two terms have vanished as u (3) instead has a zero of orderone, so equation (1.1) admits the singularity pattern (cid:0) rg , , ∞ , ∞ , , rg (cid:1) . We next turn to equation (1.3), which was obtained in [GRM93] by singularity confinement tests,though details were not given explicitly. The forward iteration mapping is given by¯ w = ¯ w (cid:18) λz + α w ′ w (cid:19) . (2.9)Say, while iterating, we arrive at a pair ¯¯ w, ¯ w given by expansions in ζ = z − z by¯¯ w = ¯¯ c + ¯¯ c ζ + ¯¯ c ζ + . . . , (2.10a)¯ w = ¯ c + ¯ c ζ + ¯ c ζ + . . . , (2.10b)with α ¯ c + λ ( z − c = 0 , α ¯ c + λ ¯ c ( z − = 0 , ¯ c = 0 , ¯¯ c = 0 . (2.11)This means that w will have a simple zero at z = z , and by direct calculation we find the following: w = λ ¯¯ c (1 − z ) (2 α ¯ c + λ ¯ c ( z − α ¯ c ζ + O ( ζ ) , (2.12a)¯ w = α ¯ c λ (1 − z ) ζ − + O ( ζ ) (2.12b)¯¯ w = λ ¯¯ c ( z −
1) (2 α ¯ c + λ ¯ c ( z − c + O ( ζ ) , (2.12c)¯¯¯ w = G (¯¯ c , ¯¯ c , ¯¯ c , ¯ c , ¯ c , ¯ c )¯¯ c (2 α ¯ c + λ ¯ c ( z − + O ( ζ ) , (2.12d)where G is a polynomial function of the generic initial data as well as z . Again, this behaviour isexceptional as a simple pole of ¯ w with ¯¯ w regular and nonzero would generically lead to ¯¯¯ w havinganother simple pole, whereas in this case a term has vanished and the iterate ¯¯¯ w is regular. Again,we summarise this observation by saying that the equation (1.3) admits the singularity pattern (cid:16) rg , ¯ ζ , , ∞ , ¯¯ ζ , rg (cid:17) , where ¯ ζ (1)0 indicates that the iterate ¯ w satisfies the condition for w to develop a simple zero, namely α ¯ c + λ ( z − c = 0 , α ¯ c + λ ¯ c ( z − = 0, and ¯¯ ζ indicates the iterate ¯¯ w = ¯¯ c + ¯¯ c ζ + ¯¯ c ζ + . . . satisfies α ¯¯ c + λ ( z + 2)¯¯ c = 0. In the previous section, we outlined certain singularity patterns admitted by the equations (1.1),(1.2) and (1.3) which involved zeroes of order one developing while iterating the systems. We nowextend these observations to higher order zeroes, and show that each of the equations admits an9nfinite family of singularity patterns with similar confinement behaviour.For equation (1.2), we have observed the singularity pattern (cid:0) rg , , ∞ − , , rg (cid:1) , which corre-sponds to v being regular and v having a zero of order one at z = z . Similarly, if v has a zero oforder two, then we pass through the following sequence of orders, which is generic until three termsvanish as v (5) becomes regular instead of a pole (with leading coefficient depending on data from¯ v ): ¯ v = O ( ζ ) , v ∼ ζ , v (1) ∼ ζ − , v (2) ∼ ζ , v (3) ∼ ζ − , v (4) ∼ ζ , v (5) = O ( ζ ) . From above, we see that equation (1.2) admits the singularity pattern (cid:0) rg , , ∞ , , ∞ , , rg (cid:1) , and because of the return to regularity and the iterate v (5) depending on the generic initial datafrom ¯ v , the singularity is confined in a similar sense to that which we observed in the case of a zero oforder one. More generally, if v has a zero of order m >
1, and ¯ v is regular, say v = c m ζ m + O ( ζ m +1 ),with c m = 0, and ¯ v = O (1), then it can be seen from the equation (1.2) that v (1) = − mqc m ζ − m − + O ( ζ − m ) , (2.13a) v (2) = − c m m ζ m + O ( ζ m +1 ) , (2.13b) v (3) = m ( m − qc m ζ − m − + O ( ζ − m ) , (2.13c)and more generally, it can be shown by induction that for k ≤ m , v (2 k ) = ( − k k ! Q k − i =0 ( m − i ) c m ζ m + O ( ζ m +1 ) , (2.14a) v (2 k +1) = ( − k Q ki =0 ( m − i ) k ! qc m ζ − m − + O ( ζ − m ) . (2.14b)What we deduce from this is that a singularity sequence beginning with ¯ v regular and v with a zeroorder m will contain a sequence of m + 1 zeroes of order m alternating with m poles of order m + 1.We know that the coefficient of ζ − m − in the iterate v (2 m +1) will vanish according to the formulae(2.14), but it turns out that the entire singular part of the expansion vanishes, so regularity isregained at the iterate v (2 m +1) . Theorem 2.1.
For each integer m > , equation (1.2) admits the singularity pattern (cid:0) rg , m , ∞ m +1 , m , ∞ m +1 , . . . , ∞ m +1 , m , ∞ m +1 , m , rg (cid:1) , (2.15) which includes m + 1 zeroes of order m alternating with m poles of order m + 1 . The proof of this theorem is provided in the appendix, along with those of similar results forthe equations (1.1) and (1.3): 10 heorem 2.2.
For each integer m > , equation (1.1) admits the singularity pattern (cid:0) rg , m , ∞ − m , ∞ , ∞ − m , . . . , ∞ k , ∞ k − m , . . . , ∞ − , ∞ m , m , rg (cid:1) , (2.16) which includes m simple poles with residues alternating between positive and negative multiples of β , which we denote ∞ j = jβz − z + O (1) . (2.17) Theorem 2.3.
Equation (1.3) admits the singularity pattern (cid:0) rg , ζ ( − m , m , ∞ , m − , ∞ , . . . , ∞ j , m − j , . . . , , ∞ m − , , ∞ m , ζ (2 m + 2) m , rg (cid:1) , (2.18) where ζ ( − m indicates that the iterate ¯ w = w ( − satisfies d k dz k ( λz ¯ w + α ¯ w ′ ) = 0 at z = z for k = 0 , ..., m − , and ζ (2 m + 2) m indicates that the iterate w (2 m +2) satisfies d k dz k (cid:16) λ ( z + m ) w (2 m +2) + α ′ w (2 m +2) (cid:17) = 0 , at z = z for k = 0 , ..., m − . We now rephrase the results of the previous section geometrically, and propose a characterisationof singularity confinement in the delay-differential setting in terms of the birational geometry ofjet spaces. Our guiding principle in developing the theory in parallel with the discrete setting willbe that of generic information loss, in particular the ways in which iterating a delay-differentialequation may result in a departure from this, and in what sense it is recovered. To explain themotivations for this analogy, we first note that a birational mapping between smooth projectivealgebraic surfaces is an isomorphism between Zariski open subsets given by the complement ofproper subvarieties that are blown down by either the mapping or its inverse. Almost all curvesare mapped bijectively to curves, and in this sense no information loss occurs generically while it-erating the corresponding discrete system. Singularities of a second-order discrete system occuringwhen curves are blown down to points may be interpreted as more information loss occurring thannormal. The system having the singularity confinement property means that, in such a case wheniterating the system results in more than the generic amount of information loss, we may composethe mapping a finite number of times to recover the generic behaviour: an isomorphism from acurve to a curve.We will formulate a concept of generic information loss for our delay-differential equations. Interms of this we will define singularity confinement as being able to, in the case when iterating thesystem results in more than generic levels of information loss, compose the iteration mapping of thesystem a finite number of times to recover the generic amount. This concept of generic informationloss has two elements: First is the amount of initial data required generically to iterate the systemforward a given number of times, which we will phrase in subsection 3.1 in terms of the orders ofjet spaces on which the systems give well-defined mappings. Second is the behaviour of subspacesunder the these mappings in terms of their codimension, which will be used to describe phenomena11nalogous to degrees of freedom being lost, which we define as ‘blow-down type’ singularities insubsection 3.2. We then outline what it means for such a singularity to be confined, and finallyverify that this geometric description fits with our analysis of the three examples, and that theyconfine all singularities in this sense.
Similarly to how second-order discrete systems are described by birational mappings between al-gebraic surfaces, we will recast our delay-differential equations as mappings between jet spaces.We consider jets associated with the trivial bundle over C with fibre P × P . We use the samecoordinate charts for P × P as in the discrete case, namely ( x, y ) , ( X, y ) , ( x, Y ) , ( X, Y ) where X = 1 /x, Y = 1 /y . The space J rz of r -jets about z is the set of equivalence classes of localholomorphic sections about some z ∈ C under the following equivalence relation. The sections σ , σ define the same r -jet if, when written in coordinates, their derivatives at z coincide up toand including order r .We will be always considering jets at z , so we omit the subscript. We will use coordinatesfor J r induced by writing sections as expansions in our coordinates for P × P . For example, if asection about z is visible in the ( x, y )-chart, it may be written in coordinates as (cid:18) x ( z ) y ( z ) (cid:19) = (cid:18) x + x ζ + x ζ + . . .y + y ζ + y ζ + . . . (cid:19) , (3.1)where ζ = z − z as before, so we have one part of J r covered by the chart with coordinates (cid:18) x x x . . . x r y y y . . . y r (cid:19) , (3.2)and J r can be thought of as four copies of C r +2 with coordinates being coefficients from expansionsof sections in the four charts for P × P , with gluing determined by that of P × P itself, namely X = 1 /x, Y = 1 /y .Consider a three-point delay-differential equation of the form (1.4) given in the introduction,with l being the highest order of derivative that appears. Similarly to how the scalar differenceequation (1.5) is recast as a QRT mapping on P × P , we let ( x, y ) = ( u, ¯ u ) and (¯ x, ¯ y ) = (¯ u, u )given by series expansions about z , so we have a mapping on sections near z , which in the ( x, y )charts for both domain and target copies of P × P is written as: (cid:18) x ( z ) y ( z ) (cid:19) (cid:18) ¯ x ( z )¯ y ( z ) (cid:19) , ¯ x = f ( x, x ′ , . . . , ∂ l x/∂z l ) + f ( x, x ′ , . . . , ∂ l x/∂z l ) yf ( x, x ′ , . . . , ∂ l x/∂z l ) + f ( x, x ′ , . . . , ∂ l x/∂z l ) y , ¯ y = x. (3.3)We now introduce a space of jets on which we consider this, corresponding to generic initial data.Consider a section written as a series expansion in one of the four coordinate charts for P × P ,for example (3.1) in the ( x, y ) chart. Denote the numerator and denominator of the function giving¯ x ( z ) in this chart by P ( z ) , Q ( z ), so for example in the ( x, y ) chart we use (3.3) and consider P = f ( x, x ′ , . . . , ∂ l x/∂z l ) + f ( x, x ′ , . . . , ∂ l x/∂z l ) y,Q = f ( x, x ′ , . . . , ∂ l x/∂z l ) + f ( x, x ′ , . . . , ∂ l x/∂z l ) y. (3.4)12ubstitute expansions giving ( x ( z ) , y ( z )) into these, to obtain formal expansions of P ( z ) , Q ( z ) about z , which we denote P ( z ) = P + P ζ + P ζ + . . . , Q ( z ) = Q + Q ζ + Q ζ + . . . , (3.5)where P , Q are polynomials in x , . . . , x l , y because of the highest order derivative appearing inthe equation (or the equivalent for an expansion of a section in another coordinate chart). Considerthe rational function P /Q on J r + l , using the transition functions between x i , X i etc. beingdefined by the P × P gluing as before, and denote its indeterminacy locus (where the numeratorand denominator simultaneously vanish) by I . We then have a well-defined map ϕ r : J r + l \ I → J r . (3.6)The reason we do not have to worry about indeterminacies of rational functions giving later coef-ficients in the expansion of P/Q to obtain a well-defined map is the following: All of the rationalfunctions giving expansions of
P/Q have denominator being a power of Q . Similarly, all rationalfunctions giving coefficients in the expansion of Q/P are powers of P . Thus if Q = 0 but P = 0,we get a well-defined expansion of Q/P , in which none of the coefficients have indeterminacies (theirdenominators cannot vanish as P = 0) so we have a well-defined a section visible in the ( ¯ X, ¯ y )chart. Similarly, if P = 0 but Q = 0, we get a well-defined expansion of Q/P , in which none ofthe coefficients have indeterminacies (their denominators cannot vanish, as P = 0). Example 3.1.
If we consider the mapping induced by equation (1.1) applied to a section visible inthe ( x, y ) chart, written as an expansion (3.1) , direct substitution yields ¯ x = ax − bx + x y x , ¯ x = bx − x x + x y x , . . . (3.7a)¯ y = x , ¯ y = x , . . . (3.7b) so when x = 0 we have a section visible in the (¯ x, ¯ y ) chart for the target bundle. Similarly, if wehave a section written in the ( X, Y ) chart as an expansion with coefficients X i , Y i , we may use thechart (¯ x, ¯ Y ) and calculate ¯ x = aX Y + bX Y + X X Y , ¯ x = 2 bX X Y − bX Y − X Y X Y , . . . (3.8a)¯ Y = X , ¯ Y = X , . . . (3.8b) so when X Y = 0 we have a section visible in the (¯ x, ¯ Y ) chart for the target bundle. Calculatingin the other charts, we find the subset I ⊂ J r +1 is defined by I = { ( x , x ) = (0 , } ∪ { ( X , X ) = (0 , } ∪ { ( x , Y ) = (0 , } ∪ { ( X , Y ) = (0 , } . (3.9) So we have, for each r ≥ , a map ϕ r : J r +1 \ I → J r . (3.10) We note that the domain J r +1 corresponds to the lowest order of jets to which the equation (1.1) gives a well-defined map from J r +1 \ I to J r . r ≥
0, a map ϕ ( k ) r = ϕ r ◦ ϕ r +1 · · · ◦ ϕ r + k − : J r + kl \ I k → J r , (3.11)defined on the Zariski open subset of J ( r + kl ) where the numerators and denominators of the rationalfunctions giving leading coefficients of successive iterates do not simultaneously vanish. Example 3.2.
To illustrate this, in the case of equation (1.1) being iterated twice, we obtain in the ( x, y ) chart rational functions giving (¯¯ x , ¯¯ y ) as ¯¯ x = a x − abx x + ax y + ax + 2 b x x − b x − bx y − bx x + x y x ( ax − bx + x y ) , (3.12a)¯¯ y = ax − bx + x y x . (3.12b) Computing the indeterminacy loci of these rational functions in all charts and taking its union with I , we obtain I = { ( x , x ) = (0 , } ∪ { ( X , X ) = (0 , } ∪ { ( x , Y ) = (0 , } ∪ { ( X , Y ) = (0 , } ∪ (cid:8) ax − bx + x y = bx − bx x + x y = 0 (cid:9) ∪ { X = 0 , X = − /b } ∪ { Y = Y = 0 } , (3.13) and we have a well-defined map ϕ (2) r = ϕ r ◦ ϕ r +1 : J r +2 \ I → J r . (3.14)We interpret this map ϕ ( k ) r in (3.11) on the set specified above as the generic behaviour ofthe system, and in particular the initial data that is required to iterate the system k times inalmost all cases. We now consider the parts of the jet spaces where the rational functions we haveconsidered above have indeterminacies. For example, if we consider a jet in the charts coming from( X, Y ) , ( ¯ X, ¯ Y ), if ( X , Y ) = (0 ,
0) then we have¯ X = 0 , ¯ X = Y bY , ¯ X = X (cid:0) Y − aY (cid:1) − bX Y X (1 + bY ) , . . . (3.15a)¯ Y = 0 , ¯ Y = X , ¯ Y = X , . . . (3.15b)and so on. By direct calculation using formal series expansions, it can be seen that as long as X = 0 , bY = 0, the jet in ( ¯ X, ¯ Y ) coordinates is determined up to the same order as the one in( X, Y ) coordinates. Thus, on the part of J r ( r ≥
1) where ( X , Y ) = (0 ,
0) but X = 0 , bY = 0,the system induces a mapping J r → J r and we have less information loss than in the genericcase. Comparing this to the discrete case, we see a parallel to the fact that indeterminacies of theiteration mappings are blown up to curves. After considering a concept of generic information loss in terms of the amount of initial data gener-ically required to iterate k times, we turn to parts of jet spaces on which the system induces mapswith more information loss. We will refer to these as blow-down type singularities, in parallel withthe discrete case where information loss corresponds to curves being blown down under iteration14appings.Consider the mapping ϕ r : J r +1 \ I → J r induced by equation (1.1) derived above. We will beinterested in the behaviour under this mapping of subvarieties defined locally by a finite number ofalgebraic constraints. For most codimension m subsets of this part of J r +1 (where r is chosen largeenough such that it includes all the variables appearing in the constraints defining the subset), theimage under ϕ r will be of codimension ≤ m in J r .For example, we can see a variety of behaviours of subspaces as follows. The subspace definedin the ( x i , y i ) chart by the single algebraic constraint y i = c , where i ≤ r + 1 and c = 0 is someconstant, is of codimension one, and its image under ϕ r is of codimension zero. Another subspacedefined by x i = c , for some i ≤ r and c again a nonzero constant, will have image under ϕ r of codimension one. The codimension two subspace where ( X , Y ) = (0 ,
0) with the rest of thecoefficients X i , Y i generic can be quickly seen from (3.15) to have image again of codimension two. Definition 3.3.
A blow-down type singularity of a delay differential equation of the form (1.4) isa codimension m subvariety of J r + l , for some r ≥ , (locally defined as the vanishing locus of anumber of polynomials in coordinates introduced above) whose image under the induced map ϕ r isof codimension greater than m . We emphasise again that this is in analogy with the discrete setting, where singularities aredefined in the sense of an increase in codimension, namely where curves are blown down to pointsunder the iteration mappings. Again we note that in the following examples, r is taken large enoughsuch that J r +1 includes all variables appearing in the algebraic constraints defining the blow-downsingularities. Example 3.4.
The equation (1.1) has a blow-down singularity in J r +1 \ I given in coordinates by x = 0 which is of codimension one (with all other x i , y i generic) but has image of codimensionthree in J r , given in coordinates as follows: { x = 0 } → (cid:8) ¯ X = 0 , ¯ X = − /b, ¯ y = 0 (cid:9) codim 1 → codim 3 Similarly, we see that the development of double and triple zeroes correspond to the following blow-down singularities: { x = 0 , x = 0 } → (cid:8) ¯ X = 0 , ¯ X = − / b, ¯ y = 0 , ¯ y = 0 (cid:9) codim 2 → codim 4 { x = 0 , x = 0 , x = 0 } → (cid:8) ¯ X = 0 , ¯ X = − / b, ¯ y = 0 , ¯ y = 0 , ¯ y = 0 (cid:9) codim 3 → codim 5 and more generally the development of a zero of order m corresponds to the following blow-downsingularity: { x i = 0 , ∀ i = 0 , . . . , m − } → (cid:8) ¯ X = 0 , ¯ X = − /mb, ¯ y i = 0 , ∀ i = 0 , . . . , m − (cid:9) codim m → codim ( m + 2)15 xample 3.5. The equation (1.2) has a blow-down singularity given in coordinates by x = 0 which is of codimension one (with all other x i , y i generic) but has image of codimension five givenin coordinates as follows: { x = 0 } → (cid:8) ¯ X = 0 , ¯ X = 0 , ¯ y = 0 , ¯ y = q ¯ X , p ¯ y = − q ¯ X (cid:9) codim 1 → codim 5 We also have a blow-down singularity corresponding to the development of a double zero { x = 0 , x = 0 } → ¯ X = 0 , ¯ X = 0 , ¯ X = 0 , ¯ y = 0 , ¯ y = 0¯ y − q ¯ X = 0 , ¯ y − p ¯ X − q ¯ X = 0 ,p ¯ X + 2 pq ¯ X ¯ X + 2 q ¯ X − q ¯ X ¯ X = 0 codim 2 → codim 8 and more generally the development of a zero of order m corresponds to the following blow-downsingularity: { x i = 0 , ∀ i = 0 , . . . , m − } → (cid:26) ¯ X i = 0 ∀ i = 0 , . . . , m, ¯ y i = 0 , ∀ i = 0 , . . . , m − F i ( ¯ X m +1 , ¯ X m +2 , . . . , ¯ y m , ¯ y m +1 , . . . ) = 0 ∀ i = m + 1 , . . . , m + 1 (cid:27) codim m → codim (3 m + 2) Here F i are polynomial in their variables that give m + 1 independent algebraic constraints, whichmay be identified by substituting series expansions for x ( z ) , y ( z ) and noting that ¯ X m +2 is the firstcoefficient in which any y i appears. Example 3.6.
The equation (1.3) has a blow-down singularity in ( ¯ x i , ¯ y j coordinates) correspondingto x ( z ) developing a zero of order one. This is given by { ( z − λ ¯ x + α ¯ x = 0 } → { x = 0 , ( z − λy + αy = 0 } codim 1 → codim 2 and more generally the development of a zero of order m corresponds to the following blow-downsingularity, which for conciseness we write in terms of derivatives of the sections, as opposed toexplicitly in terms of coefficients: (cid:26) d i dz i ( λz ¯ x ( z ) + α ¯ x ′ ( z )) | z = z = 0 ∀ i = 0 , . . . , m − (cid:27) → x i = 0 ∀ i = 0 , . . . , m − d i dz i | z = z ( λzy ( z ) + αy ′ ( z )) = 0 ∀ i = 0 , . . . , m − codim m → codim 2 m We now formulate a geometric description of the confinement type behaviour we observed in ourthree examples. Again, the analogy with the discrete case is that if, when iterating the system, wearrive at a blow-down type singularity we only need to iterate a finite number of times further torecover the generic level of information loss, both in terms of orders of jet spaces between which thesystem induces maps, and the behaviour of the singularity under these in terms of codimension.16 efinition 3.7.
Consider a three-point delay differential equation of the form (1.4) with iterationmappings ϕ r , which has a blow-down type singularity B m of codimension m . We say the singularity B m is confined if there exists some k > such that iterating the system k times induces a map from B m ⊂ J r + kl whose image is of codimension ≤ m in J r . We note that this definition captures both the recovery from the increase in codimension of B m as well as the amount of initial data required to iterate k times generically. Take B m as asubset of the same order jet space J r + kl as for the generic behaviour ϕ ( k ) r : J r + kl \ I k → J r . Weconsider accessible blow-down singularities: those that may arise when iterating the system fromregular nonzero initial data. For the three equations we consider, we first describe the set of all suchsingularities and then use our results concerning infinite families of singularity patterns to deducethat they are all confined in the above sense. The only accessible blow-down type singularities of equation (1.1) are B m = { x i = 0 ∀ i = 0 , . . . , m − } . Proof.
We will first show that the only blow-down singularities visible in the x i , y j chart are con-tained in { x = 0 } . Suppose B ⊂ J r +1 is of codimension m , so dimension d = 2( r + 1) − m ,defined locally by F = · · · = F l = 0, where F i are polynomial in x , . . . , x m , y , . . . , y m , andthat x = 0 on B . Then near p ∈ B (at which B is nonsingular) given in coordinates by p : ( x i , y j ) = ( x ∗ i , y ∗ j ), we have a parametrisation of B by d free parameters. That is, there ex-ist i , . . . , i p , j , . . . j d − p ⊂ { , . . . , r + 1 } such that we have a parametrisation s ... s p t ... t d − p x i = x ∗ i + s ... x i p = x ∗ i p + s p y j = y ∗ j + t ... y j d − p = y ∗ j d − p + t d − p (3.16a)with the rest of the variables x i , y j given by analytic functions of s , . . . , s p , t , . . . , t d − p : x i = x ∗ i + F i ( s , . . . , s p , t , . . . , t d − p ) , y j = y ∗ j + G j ( s , . . . , s p , t , . . . , t d − p ) , (3.17)for i
6∈ { i , . . . , i p } , j
6∈ { j , . . . , j d − p } , with F i , G j anaytic and zero when all s i , t j are zero, and theJacobian of this parametrisation at p is of rank d . We now show, using this parametrisation, thatthe image of B in J r under ϕ r is of dimension ≥ r − m as long as x = 0 on B . In coordinates,the mapping is of the form ¯ y n = x n , ¯ x n = y n − P n ( x , . . . , x n +1 ) x n +10 . (3.18)Here P n is a homogeneous polynomial of degree n + 1, which follows from the repeated applicationof the quotient rule in computing expressions for derivatives of ¯ x = y + ax − bx ′ x . We obtain a local17arametrisation of the image of B :¯ y i = x ∗ i + s ...¯ y i p = x ∗ i p + s p ¯ x j = y ∗ j + t + H ...¯ x j d − p = y ∗ j d − p + t d − p + H d − p (3.19)where H , . . . H d − p are analytic in s , . . . , s p (as x = 0 on B ), with the rest of the coordinates¯ y i , ¯ x j being analytic functions of the parameters. The Jacobian of this parametrisation can be seento have rank at least d −
2, with linearly independent columns corresponding to partial derivativeswith respect to s , . . . s p , t , . . . , t d − p − ( t d − p will not contribute to the rank if d − p = r + 1, i.e.if y r +1 is one of the free variables in the parametrisation of B ). The possibility that the imageis of codimension less than m has already been illustrated at the start of subsection 3.2, whereconstraints on y j may not induce constraints on the image.Similarly, if we consider a subvariety of codimension m in the chart ( X, y ) away from { X = 0 } ,we see that its image under ϕ r must be again of codimension ≤ m . This is done in exactly thesame way as above, noting that the mapping in charts is of the form¯ Y n = X n , ¯ x n = y n − P n ( X , . . . , X n +1 ) X n +10 , (3.20)where again P n is a homogeneous polynomial of degree n + 1. Regarding the part of the jet spacewith X = 0, we remark that X = 0 with y = 0 is not an accessible singularity, as for a pole todevelop while iterating, it must follow a zero. Further, the only parts of { X = 0 , y = 0 } accessiblefrom regular and nonzero initial data are those coming from one of the blow-down singularities B m . Similar calculations in the charts ( x, Y ) and ( X, Y ) show that it suffices to consider blow-down singularities visible in the ( x, y ) chart where at least x = 0. If we take x ( z ) = x m ζ m + x m +1 ζ m +1 + . . . for m > y = y + y ζ + . . . , then direct calculation shows that we have¯ X = 0 , ¯ X = − bm , ¯ X = bx m +1 − ax m b m x m − y b m , . . . (3.21)and more generally that ¯ X n = P n ( x m , . . . , x m + n , y , . . . , y n − ) b n m n x n − m − y n − b m , ¯ y n = 0 for n < m, ¯ y n = x m for n ≥ m, (3.22)where P n is polynomial in its arguments. By again considering parametrisations and their Jaco-bians, it is straightforward to show that we cannot have blow-down singularities away from x m = 0.Applying this argument inductively completes the proof that the only accessible blow-down singu-larities are as claimed.We now show how the singularity patterns pointed out in subsection 2.1 correspond to confine-ment of blow-down singularities for equation (1.1). Example 3.9.
The singularity B , which corresponds to the beginning of the singularity pattern (rg , , ∞ , ∞ , , rg) , s confined after five iterations. We calculate as we did in section 2 but keep track of orders of jetsand codimensions to find that composing the iteration on sections gives maps as follows: B ⊂ J r +5 codim( B ) = 1 ( x (0) , y (0) ) = (0 , rg) ϕ (1) : B → J r +5 codim( ϕ (1) ( B )) = 3 ( x (1) , y (1) ) = ( ∞ , ) ϕ (2) : B → J r +5 codim( ϕ (2) ( B )) = 5 ( x (2) , y (2) ) = ( ∞ , ∞ ) ϕ (3) : B → J r +3 codim( ϕ (3) ( B )) = 3 ( x (3) , y (3) ) = (0 , ∞ ) ϕ (4) : B → J r +1 codim( ϕ (4) ( B )) = 1 ( x (4) , y (4) ) = (rg , ) ϕ (5) : B → J r codim( ϕ (5) ( B )) = 0 ( x (5) , y (5) ) = (rg , rg) For each iteration, we have indicated the order of jet space to which we have well-defined mappingsfrom B , as well as codimensions of the images of B and the corresponding parts of the singularitypattern. We note that the exceptional behaviour we observed in the singularity pattern, namely thatwhen computing x (3) , three terms vanished as it developed a zero rather than a pole, is reflected inthe codimension falling from to . More generally, if we take the blow-down singularities B m as in Lemma 3.8 as subsets of J m +3+ r with the rest of the coefficients generic, from Theorem 2.2 we see that iterating the system (1.1)induces a map ϕ (2 m +3) : B m → J r , where the image of B m is a jet visible in the ( x, y ) chart. Tosee that this image is of codimension zero, we must make some observations of how the initial datafrom the section ( x (0) , y (0) ) enters into the subsequent iterates, and in particular how it is recoveredin ( x (2 m +3) , y (2 m +3) ). This will require detailed but straightforward analysis of the mapping onjets in three cases, corresponding to different points in the singularity pattern. Firstly, when thefirst pole develops and how the coefficients from ( x (0) , y (0) ) enter into X (1) , X (2) , secondly, how theinitial data is propagated through the sequence of simple poles X (1) , . . . , X (2 m ) , then how it reen-ters x (2 m +2) , x (2 m +3) after the zero develops at x (2 m +1) . The key technique for our analysis here isessentially identifying and counting free variables, which we illustrate in detail in this example.We first consider the map from ( x (0) , y (0) ) to ( X (1) , y (1) ) corresponding to the development ofthe first simple pole in the sequence. Here we omit the superscripts for conciseness, working withthe mapping in the charts ( x, y ) and ( ¯ X, ¯ y ). Beginning with initial data corresponding to B m ,namely sections in the ( x, y ) chart with x = x = . . . x m − = 0, with the rest of the coefficients x i , y j generic, by direct calculation we have¯ X = 0 , ¯ X = − mb , ¯ X n = − y n − m b + P n ( x m , . . . , x m + n − , y , . . . , y n − ) x n − m , for n ≥ y = · · · = ¯ y m − = 0 , ¯ y n = x n , for n ≥ m, where P n is polynomial in its arguments. From this, we see that the coefficients ¯ X i ≥ , ¯ y j ≥ m arealgebraically independent functions of the initial data, which follows from the way in which the freevariable y n − ( n ≥
2) appears linearly in ¯ X n but not at all in ¯ X n − and so on. In particular wehave the image of B m under a single iteration being of codimension m + 2, as noted in Example19.4. Similarly, we see that the next iterate is obtained from ¯ X i , ¯ y j above as¯¯ X = 0 , ¯¯ X = 1 b , ¯¯ X j = P j ( ¯ X , . . . , ¯ X j ) , for 2 ≤ j ≤ m + 1 , ¯¯ X n = − ¯ y n − b + Q n ( ¯ X , . . . , ¯ X n , ¯ y m . . . ¯ y n − ) , for n ≥ m + 2 , ¯¯ Y = 0 , ¯¯ Y = − mb , ¯¯ Y n = ¯ X n , for n ≥ . Here P j is again polynomial, linear in ¯ X j , and Q n is polynomial in its arguments. From this, wesee that the image of B m is of codimension m + 4, with ¯¯ X i , ¯¯ Y j having the following dependence onthe initial data x i , y j :¯¯ X = 0 , ¯¯ X = 1 b , ¯¯ X n = F n ( y , . . . , y n − , x m , . . . , x m + n − ) for n ≥ m + 2 , ¯¯ Y = 0 , ¯¯ Y = − mb , ¯¯ Y n = G n ( y , . . . , y n − , x m , . . . , x m + n − ) for n ≥ m + 2 , where, importantly, F n is linear in y n − with constant coefficient, and also linear in x m + n − withcoefficient being a constant multiple of 1 /x m .We now consider the iterates X (3) , . . . , X (2 m ) , which correspond to simple poles, and show thatwe have the same kind of dependence of coefficients on the initial data. Building on our calculation(3.15) in the charts ( X, Y ) , ( ¯ X, ¯ Y ), we see that sections with ( X , Y ) = (0 ,
0) have images underthe iteration mapping given by¯ X = 0 , ¯ X = Y bY , ¯ X n = Y n (1 + bY ) + P n ( X , . . . , X n , Y , . . . Y n − ) X n − (1 + bY ) n , ¯ Y = 0 , ¯ Y = X , ¯ Y n = X n , for n ≥ , (3.23)where P n is polynomial in its arguments, and we note that these expansions are valid for de-termining all iterates ( X (2) , Y (2) ) , . . . , ( X (2 m ) , Y (2 m ) ), as we have X ( k )1 = 0 , bY ( k )1 = 0, for k = 0 , . . . , m −
1, which we know from our explicit expressions of the residues of the simple polesin the singularity pattern, given in Theorem 2.2 . Iterating through this sequence of simple poles,we have well-defined maps J m +3+ r \ { X (1 + bY ) = 0 } → J m +3+ r , and a simple calculation usingthe Jacobian as in the proof of Lemma 3.8 shows that the image of B m cannot change codimensionin J m +3+ r under this sequence of maps, so we have the images of B m under ϕ (2) , . . . , ϕ (2 m ) are allof codimension m + 4.Further, from (3.23) and our observations of ( ¯¯ X, ¯¯ Y ) we see that for k = 2 , . . . , m , the coefficients X ( k ) n , Y ( k ) n have the same kind of dependence on the initial data, and in particular the last iteratebefore the zero develops is of the form X (2 m )0 = 0 , X (2 m )1 = 1 mb , X (2 m ) n = F (2 m ) n ( y , . . . , y n − , x m , . . . , x m + n − ) for n ≥ m + 2 ,Y (2 m )0 = 0 , Y (2 m )1 = − b , Y (2 m ) n = G (2 m ) n ( y , . . . , y n − , x m , . . . , x m + n − ) for n ≥ m + 2 , where again F n is linear in y n − with constant coefficient, and also linear in x m + n − with coefficientbeing a constant multiple of 1 /x m . 20e now consider the final step, when the map ( X (2 m ) , Y (2 m ) ) ( x (2 m +1) , Y (2 m +1) ) shows adrop in codimension of the image of B m , with the development of a zero of order m . Omiting thesuperscripts for conciseness and writing ( X (2 m ) , Y (2 m ) ) = ( X + X ζ + . . . , Y + Y ζ + . . . ), weknow that the the coefficients for the image of the B m under the iterations up to this point in thesingularity pattern must satisfy at least Y = 0 , Y = − b − , X = 0 , X = ( mb ) − . (3.24)Similarly writing ( x (2 m +1) , Y (2 m +1) ) = (¯ x + ¯ x ζ + . . . , ¯ Y + ¯ Y ζ + . . . ), we see the mapping oncoefficients from jets satisfying (3.24) gives¯ x = 0 , ¯ x = a + b mX − b Y , ¯ x = − b (cid:0) bm X + bY − mX + Y (cid:1) , ¯ x n = b ( nmX n +1 − Y n +1 ) + P n ( X , . . . , X n , Y , . . . , Y n ) , for n ≥ , ¯ Y = 0 , ¯ Y = 1 mb , ¯ Y j = X j , for j ≥ , (3.25)where we have again used P n to denote a polynomial in its arguments. We know from Theorem 2.2that if ( X (2 m ) , Y (2 m ) ) are obtained by iterating from B m , then the coefficients X i , Y j must satisfythe algebraic conditions for ¯ x , . . . , ¯ x m − given by (3.25) to all vanish, and we know exactly whatrelations must exist between the coefficients ( X (2 m ) i , Y (2 m ) j ), which have evolved through the singu-larity pattern from those defining B m . Further, from the dependence of X (2 m ) i , Y (2 m ) j on the initialdata, and the way in which X (2 m ) i , Y (2 m ) j enter into x (2 m +1) i , Y (2 m +1) j according to (3.25), we seethat the image of B m under ϕ (2 m +1) is of codimension m + 3 in the jet space corresponding to( x (2 m +1) , Y (2 m +1) ). Finally, another calculation on the exact same lines shows that after one morestep, we have the image of B m under ϕ (2 m +1) being of codimension zero. The analysis in this case proceeds in exactly the same way as the previous one, so we omit detailsfor conciseness. In particular, the following may be proved using the same techniques and approachas for Lemma 3.8:
Lemma 3.10.
The only accessible blow-down type singularities of equation (1.1) are B m = { x i = 0 ∀ i = 0 , . . . , m − } . We may also use the same techniques to examine the behaviour of blow-down singularities interms of codimension, beginning with that associated with a simple zero:
Example 3.11.
The singularity B of equation (1.2) , which corresponds to the start of the singu-larity pattern (rg , , ∞ , , rg) , is confined after four iterations, with the following behaviour under compositions of the iteration aps: B ⊂ J r +4 codim( B ) = 1 ( x (0) , y (0) ) = (0 , rg) ϕ (1) : B → J r +4 codim( ϕ (1) ( B )) = 5 ( x (1) , y (1) ) = ( ∞ , ) ϕ (2) : B → J r +4 codim( ϕ (2) ( B )) = 5 ( x (2) , y (2) ) = (0 , ∞ ) ϕ (3) : B → J r +1 codim( ϕ (3) ( B )) = 1 ( x (3) , y (3) ) = (rg , ) ϕ (4) : B → J r codim( ϕ (4) ( B )) = 0 ( x (4) , y (4) ) = (rg , rg) We note here again that the drop in codimension occurs when two terms vanish in the expansionfor x (3) as it regains regularity as opposed to having a double pole. Again, considering the blow-down singularities B m from Lemma 3.10 as subsets of J m +2+ r ,Theorem 2.1 and tracing the dependence on initial data of the iterates through the sequence usingexactly the same techniques as in the previous example, we see that we have ϕ (2 m +2) : B m → J r under which the image of B m is of codimension zero, so all accessible blow-down singularities ofequation (1.2) are confined. In this case we begin with an example, as the blow-down singularities for equation (1.3) occur notafter x develops a zero at z , but under the mapping applied to the jets in ¯ x, ¯ y coordinates satisfyingthe condition for a zero to develop. Example 3.12.
The condition on ( x, y ) for a simple zero to develop while iterating equation (1.3) ,namely B = { α ¯ x + λ ( z − x = 0 } , with the rest of the coefficients generic, corresponds to the start of the singularity pattern which wedenoted in section 2 by (cid:16) rg , ¯ ζ , , ∞ , ¯¯ ζ , rg (cid:17) . We observe a jump in codimension not from ( x (0) , y (0) ) to ( x (1) , y (1) ) , but one step earlier, and weobserve the following behaviour under compositions of the iteration maps: B ⊂ J r +5 codim( B ) = 1 ( x ( − , y ( − ) = (¯ ζ , rg) ϕ (1) : B → J r +4 codim( ϕ (1) ( B )) = 2 ( x (0) , y (0) ) = (0 , ¯ ζ ) ϕ (2) : B → J r +4 codim( ϕ (2) ( B )) = 2 ( x (1) , y (1) ) = ( ∞ , ) ϕ (3) : B → J r +3 codim( ϕ (3) ( B )) = 2 ( x (2) , y (2) ) = (¯¯ ζ , ∞ ) ϕ (4) : B → J r +1 codim( ϕ (4) ( B )) = 1 ( x (3) , y (3) ) = (rg , ¯¯ ζ ) ϕ (5) : B → J r codim( ϕ (5) ( B )) = 0 ( x (4) , y (4) ) = (rg , rg) We note here again that a drop in codimension occurs when x (3) regains regularity as opposed to asimple zero. Again by the same approach, the following may proved by local calculations in charts:22 emma 3.13.
The only accessible blow-down type singularities of equation (1.3) are B m = (cid:26) d i dz i ( λz ¯ x ( z ) + α ¯ x ′ ( z )) | z = z = 0 , ∀ i = 0 , . . . , m − (cid:27) In the same way as the other two examples, we see from Theorem 2.3 that for regarding B m asa subset of J (2 m +2) , iterating the system gives a map ϕ (2 m +3) : J (2 m +3+ r ) → J ( r ) , under which theimage of B m is of codimension zero. We now summarise our work and discuss questions that follow it naturally, again organised intotwo parts: firstly singularity analysis on the level of equations and secondly its geometric inter-pretation. On this first level, we have significantly extended previous studies of delay Painlev´eequations and discovered new confinement type behaviour, which is interesting in its own right. Inthe process we have developed techniques for the analysis of singularity patterns of arbitrary lengthand proving confinement, which we hope will be useful in tackling one of the main difficulties in thesingularity analysis of delay-differential equations. It would be interesting to adapt our methodsto other integrable delay-differential equations, for example extensions of the examples consideredin this paper such as the families generalising equation (1.2) isolated by Halburd and Korhonen byimposing Nevanlinna-theoretic integrability criteria [HK17]. Though preliminary calculations showthat these equations admit some of the same confined singularity patterns as equation (1.2) (namelythose associated with single, double and triple zeroes) it is a natural next step to determine whetherthese admit the same infinite families and whether this behaviour fits into our geometric framework.Another question that arises from our work on the level of equations relates to the use of sin-gularity analysis techniques to isolate integrability candidates. The fact that each of these threeexamples may be obtained by requiring confinement of only the simplest singularity in the familyassociated with zeroes of different orders prompts the question of whether and how this could en-sure confinement of all singularities in the family. Further, there may be applications of our resultsto the search for elliptic function solutions of degenerate cases of delay Painlev´e equations. Forexample, the a = 0 and p = 0 cases of equations (1.1) and (1.2) respectively are known [Ber17] toadmit elliptic function solutions. Degree 2 elliptic function solutions were identified with the helpof singularity analysis, and in particular that these degenerate cases admit the singularity patternsassociated with simple zeroes outlined in section 2. These patterns are compatible with ellipticfunction solutions in the sense that the numbers of poles and zeroes in a pattern are equal (countedwith multiplicity), and also that the residues of poles in the sequence sum to zero. We note thatour proofs of the infinite families of singularity patterns are also valid for the degenerate cases,and we observe the same kind of compatibility with elliptic functions in all of them, so it would beinteresting to determine whether they may be used to isolate higher degree elliptic function solutions.The other aim of this work was to initiate the geometric study of delay Painlev´e equations. Wehave put forward a geometric description of singularity confinement in these three examples, andwe hope to have worked in convincing parallel with the discrete case, and in particular capturedin our description the exceptional nature of these equations in terms of the recovery of initial datawhen a singularity is confined. By no means, however, is this geometric framework complete or23efinitive, and we hope that our ideas are refined and built upon through singularity analysis inmore examples. Acknowledgements
The author would like to express his sincere thanks to R. Halburd for valuable discussions andadvice.This research was supported by a University College London Graduate Research Scholarshipand Overseas Research Scholarship.
A Proofs of infinite families of singularity patterns
We now give proofs of the results of subsection 2.1 relating to infinite families of singularity patterns.
A.1 Proof of Theorem 2.2
For equation (1.1), our strategy is to consider a singularity pattern beginning with (rg , m ), thenderive and analyse recurrences for the coefficients in the expansions of the next (2 m + 1) iterates,to deduce that the singularity pattern is as claimed.Because the equation (1.1) is autonomous we can take without loss of generality the zero oforder m to be at the origin, and start with the formal expansions¯ u = ∞ X j =0 ¯ u j z j , (A.1a) u = ∞ X j = m u j z j , u m = 0 . (A.1b)Inserting these into the equation, we immediately see that ¯ u has a simple pole:¯ u = − mβz + O (1) . (A.2)The iterates of interest to us are u = u (0) , ¯ u = u (1) , u (2) , . . . , u (2 m ) , u (2 m +1) . By inspection of theterms on the right-hand side of the forward iteration (2.6), these will be either regular or poles oforder at most one, so we introduce the notation u ( i ) = ∞ X n = − u ( i ) n z n , (A.3)for i = 0 , . . . , m + 1, where any number of the u ( i ) n may be zero.By deriving recurrences for the coefficients u ( i ) n , we will show firstly that u ( i ) − = 0 for i =1 , . . . , m , then that u (2 m +1) − = u (2 m +1)0 = · · · = u (2 m +1) m − = 0, from which we will deduce that u (2 m +1) = O ( z m ), and in particular has a zero of order m if the rest of the initial data is generic.It will be helpful to introduce some notation to deal with the logarithmic derivative u ′ /u in theforward iteration map. 24 emma A.1. Let r be a nonzero integer. If u = P ∞ j = r u j z j with u r nonzero, then u ′ u = ∞ X n = − U n z n , where the coefficients U n are given by U − = r, U = u r +1 /u r , and so on according to the recurrence U n = 1 u r ( n + 1) u r + n +1 − n X j =1 u r + j U n − j . We first deduce from the recurrence that following the zero of order m , the next 2 m iterateshave simple poles: Proposition A.2.
The iterates u ( i ) have simple poles at z = 0 for all i = 1 , . . . , m , and we have u (2 k ) − = kβ, for k = 1 , . . . , m, (A.4a) u (2 k +1) − = ( k − m ) β for k = 0 , . . . , m. (A.4b) Proof.
We already have that u (0) − = 0 and u (1) − = − mβ . We then insert the expansions (A.3) forthe iterates u ( i ) into the relevant upshifts of the equation, making use of Lemma A.1 with r = − u ( i +1) − = u ( i − − + β, (A.5)for all i such that u ( i ) has a simple pole. Iterating this from i = 1 from the initial values for u (0) − , u (1) − ,we see that u ( i ) have simple poles for all i = 1 , . . . , m , and we obtain the formulae (A.4).It will now be helpful to introduce the following notation for the iterates: u (2 k ) = f ( k ) = ∞ X n = − f ( k ) n z n , f ( k ) n = u (2 k ) n , (A.6) u (2 k +1) = g ( k ) = ∞ X n = − g ( k ) n z n , g ( k ) n = u (2 k +1) n , (A.7)for k = 0 , . . . , m . As we now know that u (1) , . . . , u (2 m ) have simple poles at z = 0, we use LemmaA.1 to write the logarithmic derivatives of f ( k ) , g ( k − , for k = 1 , . . . m as f ( k ) ′ f ( k ) = ∞ X n = − F ( k ) n z n , g ( k ) ′ g ( k ) = ∞ X n = − G ( k ) n z n . (A.8)Further, we have from (A.4) that for k = 0 , . . . , m that F ( k ) − = kβ, G ( k ) − = ( m − k ) β , so we havethe following recursive formulae for F ( k ) n , G ( k ) n : F ( k ) n = 1 kβ ( n + 1) f ( k ) n − n X j =1 f ( k ) j − F ( k ) n − j , (A.9a) G ( k ) n = 1( k − m ) β ( n + 1) g ( k ) n − n X j =1 g ( k ) j − G ( k ) n − j , (A.9b)25alid for all k such that f ( k ) , g ( k ) have simple poles. Using this notation, the forward iteration thenleads to the recurrences, f ( k )0 = f ( k − + α − βG ( k − , (A.10a) g ( k )0 = g ( k − + α − βF ( k )0 , (A.10b)and f ( k ) n = f ( k − n − βG ( k − n , (A.11a) g ( k ) n = g ( k − n − βF ( k ) n , (A.11b)for n ≥
1, and k = 1 , . . . , m . Using (A.9) with n = 0, we see that the recurrences (A.10) are alinear system of difference equations for f ( k )0 , g ( k )0 : f ( k )0 = f ( k − + α − k − − m ) g ( k − , (A.12a) g ( k )0 = g ( k − + α − k f ( k )0 , (A.12b)subject to the initial conditions f (0)0 = 0 and g (0)0 = u (1)0 = α + ¯ u − βu /u determined by theinitial data ¯ u, u . The unique solution of (A.12) subject to these initial conditions is given by f ( k )0 = k ( α + C ) , (A.13a) g ( k )0 = ( m − k ) C, (A.13b)where C = u (1)0 /m . Similarly, after using the formula (A.9), the recurrences (A.11) become f ( k ) n = f ( k − n − k − − m ) ( n + 1) g ( k − n − n X j =1 g ( k − j − G ( k − n − j , (A.14a) g ( k ) n = g ( k − n − k ( n + 1) f ( k ) n − n X j =1 f ( k ) j − F ( k ) n − j , (A.14b)subject to the initial conditions f (0) n = 0 for n = 0 , . . . , m −
1, and g (0) n fixed by the initial data ¯ u, u .Given the solution (A.13), the n = 1 case is then a linear system of recurrences in k for f ( k )1 , g ( k )1 ,which may be solved by elementary methods. With both n = 0 , f ( k )2 , g ( k )2 can be solved, and so on. Observations of these solutions lead us to the followingproposition: Proposition A.3.
The unique solution to (A.14) subject to the initial conditions is given by ( f ( k ) n , g ( k ) n ) , n = 0 , . . . , m − , k = 0 , . . . , m of the form f ( k ) n = kP ( k ) n (A.15a) g ( k ) n = ( k − m ) Q ( k ) n , (A.15b) where P ( k ) n , Q ( k ) n are polynomial in k of degree at most n . roof. We have from formulae (A.13) that the statement is true for n = 0, so we proceed byinduction. Suppose that f ( k )0 , . . . , f ( k ) n − and g ( k )0 , . . . , g ( k ) n − are of the form (A.15). The recursiveformulae (A.9) then imply that F ( k , . . . , F ( k ) n − and G ( k )0 , . . . , G ( k ) n − are polynomial in k , of degreeat most n −
1. We then see that the following terms from (A.14) are polynomial in k of degree atmost n −
1: 1 k n X j =1 f ( k ) j − F ( k ) n − j = n X j =1 P ( k ) j − F ( k ) n − j = n − X j =0 λ j k j , (A.16a)1( k − m ) n X j =1 g ( k ) j − G ( k ) n − j = n X j =1 Q ( k ) j − G ( k ) n − j = n − X j =0 µ j k j , (A.16b)so we have f ( k +1) n = f ( k ) n − n + 1 k − m g ( k ) n + n − X j =0 µ j k j , (A.17a) g ( k +1) n = g ( k ) n − n + 1 k + 1 f ( k ) n n − X j =0 λ j k j , (A.17b)We write our ansatz (A.15) for the solution to this equation as f ( k ) n = k n X j =0 a j k j (A.18a) g ( k ) n = ( k − m ) n X j =0 b j k j . (A.18b)We note that one initial condition u (0) n = 0 is satisfied automatically, but imposing the other requiresus to set b = − u (1) n /m. (A.19)We now insert the ansatz (A.18) into the equation (A.17) and equate coefficients of powers of k toobtain a linear system in 2 n + 1 variables a , . . . , a n , b , . . . , b n :0 = a n + b n , (A.20a) λ i = ( n + 1) a i + ( i + 1) b i + n X j = i +1 ( − j − i (cid:18)(cid:18) j + 1 i (cid:19) + m (cid:18) ji (cid:19)(cid:19) b j , (A.20b) µ i = n X j = i (cid:18) ji (cid:19) a j + ( n + 1) b i , (A.20c) λ = ( n + 1) a + ( m + 1) n X j =1 ( − j b j , (A.20d) µ = n X j =0 a j + ( n + 1) b , (A.20e)27or i = 1 , . . . , n −
1. We write this as M n v = c , where v = ( a , . . . , a n , b , . . . , b n ) T , c =(0 , λ , . . . , λ n , µ , . . . , µ n ) T and M n is the square matrix of size 2 n + 1 giving the right-hand sideof the system (A.20). A simple sequence of row and column operations yields an upper-triangularmatrix and we obtain det M n = ( n !) ( m − n ) n , (A.21)where ( a ) n = Q n − i =0 ( a + i ) is the usual Pochammer symbol, so the matrix is nonsingular for n ≤ m − u (2 m +1) n = 0 for m ≥ n , and thus that u (2 m +1) = O ( z m ). A.2 Proof of Theorem 2.1
While we may proceed along the same lines as in subsection A.1, a shortcut is provided by a knownMiura-type transformation between equation (1.1) and equation (1.2). This may be easily detectedgiven the well-known transformation between the differential-difference systems that give theseequations as similarity reductions, and is proved by direct calculation:
Lemma A.4. If v solves (1.2) with parameters p, q , then u = ¯ vv solves (1.1) with parameters a = 2 p, b = − q . So, we consider a singularity pattern for equation (1.2) beginning with (¯ v, v ) = (rg , m ), and wealso assume that the zero has developed while iterating through regular and nonzero iterates, so ¯¯ v is also regular. Then under the transformation to a solution of (1.1), we have(¯ u, u ) = (¯¯ v ¯ v, ¯ vv ) = (rg , m ) , so the transformation gives us a singularity pattern for (1.1), which by Theorem 2.2 must be (cid:0) rg , m , ∞ , ∞ , ∞ , . . . , ∞ , ∞ , m , rg (cid:1) , with u ( k ) ∼ ζ − for k = 1 , . . . , m , then u (2 m +1) ∼ ζ m , and u (2 m +2) regular. So this implies thatthe iterates v ( k ) in the singularity pattern must satisfy: v ( k − v ( k ) = u ( k ) ∼ ζ − for k = 1 , . . . , m, (A.22a) v (2 m ) v (2 m +1) = u (2 m +1) ∼ ζ m , (A.22b) v (2 m +1) v (2 m +2) = u (2 m +2) = O ( ζ ) . (A.22c)Beginning with our assumption that v (0) ∼ ζ m , we see from the k = 1 case of equation (A.22a)that v (1) ∼ ζ − ( m +1) , and then using the k = 2 , . . . , m cases successively that v (2 k ) ∼ ζ m , for k = 0 , . . . , m, v (2 k +1) ∼ ζ − ( m +1) for k = 0 , . . . , m − . (A.23)Then using equations (A.22b) and (A.22c) we have that v (2 m +1) ∼ ζ and v (2 m +2) = O ( ζ ) andthe proof is complete. 28 .3 Proof of Theorem 2.3 Again, while the strategy and techniques from the proof of Theorem 2.2 are available for this case,a shortcut is provided by the following transformation between equation (1.1) and equation (1.3),which was pointed out in [GRM93]:
Lemma A.5. If w solves (1.3) with parameters λ, α , then u = ¯ w/ ¯ w solves (1.1) with parameters a = 2 λ, b = − α . Similarly to in the previous section, we consider a singularity pattern for equation (1.3) beginningwith (¯ w, w ), where d i dz i ( λz ¯ w ( z ) + α ¯ w ′ ( z )) = 0 at z = z for i = 0 , . . . , m − w ∼ ζ m . and wealso assume that the zero has developed while iterating through regular and nonzero iterates, so¯¯ w, ¯¯¯ w are also regular and nonzero. Then under the transformation to a solution of (1.1), we have(¯¯ u, ¯ u ) = ( ¯ w ¯¯¯ w , w ¯¯ w ) = (rg , m ) , so the transformation gives us a singularity pattern for (1.1), which by Theorem 2.2 must be (cid:0) rg , m , ∞ , ∞ , ∞ , . . . , ∞ , ∞ , m , rg (cid:1) , with u ( k ) ∼ ζ − for k = 0 , . . . , m −
1, then u (2 m ) ∼ ζ m , and u (2 m +2) regular. So this implies thatthe iterates w ( k ) in the singularity pattern must satisfy: w ( k +1) w ( k − = u ( k ) ∼ ζ − for k = 0 , . . . , m − , (A.24a) w (2 m +1) w (2 m − = u (2 m ) ∼ ζ m , (A.24b) w (2 m +2) w (2 m ) = u (2 m +1) = O ( ζ ) . (A.24c)Beginning with our assumptions that w (0) ∼ ζ m and w ( − is regular, we see recursively fromequation (A.24a) that w (2 k ) ∼ ζ m − k for k = 0 , . . . m − , w (2 k +1) ∼ ζ − k for k = 0 , . . . , m. (A.25)Then using equations (A.24b) and (A.24c) we see that w (2 m +1) ∼ ζ and w (2 m +2) = O ( ζ ) and theproof is complete. References [Ber17] Bjorn K. Berntson,
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