Discrete-Time Goldfishing
aa r X i v : . [ n li n . S I] A ug Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2011), 082, 35 pages Discrete-Time Goldf ishing ⋆ Francesco CALOGEROPhysics Department, University of Rome “La Sapienza”,Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy
E-mail: [email protected], [email protected]
Received May 04, 2011, in final form July 29, 2011; Published online August 23, 2011http://dx.doi.org/10.3842/SIGMA.2011.082
Abstract.
The original continuous-time “goldfish” dynamical system is characterized bytwo neat formulas, the first of which provides the N Newtonian equations of motion of thisdynamical system, while the second provides the solution of the corresponding initial-valueproblem. Several other, more general, solvable dynamical systems “of goldfish type” havebeen identified over time, featuring, in the right-hand (“forces”) side of their Newtonianequations of motion, in addition to other contributions, a velocity-dependent term such asthat appearing in the right-hand side of the first formula mentioned above. The solvable character of these models allows detailed analyses of their behavior, which in some cases isquite remarkable (for instance isochronous or asymptotically isochronous ). In this paper weintroduce and discuss various discrete-time dynamical systems, which are as well solvable ,which also display interesting behaviors (including isochrony and asymptotic isochrony ) andwhich reduce to dynamical systems of goldfish type in the limit when the discrete-time independent variable ℓ = 0 , , , . . . becomes the standard continuous-time independentvariable t , 0 ≤ t < ∞ . Key words: nonlinear discrete-time dynamical systems; integrable and solvable maps;isochronous discrete-time dynamical systems; discrete-time dynamical systems of goldfishtype
The original “goldfish” dynamical system [1, 2] is characterized by the system of N Newtonianequations of motion¨ z n = N X k =1 ,k = n z n ˙ z k z n − z k , n = 1 , . . . , N, (1.1a)and by the following neat prescription yielding the solution of the corresponding initial-valueproblem: the N values of the dependent variables z n ≡ z n ( t ) at time t are the N solutions ofthe algebraic equation (for the unknown z ) N X k =1 ˙ z k (0) z − z k (0) = 1 t , (1.1b)i.e. the N roots of the polynomial equation of degree N in the variable z that obtains bymultiplying this formula by the polynomial N Q j =1 [ z − z j (0)]. ⋆ F. Calogero
Notation 1.1.
Here and hereafter N is an arbitrary positive integer (generally N ≥ t (“continuoustime”), and the N dependent variables z n ≡ z n ( t ) may be interpreted as the coordinates of N point-like unit-mass particles – hence ˙ z n denote their velocities and ¨ z n their accelerations, consis-tently with the interpretation of (1.1a) as a set of Newtonian equations of motion with velocity-dependent forces. The indices – such as n , m , j , k – generally run from 1 to N (below, asa convenient reminder, we often indicate this explicitly; as well as the exceptions to this rule).Hereafter we denote as “dynamical system of goldfish type” any dynamical system charac-terized by Newtonian equations of motion featuring in their right-hand sides – which have, inthe Newtonian context, the significance of “forces” – a velocity-dependent term such as thatappearing in the right-hand side of (1.1a) (of course in addition to other terms). Let us alsoemphasize that the dynamical system (1.1) is the simplest model belonging to the Ruijsenaars–Schneider integrable class [3, 4].For instance a simple extension of the above model (reducing to it for ω = 0) is characterizedby the Newtonian equations of motion:¨ z n = (1 − α ) iω ˙ z n + α ( α − ω z n + N X m =1 ,m = n
2( ˙ z n + iαωz n )( ˙ z m + iαωz m ) z n − z m , n = 1 , . . . , N. (1.2a)The corresponding solution of the initial-value problem is again given by a simple rule: the N values of the dependent variables z n ≡ z n ( t ) at time t are related by the formula z n ( t ) = ζ n ( t ) exp( − iαωt ) , n = 1 , . . . , N, (1.2b)to the N solutions ζ n ( t ) of the algebraic equation (for the unknown ζ ) N X k =1 ˙ z k (0) + iαωz k (0) ζ − z k (0) = iω exp( iωt ) − N in ζ after multiplication bythe polynomial N Q j =1 [ ζ − z j (0)]); or, equivalently but more directly, the N values of the dependentvariables z n ≡ z n ( t ) at time t are the N solutions z n ( t ) of the algebraic equation (for theunknown z ) N X k =1 ˙ z k (0) + iαωz k (0) z − z k (0) exp( − iαωt ) = iω exp( iαωt )exp( iωt ) − . (1.2d) Notation 1.2.
Here and hereafter i is the imaginary unit, i = − ω is an arbitrary constant(dimensionally, an inverse time), and α is an arbitrary (dimensionless) constant. Clearly – unlessboth ω and αω are both imaginary, Re( ω ) = Re( αω ) = 0 – the time-evolution of this systemtakes place in the complex z -plane, i.e. the dependent variables z n ≡ z n ( t ) are complex ; butit may as well be viewed as describing the evolution of N point-like particles moving in the real xy -plane – whose positions at time t are characterized by the ( real ) Cartesian coordinates x n ≡ x n ( t ) , y n ≡ y n ( t ) – by setting z n ( t ) = x n ( t ) + iy n ( t ); and one of the remarkable featuresof the resulting real model is the possibility to write its Newtonian equations of motion in covariant – i.e., rotation-invariant – form, see Chapter 4 of [4]. Hereafter we generally refer tothis model, and its generalizations, see below, in their complex versions.iscrete-Time Goldfishing 3 Remark 1.1.
Clearly for ω = 0 this model, (1.2), reduces to the previous model (1.1). For ω = 0the time evolution of this model, (1.2), depends mainly on the values of the two constants ω and αω , as displayed by its solution, see (1.2d). If both these constants are real , Im( ω ) =Im( αω ) = 0 (hence as well Im( α ) = 0), the time evolution of this model is confined , indeed completely periodic if the real number α is rational , while if α is irrational it is multiply periodic ,being a nonlinear superposition of two periodic evolutions with the two noncongruent periods T = 2 π/ | ω | and T /α . Note that these outcomes obtain for generic initial data: hence, for α rational , α = q/p with q and p coprime integers (and p > isochronous , its generic solutions being completely periodic with period pT – or possibly with a period whichis a, generally small, integer multiple of pT : indeed, when the equation (1.2d) is itself periodicwith period pT , the unordered set of its N roots is clearly periodic with the same period pT ,but the periodicity of the time-evolution of each individual coordinate z n ( t ) may then be a,generally small, integer multiple of pT due to the possibility that different roots get exchangedthrough the time evolution (for a discussion of this phenomenology – including a justificationof the assertion that the relevant integer multiple of pT is generally small – see [5]).On the other hand, if ω is real but α is imaginary, say αω = iγ with γ real and nonvanishing ,then clearly in the remote future – i.e., as t → ∞ , and up to relative corrections of orderexp( −| γ | t ) – all the N coordinates z n ( t ) tend to the origin, z n ( ∞ ) = 0 , if γ <
0, while if γ > ω is not real , Im( ω ) = 0, then in the remote future (i.e., as t → ∞ , and up torelative corrections of order exp( −| Im( ω ) | t )) the N solutions of (1.2c) become asymptotically, ifIm( ω ) >
0, the N solutions ζ n = ζ n ( ∞ ) of the time-independent polynomial equation of order N in ζ N X k =1 ˙ z k (0) + iαωz k (0) ζ − z k (0) = − iω, while if instead Im( ω ) < time-independent polynomial equation N X k =1 ˙ z k (0) + iαωz k (0) ζ − z k (0) = 0 , hence N − N − N − ζ ) and one of them approaches asymptotically the diverging coordinate ζ asy ( t ) = exp( iωt ) N X k =1 [ ˙ z k (0) + iαωz k (0)] . Note that this implies (see (1.2b)) that, if Im( ω ) > αω is real , αω = ρ with ρ real and nonvanishing , then the model (1.2) is asymptotically isochronous , its generic solutions beco-ming, in the remote future, completely periodic with period 2 π/ | ρ | , up to corrections vanishingexponentially as t → ∞ (for a more detailed discussion of the notion of asymptotic isochrony see Chapter 6, entitled “Asymptotically isochronous systems”, of [6]).For ω = 0 (i.e., when the model (1.2) reduces to (1.1)) it is possible to restrict considerationto real dependent variables z n , but even then it is more interesting not to do so, so that thetime evolution takes place in the plane rather than on the real line: see the remarkable behaviorof this dynamical system in this case (“the game of musical chairs”), as detailed in Section 4.2.4of [4]. Hence let us reiterate that we always consider the dependent variables z n to be complex F. Calogeronumbers, both in the continuous-time case, z n ≡ z n ( t ), 0 ≤ t < ∞ , and (see below) in the discrete-time case, z n ≡ z n ( ℓ ), ℓ = 0 , , , . . . .Another large class of solvable dynamical systems “of goldfish type” is characterized by theequations of motion¨ z n = a ˙ z n + a + a z n − N − a z n + N X m =1 , m = n ( z n − z m ) − (cid:2) z n ˙ z m + ( a + a z n )( ˙ z n + ˙ z m ) + a z n ( ˙ z n z m + ˙ z m z n ) + 2 (cid:0) a + a z n + a z n + a z n (cid:1)(cid:3) ,n = 1 , . . . , N, (1.3)featuring 10 arbitrary constants (see [4, equation (2.3.3-2)]). In this case the solvability isachieved by identifying the N dependent variables z n ( t ) with the N roots of a time-dependentpolynomial ψ ( z, t ) of degree N in z satisfying a linear second-order PDE in the two independentvariables z and t .For an explanation of the origin of the name “goldfish” attributed to these models see Sec-tion 1.N of [6] and the literature cited there. In this book [6] (see in particular its Section 4.2.2,entitled “Goldfishing”, and the papers referred to there) several other solvable models “ofgoldfish type” are reported, including isochronous ones (i.e., models featuring solutions whichare completely periodic with a period independent of the initial data ). A few additional modelsof goldfish type have been identified more recently [7, 8, 9].The most remarkable aspect of these dynamical systems is their solvability , namely the possi-bility to solve their initial-value problems by algebraic operations, amounting generally to findingthe N eigenvalues of an N × N explicitly known time-dependent matrix (see below), or equiva-lently to finding the N roots of an explicitly known time-dependent polynomial of degree N (seefor instance (1.1b) and (1.2d)). Quite interesting is also the identification of multiply periodic , completely periodic , or even isochronous or asymptotically isochronous cases.In the present paper we present various discrete-time dynamical systems “of goldfish type”,so denoted because all these models reduce, in the limit when the discrete-time independentvariable ℓ = 0 , , , . . . becomes continuous , to continuous-time dynamical systems of goldfishtype. All these models are moreover solvable , i.e. the solution of their initial value problems canbe achieved by finding the N eigenvalues z n ( ℓ ) of N × N matrices explicitly known in termsof the initial data and of the discrete-time independent variable ℓ ; or equivalently by findingthe N roots z n ( ℓ ) of a polynomial, of degree N in the complex variable z , as well explicitlyknown in terms of the initial data and of the discrete-time independent variable ℓ . Some ofthese models feature interesting behaviors, even isochrony or asymptotic isochrony . Two ofthese models (see Subsection 2.1 and 2.2) were treated in the paper [10], which has not beenpublished because – after it was submitted for publication but before getting any feedback –new solvable models were identified and it was therefore considered preferable to report all thesemodels in a single paper, this one. The main properties of each of these discrete-time modelsare reported in Section 2, and proven in Section 3. These properties include the display of theequations of motion of these discrete-time models, the solution of their initial-value problems,a terse discussion (for the first three models) of their behavior including the possibility thatfor special values of some of their parameters they possesses periodic or multiperiodic solutionsor even display isochrony or asymptotic isochrony , and some mention of their continuous-time limits. Section 4 entitled “Outlook” concludes the paper: in it a general framework is outlinedwhich might allow the identification of additional solvable discrete-time models. And somemathematical developments are confined to two appendices.These findings are congruent with the recent surge of interest for discrete-time evolutions –in particular, such evolutions which are in some sense integrable or even solvable . Given thelarge body of research devoted to these topics over the last two decades, our reference to theiscrete-Time Goldfishing 5relevant literature shall be limited to citing the following surveys: [11, 12, 13, 14, 15]. Buta special mention must be made of the seminal papers by Nijhoff, Ragnisco, Kuznetsov andPang, see [16] as well as the earlier paper [17], where discrete-time versions were introduced ofthe well-known integrable “Ruijsenaars–Schneider” and “Calogero–Moser” dynamical systems;as well as of the paper by Suris [18], which treats specifically a discrete-time version of theoriginal goldfish model. Some results of these papers refer to models whose equations of motionfeature trigonometric/hyperbolic or even elliptic functions, and are therefore more general thanthose treated in the present paper, whose equations of motion only feature rational functions(see below); on the other hand the findings reported below include more general models thanthose previously treated, demonstrate the solvability of these models by an approach somewhatdifferent from those previously employed, and, most significantly, display the possible emer-gence of remarkable phenomenologies – including periodicity and even isochrony or asymptoticisochrony , see below – not previously identified for this kind of discrete-time dynamical systems.Let us also mention that the approach developed below also allows to identify and investi-gate discrete-time variants of another class of solvable continuous-time dynamical systems, theprototype of which is characterized by the Newtonian equations of motion¨ z n = N X k =1 , k = n c ( z n − z k ) , n = 1 , . . . , N (with c an arbitrary constant), instead of (1.1a). But in this paper we merely indicate, at theappropriate point, how to proceed in this direction, postponing a complete treatment of thisdevelopment to a separate paper.And let us finally pay tribute to Olshanetsky and Perelomov who were the first to show, morethan 35 years ago, that the time-evolution of a nontrivial many-body system could be usefullyidentified with the evolution of the eigenvalues of a matrix itself evolving in a much simpler,explicitly solvable, manner: see [19] and their other papers referred to in Section 2.1.3.2 of [4],entitled “The technique of solution of Olshanetsky and Perelomov”. The present paper extendstheir approach to the discrete-time context. In this section we report the main results of this paper; they are then proven in Section 3.
Notation 2.1.
Hereafter the dependent variables are indicated again as z n , but they are nowfunctions, z n ≡ z n ( ℓ ), of the discrete-time variable ℓ taking the integer values ℓ = 0 , , , . . . ; andsuperimposed tildes indicate generally a unit increase of the independent variable ℓ , for instance˜ z n ≡ z n ( ℓ + 1), e ˜ z n ≡ z n ( ℓ + 2). Hereafter δ nm is the standard Kronecker symbol, δ nm = 1 if n = m , δ nm = 0 if n = m , and underlined quantities are N -vectors, for instance z ≡ ( z , . . . , z N ).For the remaining notation we refer to Notation 1.1, see above, and to specific indications givencase-by-case below.As reported in this section and explained in Sections 3 and 4, the solvable models consideredin this paper generally feature three equivalent versions of the second-order “equations of mo-tion” characterizing their evolution in discrete-time . The treatment of the first model given inSubsections 2.1 and 3.1 is somewhat more detailed than that provided for the other models inthe subsequent subsections where, to avoid repetitions, we often refer to the treatments providedin Subsections 2.1 and 3.1. And already in this section, as well as in Section 3, we often takeadvantage – to simplify the presentation of some results – of identities and lemmata collectedin Appendix A. F. Calogero The first model is defined by the following second-order discrete-time equations of motion: the N values of the twice-updated variables e ˜ z n ≡ z n ( ℓ + 2) are given, in terms of the 2 N values ofthe variables z m ≡ z m ( ℓ ), ˜ z m ≡ z m ( ℓ + 1), by the N roots of the following ( single ) algebraicequation in the unknown z , N X k =1 (cid:18) ˜ z k − az k z − a ˜ z k (cid:19) N Y j =1 , j = k (cid:18) ˜ z k − az j ˜ z k − ˜ z j (cid:19) = 1 , (2.1a)which clearly amounts to a polynomial equation of degree N in this variable z (as it is imme-diately seen by multiplying this equation by the polynomial N Q m =1 ( z − a ˜ z m )). Here and below a isan arbitrary (dimensionless, nonvanishing) constant. A neater version of this formula is easilyobtained by multiplying it by a and by then using the identity (A.10) with η n = a ˜ z n , ζ n = a z n , n = 1 , . . . , N . It reads N Y j =1 (cid:18) z − a z j z − a ˜ z j (cid:19) = 1 + a. (2.1b)An equivalent formulation of this model is provided by the following system of N polynomialequations of degree N for the twice-updated coordinates e ˜ z n ≡ z n ( ℓ + 2): N X k =1 e ˜ z k − a ˜ z k ˜ z k − az n ! N Y j =1 , j = k e ˜ z j − a ˜ z k ˜ z j − ˜ z k ! = a N − , n = 1 , . . . , N. (2.2a)Again, a neater version of this formula is easily obtained by dividing it by a and by then usingthe identity (A.10), now with z = a z n , η n = a ˜ z n , ζ n = e ˜ z n , n = 1 , . . . , N . It reads N Y j =1 e ˜ z j − a z n ˜ z j − az n ! = (1 + a ) a N − , n = 1 , . . . , N. (2.2b)And a third, equivalent version of this model is provided by the following system of N polynomial equations of degree N for the twice-updated coordinates e ˜ z n ≡ z n ( ℓ + 2): N Y j =1 e ˜ z j − a ˜ z n az j − ˜ z n ! = − a N − , n = 1 , . . . , N. (2.3)The similarities and differences among these three sets of “equations of motion”, (2.1), (2.2)and (2.3), are remarkable: let us reemphasize that they in fact yield the same evolution indiscrete-time of the N coordinates z n ≡ z n ( ℓ ). Particularly remarkable is their similarity in thespecial a = 1 case, when the 3 versions (2.1b), (2.2b) and (2.3) of the equations of motion readas follows: N Y j =1 e ˜ z n − z j e ˜ z n − ˜ z j ! = 2 , n = 1 , . . . , N, N Y j =1 e ˜ z j − z n ˜ z j − z n ! = 2 , n = 1 , . . . , N, iscrete-Time Goldfishing 7 N Y j =1 , j = n e ˜ z j − ˜ z n z j − ˜ z n ! = − , n = 1 , . . . , N. The last of these three systems coincides with equation (1.8) of [18].
Remark 2.1.
This model, see (2.1), (2.2) and (2.3) – as the original goldfish model, see (1.2) –is invariant under an arbitrary rescaling of the dependent variables, z n ⇒ cz n with c an arbitraryconstant ; including the special case c = exp( iγ ) with γ an arbitrary real constant, correspondingto an overall rotation around the origin in the complex z -plane.The solution of the initial-value problem for this model is given by the following Proposition 2.1.
The N values z n ( ℓ ) of the dependent variables at the discrete time ℓ are the N eigenvalues of the N × N matrix U nm ( ℓ ) = δ nm z n (0) a ℓ + v m (0) a ℓ − a − , n, m = 1 , . . . , N (2.4a) with v m ≡ v m ( z, ˜ z ) = N Q j =1 (˜ z j − az m ) N Q j =1 , j = m [ a ( z j − z m )] , m = 1 , . . . , N, (2.4b) where of course v m (0) indicates the value of v m ( z, ˜ z ) corresponding to the initial data z = z (0) , ˜ z = ˜ z (0) ≡ z (1) .A neater, equivalent formulation of this finding – obtained from (2.4) via Lemma A.4 with ζ n = z n (0) a ℓ and η m = v m (0)( a ℓ − / ( a − – states that the N coordinates z n ( ℓ ) are the N solutions of the following algebraic equation in z : N X k =1 (cid:20) z k (1) − az k (0) z − a ℓ z k (0) (cid:21) N Y j =1 , j = k (cid:20) z j (1) − az k (0) az j (0) − az k (0) (cid:21) = a − a ℓ − . And another, even neater, equivalent formulation – obtained from this via the identity (A.10) with z replaced by za − ℓ , η k = az k (0) , ζ j = ˜ z j (0) = z j (1) – states that the coordinates z n ( ℓ ) arethe N solutions of the following algebraic equation in z : N Y k =1 (cid:20) z − a ℓ − z k (1) z − a ℓ z k (0) (cid:21) = a ℓ − − a ℓ − . (2.5) The last two equations become of course polynomial equations of degree N in z after multiplica-tion by the product N Q j =1 [ z − a ℓ z j (0)] . These formulas are also valid for a = 1 (by taking the obvious limit, i.e. replacing ( a p − / ( a −
1) with p ). If instead | a | <
1, then clearly for all ( positive ) values of ℓ the matrix U ( ℓ )is bounded and U nm ( ∞ ) = v m (0) / (1 − a ); hence for all values of ℓ the N coordinates z n ( ℓ ) arebounded and N − ℓ → ∞ while one of them tends to the value z asy = (1 − a ) − N X k =1 v k (0) = 11 − a N X k =1 [ z k (1) − az k (0)] , F. Calogerosee (2.4b) and the identity (A.11) (with η k = a z k (0), ζ j = ˜ z j (0) = z j (1)). If a = 1 but it has unit modulus, a = exp(2 πiλ ) (2.6)with λ real and not integer , then clearly the matrix U ( ℓ ) is again, for all values of ℓ , bounded;and if moreover λ is a (strictly, i.e. non integer) rational number, λ = KL , a = exp (cid:18) πiKL (cid:19) , (2.7)with K and L two coprime integers and L >
1, then clearly the matrix U ( ℓ ) is periodic withperiod L , U ( l + L ) = U ( l ) , hence the (unordered) set of its N eigenvalues z n ( ℓ ) is as well periodic with period L . Thisshows that in this case, see (2.7), the discrete-time goldfish model, see (2.1) or (2.2) or (2.3),is isochronous . On the other hand if λ is real and irrational , then clearly the time evolution ofthis discrete-time dynamical system is not periodic : indeed, while the right-hand side of (2.4a)(with (2.6) and λ real and irrational ) is periodic (with unit period) as a function of the real variable τ = λℓ, clearly it is not periodic as a function of the variable ℓ taking the integervalues ℓ = 0 , , , . . . . (We made this analysis, for convenience, referring to the coordinates z n ( ℓ )as the eigenvalues of U ( ℓ ), see (2.4); of course an analogous discussion could be made on thebasis of the alternative identification of the coordinates z n ( ℓ ) as the N roots of the polynomialequation (2.5) – whose similarity with (1.2d) is in any case to be noted, see below.)To explore the transition from the discrete-time independent variable ℓ to the continuous-time variable t one makes the formal replacements ℓ = ⇒ tε , ℓ + 1 = ⇒ t + εε , ℓ + 2 = ⇒ t + 2 εε , (2.8a) a = ⇒ − iωε, (2.8b)and (with a slight abuse of notation) z n ( ℓ ) = ⇒ z n ( t ) , ˜ z n ( ℓ ) ≡ z n ( ℓ + 1) = ⇒ z n ( t ) + ε ˙ z n ( t ) + ε z n ( t ) + O (cid:0) ε (cid:1) , e ˜ z n ( ℓ ) ≡ z n ( ℓ + 2) = ⇒ z n ( t ) + 2 ε ˙ z n ( t ) + 2 ε ¨ z n ( t ) + O (cid:0) ε (cid:1) , (2.8c)with ε infinitesimal. It is then a matter of standard, if a bit cumbersome, algebra, to verifythat the insertion of this ansatz , see (2.8b) and (2.8c), in (2.1b) or (2.2b) or (2.3) yields a trivialidentity to order ε = 1, while to order ε it reproduces (1.2a) with α = 1 , reading¨ z n = − iω ˙ z n + N X m =1 , m = n
2( ˙ z n + iωz n )( ˙ z m + iωz m ) z n − z m , n = 1 , . . . , N. (2.9a)Likewise, the discrete-time solution formula (2.5) becomes, in the continuous-time limit, N X k =1 ˙ z k (0) + iωz k (0) z − z k (0) exp( − iωt ) = iω − exp( − iωt ) ≡ iω exp( iωt )exp( iωt ) − , (2.9b)which coincides with (1.2d) with α = 1. A terse outline of the derivation of these resultsis provided at the end of Subsection 3.1. To higher order in ε one would obtain additionalrelations satisfied by the solution z n ( t ) of this continuous-time goldfish model, which mightalternatively be obtained by differentiating its equations of motion (2.9a).iscrete-Time Goldfishing 9 Remark 2.2.
Clearly, for ω real and nonvanishing, this continuous-time model, (2.9a), is isochronous : see (2.9b) and/or Remark 1.1. This is consistent with the fact that the limitingreplacement (2.8b) can be considered to obtain from (2.7) – entailing isochrony of the discrete-time model – by identifying εω with 2 πK/L in the context of the replacement (see (2.8)) ofthe unit interval in the discrete-time model with the infinitesimal time interval ε to make thetransition to the continuous-time case. Remark 2.3.
At every step of the discrete-time evolution the N values of the twice-updatedvariables e ˜ z n ≡ z n ( ℓ + 2) are given, in terms of the N unupdated variables z m ≡ z m ( ℓ ) and the N once-updated variables ˜ z m ≡ z m ( ℓ + 1), as the N roots of a polynomial, of degree N in its argu-ment z , whose coefficients are explicitly defined in terms of the 2 N unupdated and once-updatedvariables: see (2.1) or (2.2) or (2.3) (hereafter – within this important Remark 2.3 – we generallyidentify, for simplicity, this model only via the version (2.1) of its equations of motion). Henceat every step of this discrete-time evolution the unordered set of N twice-updated variables e ˜ z n is uniquely determined, but not the value of each of them. This implies a qualitative differenceamong the continuous-time respectively the discrete-time evolutions described by the equationsof motions (2.9a) (or, more generally, (1.2a) and (1.3)) respectively by (2.1): in contrast tothe continuous-time case, the discrete-time evolution (2.1) is only deterministic in terms of the unordered set of N coordinates z m ( ℓ ), but not for each individual coordinate z n ( ℓ ). Indeed the continuous-time Newtonian equations of motion, see for instance (1.2a), determine uniquely thevalue of the acceleration ¨ z n ( t ) of the n -th moving point in terms of the N positions z m ( t ) andthe N speeds ˙ z m ( t ) of all moving points; and correspondingly, while the solution formula (1.2d)determines only the unordered set of N values z n ( t ) as the N roots of a polynomial of degree N ,the value of each individual coordinate z n ( t ) gets then uniquely determined by continuity inthe time variable t . This latter mechanism to identify uniquely the value of the coordinate of each moving point is instead missing for the discrete-time evolution (2.1). On the other handit is clear that there are appropriate ranges of values of the parameter a and of the 2 N initialdata z m (0), ˜ z m (0) ≡ z m (1) – with a sufficiently close to unity , the N initial coordinates z m (0) allsufficiently well separated among themselves, and each ˜ z m (0) ≡ z m (1) sufficiently close to thecorresponding z m (0), see (2.8) – which cause the evolution yielded by the discrete-time goldfishmodel (2.1) to mimic closely that yielded by the continuous-time goldfish model (2.9), providedat every step of the discrete-time evolution the appropriate identification is made of the valueof each twice-updated coordinate e ˜ z n ≡ z n ( ℓ + 2) (among the unordered set of N values yieldedby the discrete-time equations of motion) by an argument of contiguity with ˜ z n ≡ z n ( ℓ + 1) and z n ≡ z n ( ℓ ); and likewise an appropriate identification is made by contiguity of each coordinate z n ( ℓ + 1) with the corresponding coordinate z n ( ℓ ) (among the unordered set of N values yieldedby Proposition 2.1) – these arguments of contiguity taking the place of the continuity of z n ( t ) asfunction of t applicable in the continuous-time case. But the contiguity argument breaks down ifthe positions at time ℓ of two different points, z n ( ℓ ) and z m ( ℓ ) with n = m , get too close to eachother, corresponding to a quasi-collision, or even coincide, corresponding to an actual collision;which is however not featured by the generic solution of the discrete-time model (2.1) – nor ofthe standard goldfish models (1.2a) or (1.3) – clearly emerging only for a set of initial conditions z n (0), ˜ z (0) ≡ z n (1) having unit codimension in the 2 N -dimensional (complex) phase space z , ˜ z .This important remark is applicable to all the discrete-time models considered below, al-though it will not be repeated. Remark 2.4.
Several of the formulas written above (in this section) simplify somewhat via thefollowing replacement of the dependent variables: z n ( ℓ ) ⇒ a ℓ z n ( ℓ ) , n = 1 , . . . , N. N Y j =1 e ˜ z n − z j e ˜ z n − ˜ z j ! = 1 + a, n = 1 , . . . , N, N Y j =1 e ˜ z j − z n ˜ z j − z n ! = 1 + aa , n = 1 , . . . , N, N Y j =1 e ˜ z j − ˜ z n z j − ˜ z n ! = − a , n = 1 , . . . , N, and correspondingly the formula (2.5) providing the solution of the initial-value problem reads N Y k =1 (cid:20) z n ( ℓ ) − a − z k (1) z n ( ℓ ) − z k (0) (cid:21) = a ℓ − − a ℓ − , n = 1 , . . . , N. Somewhat analogous remarks are applicable to all the discrete-time models considered below;their explicit implementation is left to the interested reader.
In this subsection we treat rather tersely a discrete-time dynamical system that generalizesthe discrete-time goldfish model described in the preceding Subsection 2.1. This generalizationamounts to the presence of an additional free parameter, b : indeed, for b = 0 one reobtains themodel treated in the preceding Subsection 2.1 (hence in this subsection we assume that b doesnot vanish, b = 0).The three equivalent versions of the equations of motion of this model read as follows. Thefirst version identifies the twice updated coordinates z n ( ℓ + 2) as the N solution of the followingequation in z (amounting to the identification of the N roots of a polynomial of degree N inthis variable): N X k =1 (cid:18) ˜ z k − az k z − a ˜ z k (cid:19) (cid:18) b ˜ z k bz k (cid:19) N Y j =1 , j = k (cid:20)(cid:18) ˜ z k − az j ˜ z k − ˜ z j (cid:19) (cid:18) b ˜ z j /a bz j (cid:19)(cid:21) = 1 . (2.10a)The second and third versions consist of the following two systems: N X k =1 e ˜ z k − a ˜ z k ˜ z k − az n ! (cid:18) bz n b ˜ z k (cid:19) N Y j =1 , j = k e ˜ z j − a ˜ z k ˜ z j − ˜ z k ! = a N − , n = 1 , . . . , N ; (2.10b) N Y j =1 e ˜ z j − a ˜ z n az j − ˜ z n ! N Y j =1 , j = n (cid:18) bz j b ˜ z j /a (cid:19) = a N − (1 + b ˜ z n )(1 + bz n ) , n = 1 , . . . , N. (2.10c) Remark 2.5.
This model – as the original goldfish model (1.2a), and as the model treatedabove, see Remark 2.1 – is invariant under a rescaling of the dependent variables, z n ⇒ cz n with c an arbitrary constant ; but only provided the parameter b is also rescaled, b ⇒ b/c .The solution of this model is provided by an analog of (the first part of) Proposition 2.1,reading as follows:iscrete-Time Goldfishing 11 Proposition 2.2.
The N values z n ( ℓ ) of the dependent variables at the discrete time ℓ are the N eigenvalues of the N × N matrix U ( ℓ ) = U (0)[ aI + bV (0)] ℓ + V (0)[( a − I + bV (0)] − (cid:8) [ aI + bV (0)] ℓ − I (cid:9) , (2.11a) where again ( see (2.4)) U (0) = diag[ z n (0)] , U nm (0) = δ nm z n (0) , (2.11b) while the N × N matrix V (0) is now defined componentwise as follows: [ V (0)] nm = v m (0)1 + bz m (0) , n, m = 1 , . . . , N, (2.11c) with the quantities v m (0) defined again as in Subsection see (2.4b) and the sentence followingthis formula ) . Let us recall that I is the N × N unit matrix ( whose presence in (2.11a) , however,might well be considered pleonastic ) . As evidenced by a comparison of (2.11a) with (2.4a), the behavior of the solutions of thismodel (with b = 0) are less simple than those of the model discussed in the preceding Subsec-tion 2.1. In particular a confined behavior emerges only, see (2.11a), from initial data z n (0),˜ z n (0) ≡ z n (1) implying, via (2.11c) and (2.4b), that all the N eigenvalues of the N × N matrix aI + bV (0) have modulus not larger than unity , reading exp( − q n + 2 πir n ) with the numbers q n and r n real and the N numbers q n nonnegative , q n ≥
0. If moreover the N numbers q n allvanish and the N numbers r n are all rational , the behavior is periodic (but not isochronous ,since these numbers, q n and r n , generally depend on the initial data; see (2.11c) and (2.4b)).While, if some of (but not all) the N numbers q n are positive , and none is negative , then thephenomenology we just described (corresponding to the q n ’s all vanishing) emerges only asymp-totically , as ℓ → ∞ , up to corrections of order exp( − qℓ ) with q the smallest of the nonvanishingnumbers q n – provided all those r n ’s are rational whose corresponding q n vanish .Let us finally mention that, also for this second model, a transition from the discrete-time independent variable ℓ to the continuous-time variable t can be performed (as tersely outlinedat the end of Subsection 3.2); but the continuous-time goldfish-type model obtained in thismanner turned out to be, to the best of our knowledge, new , hence it seemed appropriate todevote a separate paper to it, see [8]. The third model is another one-parameter extension of the model treated in Subsection 2.1(different from that treated in the preceding Subsection 2.2). Again its discrete-time equationsof motion can be presented in three equivalent versions.The first is characterized by this prescription: the twice-updated N coordinates e ˜ z n ≡ z n ( ℓ +2)are the N roots of the following equation in the variable z , N X k =1 (cid:18) ˜ z k − a + z k z − a + ˜ z k (cid:19) N Y j =1 , j = k (cid:18) ˜ z k − a + z j ˜ z k − ˜ z j (cid:19) = 1 a − , (2.12a)amounting again to the determination of the N roots of a polynomial of degree N in the vari-able z . Here and below a + and a − are 2 arbitrary constants. Remark 2.6.
As entailed by a comparison of these discrete-time second-order equations ofmotion with those of the first model, see (2.1a), this third model coincides, for a − = 1, with thefirst model with a = a + .2 F. CalogeroA neater formulation of these equations of motion reads (after multiplication by a + , via (A.10)with ζ k = a z k , η k = a + ˜ z k ) as follows: N Y j =1 (cid:18) z − a z j z − a + ˜ z j (cid:19) = a − + a + a − . (2.12b)An equivalent , second formulation of this model is provided by the following system of N polynomial equations for the twice-updated coordinates e ˜ z n ≡ z n ( ℓ + 2): N X k =1 e ˜ z k − a + ˜ z k ˜ z k − a + z n ! N Y j =1 , j = k a + ˜ z k − e ˜ z j ˜ z k − ˜ z j ! = a − a N − , n = 1 , . . . , N ; (2.13a)and a neater version of these equations of motion reads (again via (A.10), but now with ζ k = e ˜ z k , η k = a + ˜ z k and z replaced by a z n ) N Y j =1 e ˜ z j − a z n ˜ z j − a + z n ! = ( a + + a − ) a N − , n = 1 , . . . , N. (2.13b)And a third, also equivalent , version of these equations of motion reads as follows: N Y j =1 e ˜ z j − a + ˜ z n a + z j − ˜ z n ! = − a − a N − , n = 1 , . . . , N. (2.14) Remark 2.7.
Remark 2.1 also holds for this model.The solution of the initial-value problem for this discrete-time dynamical system is providedby the following
Proposition 2.3.
The N coordinates z n ( ℓ ) are the N eigenvalues of the N × N matrix U ( ℓ ) = ( a + ) ℓ C + + ( a − ) ℓ C − , (2.15a) where the two constant ( i.e., ℓ -independent ) N × N matrices C + and C − are defined in termsof the N initial data z n (0) and ˜ z (0) ≡ z n (1) by the formula C ± = ± ( a + − a − ) − [ U (1) − a ∓ U (0)] (2.15b) with the two matrices U (0) and U (1) defined componentwise as follows: [ U (0)] nm = δ nm z n (0) , (2.15c)[ U (1)] nm = a − N + N X k =1 z k (1) − a + N X k =1 , k = n z k (0) × N Y j =1 , j = m (cid:20) a + z m (0) − z j (1) z m (0) − z j (0) (cid:21) , n, m = 1 , . . . , N. (2.15d) Note that we are, for simplicity, assuming that the two coupling constants a ± are different, a + = a − ( see (2.15b)) . iscrete-Time Goldfishing 13It is plain from these formulas that, if the two “coupling constants” a ± (are different and)are conveniently written as follows, a ± = exp( − q ± + 2 πir ± ) (2.16)with q ± and r ± real , then, if the two numbers q ± are both nonnegative , q ± ≥
0, the timeevolution of the N × N matrix U ( ℓ ) is bounded for all values of the discrete-time independentvariable ℓ = 0 , , , . . . , hence its N eigenvalues z n ( ℓ ) are all as well bounded (the motion isconfined); if in particular the two numbers q ± both vanish, q ± = 0, and the two numbers r ± are both rational numbers, r ± = K ± /L ± with K + , L + and K − , L − coprime integers (and,for definiteness, L ± > discrete-time evolution of the matrix U ( ℓ ) is periodic (witha period L independent of the initial data, being the minimum common multiple of L + and L − , L = mcm[ L + , L − ]) hence the discrete-time dynamical system (2.12) is isochronous ; while if, ofthe two numbers q ± , one vanishes and the other is positive, q + = 0, q − = q > q − = 0, q + = q >
0, and r + respectively r − are rational numbers, then the isochronous behavior(with period L + respectively L − ) only emerges asymptotically , as ℓ → ∞ , up to corrections oforder exp( − qℓ ). While clearly if q + and q − are both positive entailing (see (2.16)) | a ± | < U ( ℓ ) , hence as well all it eigenvalues z n ( t ), vanish asymptotically(as t → ∞ ): z n ( ∞ ) = 0, n = 1 , . . . , N .Finally let us mention the transition from this discrete-time model to its continuous-time counterpart. The treatment is completely analogous to that detailed at the end of Subsection 2.1;except that now (2.8b) must be replaced by a ± = ⇒ − iω ± ε with ω + = αω, ω − = ( α − ω. It is then easily seen that again, at order ε = 1, one gets from (2.12b) or (2.13b) or (2.14) a trivialidentity, while at order ε one gets the continuous-time goldfish equations of motion (1.2a); andthe solution of this model, see Proposition 2.3, reproduces in this continuous-time limit theprescription (1.2d). The fourth model is also characterized by three equivalent versions of its discrete-time equationsof motion. The first consists of the following prescription: the twice-updated N coordinates e ˜ z n ≡ z n ( ℓ + 2) are the N roots of the following equation in the variable z , N X k =1 (cid:20) ˆ g k ( z, ˜ z ) z − a ˜ z k − b (cid:21) = 1 γ , (2.17a)where the N quantities ˆ g k ( z, ˜ z ) are defined as follows:ˆ g n ( z, ˜ z ) = a − N ( η ˜ z n + β ) N Y j =1 (cid:18) ˜ z n − az j − bηz j + β (cid:19) × N Y j =1 , j = n (cid:18) η ˜ z j + aβ − bη ˜ z n − ˜ z j (cid:19) , n = 1 , . . . , N. (2.17b)4 F. CalogeroThroughout this subsection, the following Subsection 3.4, and Appendix B, a = α + ηρ − γ , b = βρ − γ , (2.17c)entailing αβ = aβ − bη, ηρ = ( a − α )(1 − γ ) , βρ = b (1 − γ ) , (2.17d)where α , β , γ , η and ρ are 5 arbitrary constants (but the 3 constants β , η , ρ only enter as βρ and ηρ , hence any one of these three constants could be replaced by unity without significantloss of generality); in the following we use interchangeably these constants in order to simplifysome formulas.An equivalent formulation of these discrete-time equations of motion reads as follows: N X k =1 " ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) ( η ˜ z k + β )(˜ z k − az n − b ) = 1( ηz n + β ) , n = 1 , . . . , N, (2.18a)with ˇ g n (cid:0) ˜ z, e ˜ z (cid:1) = a − N γ (cid:0)e ˜ z n − a ˜ z n − b (cid:1) N Y j =1 , j = k e ˜ z j − a ˜ z n − b ˜ z j − ˜ z n ! , n = 1 , . . . , N. (2.18b)It is a matter of trivial algebra to rewrite these equations of motion, (2.18), as follows: N X k =1 η ˜ z k + β )(˜ z k − az n − b ) N Q j =1 (cid:0)e ˜ z j − a ˜ z k − b (cid:1) N Q j =1 , j = k (cid:0) ˜ z j − ˜ z k (cid:1) = γa N − ηz n + β , n = 1 , . . . , N. (2.19a)And, as shown at the end of Appendix B, a neater version of this system of equations of motionthen reads as follows: N Y j =1 "e ˜ z j − a z n − b (1 + a )˜ z j − az n − b − N Y j =1 η e ˜ z j + αβη ˜ z k + β ! = γa N − ηaz n + β + ηbηz n + β ,n = 1 , . . . , N. (2.19b)And a third, equivalent formulation of these equations of motion reads as follows:ˇ g n (cid:0) ˜ z, e ˜ z (cid:1) = ˆ g n ( z, ˜ z ) , n = 1 , . . . , N, (2.20)with ˇ g n (cid:0) ˜ z, e ˜ z (cid:1) respectively ˆ g n ( z, ˜ z ) defined by (2.18b) respectively (2.17b). Remark 2.8.
Above and below we assume for simplicity that the parameters characterizingthis model have generic values, for instance γ = 0 and γ = 1 (see (2.17a) and (2.17c)) and a = 0(see (2.18b)). Remark 2.9.
This model – as the original goldfish model (1.2a), and as the models treatedabove, see Remarks 2.1, 2.5 and 2.6 – is invariant under a rescaling of the dependent variables, z n ⇒ cz n with c an arbitrary constant ; but only provided the parameter β – hence as well theparameter b , see (2.17c) – is also rescaled, β ⇒ cβ , b ⇒ cb .The solution of the initial-value problem for this discrete-time dynamical system is providedby the followingiscrete-Time Goldfishing 15 Proposition 2.4.
The N coordinates z n ( ℓ ) are the N eigenvalues of the N × N matrix U ( ℓ ) = U (0) P (0 , ℓ −
1) + ℓ X k =1 (cid:2) ( Bγ k − + b ) P ( k, ℓ − (cid:3) , (2.21a) where the N × N matrix P ( ℓ , ℓ ) is defined as follows P ( ℓ , ℓ ) = ℓ Y j = ℓ ( Aγ j + a ) , (2.21b) and the two ℓ -independent N × N matrices A and B are defined as follows A = ηV (0) − ηβ b, B = βV (0) − b. (2.21c) Here and throughout we use the convention that ( for arbitrary finite X j ) ℓ Q j = ℓ X j = I if ℓ > ℓ and k P k = k X j = 0 if k > k . As for the two N × N matrices U (0) and V (0) , they are defined interms of the N initial data z n (0) and ˜ z n (0) ≡ z n (1) as follows: U (0) = Z (0) = diag[ z n (0)] ,V (0) = [ ηZ (0) + β ] − (cid:8) M (0) Z (1)[ M (0)] − − αZ (0) (cid:9) , with the N × N matrices Z ( ℓ ) and M (0) defined, componentwise, as follows: Z ( ℓ ) = diag[ z n ( ℓ )]; Z nm ( ℓ ) = δ nm z n ( ℓ ) , n, m = 1 , . . . , N,M nm (0) = ˆ g m (0) z m (1) − az n (0) − b , n, m = 1 , . . . , N, where the notation ˆ g m (0) is an abbreviation for ˆ g m ( z, ˜ z ) , see (2.17b) , evaluated at z = z (0) , ˜ z = ˜ z (0) ≡ z (1) . Note that Lemma A.5 ( with f n = 1 , g m = ˆ g m (0) , ξ m = z m (1) , η n = az n (0) + b ) entails the following componentwise definition of the inverse matrix [ M (0)] − : (cid:8) [ M (0)] − (cid:9) nm = a − N (cid:20) z n (1) − az m (0) − b ˆ g n (0) (cid:21) N Y j =1 , j = n (cid:20) z j (1) − az m (0) − bz j (1) − z n (1) (cid:21) × N Y j =1 , j = m (cid:20) z n (1) − az j (0) − bz m (0) − z j (0) (cid:21) , n, m = 1 , . . . , N, hence an explicit expression of the N × N matrix V (0) reads, componentwise, as follows: V nm (0) = − αz n (0) ηz n (0) + β δ nm + a − N ηz n (0) + β N X k =1 z k (1) N Y j =1 , j = k (cid:20) z j (1) − az m (0) − bz j (1) − z k (1) (cid:21) × N Q j =1 , j = n [ z k (1) − az j (0) − b ] Q j =1 , j = m [ z m (0) − z j (0)] . Remark 2.10.
It is relevant to this expression, (2.21), of the N × N matrix U ( ℓ ) – whose N eigenvalues provide the N coordinates z n ( ℓ ) – that (2.21b) and (2.21c) entail P ( ℓ , ℓ ) = Q diag[ p n ( ℓ , ℓ )] Q − , A = Q diag[ a n ] Q − , B = Q diag[ b n ] Q − ,p n ( ℓ , ℓ ) = ℓ Y j = ℓ ( a n γ j + a ) , a n = ηv n − ηβ b, b n = βv n − b, n = 1 , . . . , N, with v n the N ( ℓ -independent) eigenvalues of the N × N matrix V (0) and Q the corresponding( ℓ -independent) diagonalizing matrix, V (0) = Q diag( v n ) Q − . And let us mention that, also for this fourth model, a transition from the discrete-time independent variable ℓ to the continuous-time variable t can be performed (see the end ofSubsection 3.4). And, as in the case of the second model, also in this case the continuous-time goldfish model thereby obtained turned out to be, to the best of our knowledge, new . Hence itseemed appropriate to devote to this model a separate paper [9]. In this section we prove the findings reported in the preceding Section 2.The basic strategy to obtain all these results goes as follows. The starting point is a solvable system of two matrix first-order discrete-time
ODEs, say˜ U = F ( U, V ) , ˜ V = F ( U, V ) , (3.1)where ℓ = 0 , , , . . . is the discrete-time independent variable, the two dependent variables U ≡ U ( ℓ ), V ≡ V ( ℓ ) are N × N matrices and of course superimposed tildes denote the updatingof the discrete-time , ˜ U ≡ U ( ℓ + 1), ˜ V ≡ V ( ℓ + 1). The solvable character of this matrix systementails the possibility to obtain explicitly the solution of its initial-value problem. Four caseswhen this is possible – corresponding to 4 simple assignments of the functions F ( U, V ) and F ( U, V ) – are treated in the following 4 subsections. Note that the two functions F ( U, V ), F ( U, V ) are assumed to depend on no other matrix besides U and V (and the unit matrix I );they may of course feature some scalar constants, and the order in which the two, generallynoncommuting, matrices U and V appear in their definition is of course relevant: see below.One assumes moreover that the N × N matrix U ≡ U ( ℓ ) is diagonalizable and denotes as R ≡ R ( ℓ ) the diagonalizing N × N matrix: U ≡ RZR − , U ( ℓ ) ≡ R ( ℓ ) Z ( ℓ )[ R ( ℓ )] − , (3.2a) Z = diag[ z n ] , Z ( ℓ ) = diag[ z n ( ℓ )] , (3.2b)where the notation z n ( ℓ ) for the N eigenvalues of the N × N matrix U ≡ U ( ℓ ) shall be justified bythe identification, see below, of these quantities with the dependent variables of the discrete-time dynamical systems introduced above. Remark 3.1.
These formulas entail that the matrix R ( ℓ ) is defined up to right-multiplicationby an arbitrary diagonal matrix D ( ℓ ), R ( ℓ ) ⇒ R ( ℓ ) D ( ℓ ).Next we introduce the two matrices M ( ℓ ) and Y ( ℓ ) defined as follows: M = R − ˜ R, M ( ℓ ) = [ R ( ℓ )] − R ( ℓ + 1) , (3.3a)iscrete-Time Goldfishing 17 V = RY ˜ R − , V ( ℓ ) = R ( ℓ ) Y ( ℓ )[ R ( ℓ + 1)] − , (3.3b)so that V = RY M − R − , V ( ℓ ) = R ( ℓ ) Y ( ℓ )[ M ( ℓ )] − [ R ( ℓ )] − . (3.3c) Remark 3.2.
The element of freedom in the definition of the matrix R ( ℓ ), see Remark 3.1,entails that the matrix M ( ℓ ) is defined up to the “gauge transformation” resulting by insertingin its definition (3.3a) the N × N matrix [ R ( ℓ ) D ( ℓ )] − = [ D ( ℓ )] − [ R ( ℓ )] − in place of the matrix[ R ( ℓ )] − (and of course R ( ℓ + 1) D ( ℓ + 1) in place of R ( ℓ + 1)): hence, as a consequence of the arbitrary nature of the diagonal matrix D ( ℓ ) , out of the N elements of the N × N matrix M ≡ M ( ℓ ) only N − N are significant. Likewise for the matrix Y ≡ Y ( ℓ ).One then, by inserting (3.2a) and (3.3c) in (3.1), obtains the following system of two first-order discrete-time N × N matrix evolution equations: M ˜ Z = F (cid:0) Z, Y M − (cid:1) M, M ˜ Y = F (cid:0) Z, Y M − (cid:1) M ˜ M ; (3.4)and from these two matrix equations, by making a convenient ansatz for the two matrices M ≡ M ( ℓ ) and Y ≡ Y ( ℓ ) in terms of the 2 N quantities z n ≡ z n ( ℓ ) and ˜ z n ≡ z n ( ℓ + 1) – an ansatz which must of course be consistent with these two matrix evolution equations – one obtainsa system of N second-order discrete-time evolution equations for the N coordinates z n ≡ z n ( ℓ ).This last step is of course only possible for special assignments, in the discrete-time matrixevolution equations (3.1), of the two matrix functions F ( U, V ) and F ( U, V ), see below.The discrete-time dynamical system thereby obtained is then solvable , since the quantities z n ≡ z n ( ℓ ) are the N eigenvalues of the N × N matrix U ≡ U ( ℓ ) which, as solution of the,assumedly solvable , matrix evolution system (3.1), can be explicitly evaluated. How this worksout is shown in detail in the following subsections: in more detail in Subsection 3.1, where thesimplest case is treated.Let us also mention, once and for all, that in the following we will conveniently assume thatthe matrix U is initially diagonal: U (0) = Z (0) ≡ diag[ z n (0)] , (3.5a)implying (up to the ambiguity mentioned above, see Remark 3.1) R (0) = I. (3.5b)Here and throughout I is the N × N unit matrix, i.e., componentwise, I nm = δ nm . The point of departure to obtain the findings reported in Subsection 2.1 is the following discrete-time first-order, linear, matrix system (see (3.1)):˜ U = aU + V, ˜ V = V, (3.6a)where a is an arbitrary scalar constant. Note that the second of these two ODEs entails that inthis case V is a constant (i.e., ℓ -independent) N × N matrix, V ( ℓ ) = V (0). It is plain that thesolution of the corresponding initial-value problem for the N × N matrix U reads U ( ℓ ) = U (0) a ℓ + V (0) a ℓ − a − . (3.6b)8 F. CalogeroLet us now proceed as indicated in the first part of Section 3. It is then easily seen(via (3.2) and (3.3)) that the first of the two discrete-time matrix evolution equations (3.6a)yields (see (3.4)) the matrix equation M ˜ Z − aZM = Y, (3.7a)namely, componentwise, M nm = Y nm ˜ z m − az n , n, m = 1 , . . . , N. (3.7b)Likewise, the second of the two discrete-time matrix evolution equations (3.6a) yields thematrix relation Y ˜ M = M ˜ Y .
Via (3.7b) this matrix equation implies the following N relations: N X k =1 n Y nk ˜ Y km h(cid:0)e ˜ z m − a ˜ z k (cid:1) − − (˜ z k − az n ) − io = 0 , n, m = 1 , . . . , N. (3.8)This derivation shows that this system of N discrete-time equations of motion is equivalentto the solvable equation of motion (3.6a) for the N × N matrix U ; hence it is just as solvable . Notethat the dependent variables are now the N coordinates z n and the N matrix elements Y nm (of which only N ( N −
1) are significant, see Remark 3.2; so the number of equations and thenumber of dependent variables tally). To obtain a model that qualifies as discrete-time analogof the continuous-time goldfish model (2.9a) we need to distill from this system a set of only N equations of motion involving only the N coordinates z n . The standard trick to do so (see, forinstance, Section 4.2.2 entitled “Goldfishing” of [6]) is to identify – if possible – an ansatz whichexpresses the N components of the matrix Y in terms of the 2 N quantities z n , ˜ z n , yielding N equations of motion involving only the N coordinates z n , ˜ z n and e ˜ z n – to be interpretedas equations of motion of the discrete-time goldfish – and implying that the N equations ofmotion (3.8) are all satisfied, thanks to these very equations of motion.An educated guess for such an ansatz reads as follows: Y nm = g m , n, m = 1 , . . . , N. (3.9)Note that we reserve at this stage the option to assign the N quantities g m .Via this ansatz the equations (3.8) become N X k =1 (cid:18) g k e ˜ z m − a ˜ z k − g k ˜ z k − az n (cid:19) = 0 , n, m = 1 , . . . , N, (3.10a)hence they amount to the following 2 systems, each involving only N equations: N X k =1 (cid:18) g k ˜ z k − az n (cid:19) = 1 , n = 1 , . . . , N, (3.10b) N X k =1 (cid:18) g k e ˜ z n − a ˜ z k (cid:19) = 1 , n = 1 , . . . , N. (3.10c)The unit in the right-hand sides could of course be replaced by an arbitrary constant c – ofcourse the same constant in (3.10b) and (3.10c) – but this would merely entail an irrelevantrescaling of g k by c ; see below.iscrete-Time Goldfishing 19These are now two sets of N equations, each featuring linearly the N quantities g k , thatwe like to eliminate in order to obtain a set of N “equations of motion” determining the twiceupdated coordinates e ˜ z n ≡ z n ( ℓ +2) in terms of the 2 N coordinates z n ≡ z n ( ℓ ) and ˜ z n ≡ z n ( ℓ +1).There are three alternative strategies to achieve this goal. One can solve the first linear systemthereby obtaining g k as a function of z and ˜ z, and then insert this expression g k ≡ ˆ g k ( z, ˜ z ) inthe second system; alternatively, one can solve the second linear system, thereby obtaining g k as a function of ˜ z and e ˜ z , and then insert this expression g k ≡ ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) in the first system; orone can equate the two expressions of g k obtained solving the first, respectively the second,system, i.e. write ˆ g k ( z, ˜ z ) = ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) . Clearly the three sets of equations of motion obtainedin this manner are equivalent , i.e. they characterize the same discrete-time evolution of the N coordinates z n ≡ z n ( ℓ ); but they may seem quite different (indeed, see (2.1), (2.2) and (2.3)).Note that we introduced a superimposed decoration on the functions ˆ g k ( z, ˜ z ) respectively ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) to emphasize that the functional dependence on their arguments is generally different, as impliedby their definitions as solutions of (3.10b) respectively of (3.10c).Let us first of all see what the first approach yields. From (3.10b) one obtains (via Lemma A.1reported in Appendix A, with ξ k = ˜ z k , η n = az n , c = 1) the following expression of g k ≡ ˆ g k ( z, ˜ z ):ˆ g k ( z, ˜ z ) = (˜ z k − az k ) N Y j =1 , j = k (cid:18) ˜ z k − az j ˜ z k − ˜ z j (cid:19) , k = 1 , . . . , N. (3.11)The insertion of this expression of g k in (3.10c) yields the equations of motions (2.1a).Likewise, the second approach yields, from (3.10c) (again via Lemma A.1, but now with ξ k = a ˜ z k , η n = e ˜ z n , c = −
1) the following expression of g k ≡ ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) :ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) = a − N (cid:0)e ˜ z k − a ˜ z k (cid:1) N Y j =1 , j = k e ˜ z j − a ˜ z k ˜ z j − ˜ z k ! , k = 1 , . . . , N. (3.12)The second version, (2.2a), of the discrete-time equations of motion follows by inserting thisexpression of g k in (3.10b).And the third approach yields, by equating (3.11) to (3.12), the third version, (2.3), of the discrete-time equations of motion.We have seen that the solutions z n ( ℓ ) of these discrete-time equations of motion are providedby the eigenvalues of the N × N matrix U ( ℓ ), see (3.6b). To prove Proposition 2.1 we mustnow obtain from (3.6b) (also taking advantage of the ansatz (3.9)) the expression (2.4) of thismatrix in terms of the initial data z n (0), ˜ z n (0) ≡ z n (1) of the discrete-time dynamical system.This requires that we express the two matrices U (0) and V (0) appearing in the right-handside of (3.6b) in terms of the initial data z n (0), ˜ z n (0) ≡ z n (1).The expression of U (0) is an immediate consequence of (3.5a):[ U (0)] nm = δ nm z n (0) . (3.13)To obtain V (0) we note first of all that (3.3c) and (3.5b) imply V (0) = Y (0)[ M (0)] − , while (3.7b) with the ansatz (3.9) implies (at ℓ = 0) M nm (0) = ˆ g m (0) z m (1) − az n (0) , n, m = 1 , . . . , N, where of course ˆ g m (0) stands for ˆ g m ( z, ˜ z ), see (3.11), evaluated at z = z (0), ˜ z = ˜ z (0) ≡ z (1).0 F. CalogeroWe then evaluate the matrix [ M (0)] − via Lemma A.5 (with ξ m = z m (1) , η n = a z n (0), f n = 1 and g m = ˆ g m (0)) and, using again the ansatz (3.9) (at ℓ = 0), we obtain the followingexpression of the N × N matrix V (0):[ V (0)] nm = v m (0) u nm , n, m = 1 , . . . , N, with v m (0) defined as in Subsection 2.1 (see (2.4b) and the sentence following this formula) and u nm = N X k =1 N Q j =1 , j = m [ z k (1) − az j (0)] N Q j =1 , j = k [ z k (1) − z j (1)] , n, m = 1 , . . . , N. But the identity (A.8) (with η k = z k (1), ζ j = az j (0)) entails u nm = 1, hence[ V (0)] nm = v m (0) , n, m = 1 , . . . , N. (3.14)The insertion of these expressions of U (0) and V (0), (3.13) and (3.14), in (3.6b) yields (2.4),thereby completing the proof of Proposition 2.1.Let us now provide a terse treatment of the transition from the discrete-time equations ofmotion (2.1b), which we conveniently re-write here as follows, N Y j =1 e ˜ z n − a z j e ˜ z n − a ˜ z j ! = 1 + a, n = 1 , . . . , N, (3.15)to the continuous-time case, see (2.9a). It is then appropriate to treat separately the factor with j = n in the product appearing in the left-hand side of (3.15), and all the other factors with j = n . The basic equations are (2.8), entailing e ˜ z n − a z n = 2 ε ( ˙ z n + iωz n ) + ε (cid:0) z n + ω z n (cid:1) + O (cid:0) ǫ (cid:1) , e ˜ z n − a ˜ z n = ε ( ˙ z n + iωz n ) + ε z n + 2 iω ˙ z n ) + O (cid:0) ǫ (cid:1) , e ˜ z n − a z j = z n − z j + 2 ε ( ˙ z n + iωz j ) + O (cid:0) ε (cid:1) , j = n, e ˜ z n − a ˜ z j = z n − z j + ε (2 ˙ z n − ˙ z j + iωz j ) + O (cid:0) ε (cid:1) , j = n. Hence, after a little algebra, e ˜ z n − a z n e ˜ z n − a ˜ z n = 2 + ε − ¨ z n − iω ˙ z n + ω z n ˙ z n + iωz n + O (cid:0) ε (cid:1) , (3.16a) e ˜ z n − a z j e ˜ z n − a ˜ z j = 1 + ε ˙ z j + iωz j z n − z j + O (cid:0) ε (cid:1) , j = n, (3.16b)implying N Y j =1 e ˜ z n − a z j e ˜ z n − a ˜ z j ! = (cid:18) ε − ¨ z n − iω ˙ z n + ω z n ˙ z n + iωz n (cid:19) × ε N X j =1 , j = n (cid:18) ˙ z j + iωz j z n − z j (cid:19) + O (cid:0) ε (cid:1) . (3.16c)iscrete-Time Goldfishing 21While of course1 + a = 2 − iωε, (3.16d)see (2.8b). It is then clear that the insertion of these two formulas, (3.16c) and (3.16d), in (3.15)yields, at order ε = 1 , the trivial identity 2 = 2, and at order ε the equations of motion of the continuous-time goldfish model (2.9a).In an analogous manner one reobtains (2.9a) from (2.2b) or from (2.3).Let us also show that (2.5), which we rewrite here conveniently as follows, N Y k =1 (cid:20) z − a ℓ z k (1) a − z − a ℓ z k (0) (cid:21) = a ℓ a − − a ℓ − , (3.17)yields, in the continuous-time limit, (2.9b). Indeed the relation a = 1 − iεω (see (2.8b)) entails a − = 1 + iεω + O (cid:0) ε (cid:1) ,a ℓ = exp( − iωt ) (cid:18) − ε ω tt (cid:19) + O (cid:0) ε (cid:1) (via the first of the three relations (2.8a)), and z k (1) = z k (0) + ε ˙ z (0) + O (cid:0) ε (cid:1) (via the second of the three relations (2.8c), with ℓ = 0). Via these three relations (3.17) becomes N Y k =1 " z − exp( − iωt ) (cid:0) − εω t/ (cid:1) { z k (0) + ε [ ˙ z k (0) + iωz k (0)] } + O (cid:0) ε (cid:1) z − exp( − iωt ) (cid:0) − εω t/ (cid:1) z k (0) + O (cid:0) ε (cid:1) = exp( − iωt ) (cid:0) − εω t/ (cid:1) (1 + iεω ) − O (cid:0) ε (cid:1) exp( − iωt ) (cid:0) − εω t/ (cid:1) − O ( ε ) , i.e. (dividing each numerator by the corresponding denominator) N Y k =1 (cid:20) − ε [ ˙ z k (0) + iωz k (0)] z − exp( − iωt ) z k (0) + O (cid:0) ε (cid:1)(cid:21) = 1 + ε iω exp( − iωt ) − O (cid:0) ε (cid:1) . Clearly to order ε = 1 this yields the trivial identity 1 = 1 , and to order ε just the formula (2.9b).Let us end this subsection by pointing out that there is another ansatz that allows to trans-form the system of N equations (3.8) into two separate systems of N equations, but only inthe special case a = 1. This alternative ansatz reads (instead of (3.9)) Y nm = f m ˜ z m − z n , n, m = 1 , . . . , N, entailing (but only provided a = 1) the replacement of the system of N equations (3.8) withthe following two systems of N equations: N X k =1 (cid:20) f k (˜ z k − z n ) (cid:21) = 1 , n = 1 , . . . , N, N X k =1 " f k (cid:0)e ˜ z n − ˜ z k (cid:1) = 1 , n = 1 , . . . , N. But, as indicated at the end of Section 1, we postpone the treatment of the corresponding classof discrete-time dynamical systems to a separate paper.2 F. Calogero
The proof of the findings reported in Subsection 2.2 is analogous to that provided above, seeSubsection 3.1, so our treatment in this subsection is quite terse, being limited to indicate thechanges with respect to that reported in the preceding Subsection 3.1. Now the system of matrixevolution equations (3.6a) is generalized to read˜ U = U ( aI + bV ) + V, ˜ V = V ; (3.18)hence its solution is given by (2.11a). Clearly this evolution equation, (3.18), respectively itssolution, (2.11a), reduce to (3.6a) respectively to (3.6b) when b vanishes.The rest of the treatment is analogous. (3.7a) is now generalized to read M ˜ Z − aZM = ( I + bZ ) Y, hence it yields, in place of (3.7b), M nm = (cid:18) bz n ˜ z m − az n (cid:19) Y nm , n, m = 1 , . . . , N. In place of (3.10a) (again via the ansatz (3.9)) one now has N X k =1 " g k (1 + b ˜ z k ) (cid:0)e ˜ z m − a ˜ z k (cid:1) − g k (1 + bz n )˜ z k − az n = 0 , n, m = 1 , . . . , N, (3.19a)hence in place of (3.10b) and (3.10c) one gets the two sets of N equations N X k =1 (cid:20) g k ˜ z k − az n (cid:21) = 11 + bz n , n = 1 , . . . , N, (3.19b) N X k =1 (cid:20) g k (1 + b ˜ z k ) e ˜ z n − a ˜ z k (cid:21) = 1 , n = 1 , . . . , N. (3.19c)By solving the first set one obtains (via Lemma A.2, with ξ k = ˜ z k , η n = az n , c n = 1 / (1+ bz n ),and then the identity (A.7) with z = − a/b , η k = az k , ζ j = ˜ z j ) the following expression of g k ≡ ˆ g k ( z, ˜ z ):ˆ g k ( z, ˜ z ) = (cid:18) ˜ z k − az k bz k (cid:19) N Y j =1 , j = k (cid:20)(cid:18) ˜ z k − az j ˜ z k − ˜ z j (cid:19) (cid:18) b ˜ z j /a bz j (cid:19)(cid:21) . (3.20a)By solving instead the second set one obtains (via Lemma A.1, with ξ k = − a ˜ z k , η n = − e ˜ z n , c = 1 and g k replaced by g k (1 + b ˜ z k )) the following expression of g k ≡ ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) :ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) = a − N e ˜ z k − a ˜ z k b ˜ z k ! N Y j =1 , j = k e ˜ z j − a ˜ z k ˜ z j − ˜ z k ! . (3.20b)The three versions, (2.10), of the equations of motion reported in Subsection 2.2 then followby inserting (3.20a) in (3.19c), by inserting (3.20b) in (3.19b), and by equating (3.20a) to (3.20b).Next, let us prove Proposition 2.2. One proceeds again in close analogy to the treatmentof the preceding Subsection 3.1, hence we only mention where the treatment here differs fromthat provided there. It is easily seen that the expression of U (0) is the same as that giventhere, see (3.13), while the expression of V (0) (because one must now use Lemma A.5 withiscrete-Time Goldfishing 23 f n = 1 + bz n (0) rather than f n = 1) is now given by (2.11c). The insertion of these expressionsof U (0) and V (0) in (3.18) reproduce (2.11), thereby proving Proposition 2.2.Let us end this subsection by outlining what happens in the continuous-time limit whichobtains by setting a = 1 + εbη, V (0) = εB with ε infinitesimal, and correspondingly replacing the discrete-time matrix evolution equa-tion (3.18) with the matrix ODE˙ U = bU ( ηI + B ) + B, (3.21a)the solution of which reads U ( t ) = U (0) exp[ b ( ηI + B ) t ] + B [ b ( ηI + B )] − { exp[ b ( ηI + B ) t ] − I } . (3.21b)As already mentioned in Subsection 2.2, the ( continuous-time ) goldfish-type model obtainableby focussing appropriately on the evolution of the N eigenvalues z n ( t ) of this N × N matrix U ( t )(evolving according to (3.21a)) was, to the best of our knowledge, new , when the solvable matrixevolution equation (3.21a) was identified as continuous-time limit of (3.18); its treatment isprovided in [8]. The starting point is the following linear system of two discrete-time matrix evolution equations:˜ U = a + U + βV, ˜ V = a − V, (3.22a)where the 3 constants a ± , β are a priori arbitrary ( β = 0). It is easily seen that the solution ofthe initial-value problem for U reads as follows (with an analogous formula for V ): U ( ℓ ) = a ℓ + C + + a ℓ − C − , (3.22b)and the two constant matrices C ± given by (2.15b). Remark 3.3.
This solution U ( ℓ ) of the initial-value problem for the discrete-time N × N matrix evolution equation (3.22a) depends only on the 2 constants a ± : see (3.22b) and (2.15b).Indeed the system of two first-order discrete-time evolution equations (3.22a) is easily seen tocorrespond to the single second-order evolution equation e ˜ U − ( a + + a − ) ˜ U + a + a − U = 0 , from which the constant β has disappeared (but note that this second-order matrix ODE obtainsonly if β = 0; indeed if β = 0, U satisfies a first-order evolution equation, see the first of thetwo equations (3.22a). Remark 3.4.
For a + = a , β = 1, a − = 1, the system (3.22a) coincides with (3.6a) hence thismodel reduces to the first model, confirming Remark 2.6.We then proceed as in the first part of Section 3. It is then easily seen that the two matrixevolution equations (3.22a) become M ˜ Z = a + ZM + βY,M ˜ Y = a − Y ˜ M . M nm = βY nm ˜ z m − a + z n , n, m = 1 , . . . , N, (3.23)and using this formula it is easily seen that the second can be written, componentwise, as follows: N X k =1 (cid:20) Y nk ˜ Y km (cid:18) a − e ˜ z m − a + ˜ z k − z k − a + z n (cid:19)(cid:21) = 0 , n, m = 1 , . . . , N. (3.24)At this point we use again the ansatz (3.9) for the matrix Y nm , the consistency of whichis vindicated by the subsequent developments. Here the N quantities g m are again a pri-ori arbitrary; they shall be determined as functions of the 2 N un-updated and once-updatedcoordinates z n ≡ z n ( ℓ ) and ˜ z n ≡ z n ( ℓ + 1), or alternatively of the once and twice updated coor-dinates ˜ z n ≡ z n ( ℓ + 1) and e ˜ z n ≡ z n ( ℓ + 2), see below. It is indeed immediately seen that via the ansatz (3.9) the system of N equations (3.24) becomes N X k =1 (cid:18) a − g k e ˜ z m − a + ˜ z k − g k ˜ z k − a + z n (cid:19) = 0 , n, m = 1 , . . . , N ; (3.25a)hence it can be replaced by the following two separated systems of only N equations: N X k =1 (cid:18) g k ˜ z k − a + z n (cid:19) = 1 , n = 1 , . . . , N, (3.25b) N X k =1 (cid:18) g k e ˜ z n − a + ˜ z k (cid:19) = 1 a − , n = 1 , . . . , N. (3.25c)The first, (3.25b), of these two systems defines uniquely the N quantities g k ≡ ˆ g k ( z, ˜ z ),yielding again, via (A.13), the expression (3.11) (with a replaced by a + ):ˆ g k ( z, ˜ z ) = (˜ z k − a + z k ) N Y j =1 , j = k (cid:18) ˜ z k − a + z j ˜ z k − ˜ z j (cid:19) , k = 1 , . . . , N. (3.26)Insertion of this expression in the second, (3.25c), of the two systems written just above thenyields the evolution equation (2.12a). The identification of the third discrete-time dynamicalsystem of goldfish type, see (2.12a), is thereby accomplished.The second version, (2.13a), of this model obtains by solving for g k ≡ ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) the se-cond, (3.25c), of the two systems written above (via Lemma A.1, with ξ k = a + ˜ z k , η n = e ˜ z n and c = − /a − ), thereby obtainingˇ g k (cid:0) ˜ z, e ˜ z (cid:1) = e ˜ z k − a + ˜ z k a − ! N Y j =1 , j = k a + ˜ z k − e ˜ z j a + ˜ z k − a + ˜ z j ! , k = 1 , . . . , N ; (3.27)and by then inserting this expression of g k in (3.25b).And the third version, (2.14), of this model obtains by equating (3.26) to (3.27).As for the proof of Proposition 2.3, it follows immediately from the above treatment, seein particular (2.15); there only remains to justify the identification of the two matrices U (0)and U (1), see (2.15c) and (2.15d).The first of the two formulas, (2.15c), is just (3.5a).iscrete-Time Goldfishing 25To prove the second, (2.15d), we note that (3.5a) and (3.3a) entail (for ℓ = 0) R (1) = M (0) , (3.28)while (3.2) (with ℓ = 0) reads U (1) = R (1) Z (1)[ R (1)] − . (3.29a)Hence (via (3.28)) U (1) = M (0) Z (1)[ M (0)] − . (3.29b)We now note that (3.23) and the ansatz (3.9) imply that the N × N matrix M (0) is definedcomponentwise as follows: M nm (0) = g m (0) z m (1) − a + z n (0) , n, m = 1 , . . . , N, (3.30a)with (see (3.26)) g m (0) = [ z m (1) − a + z m (0)] N Y j =1 , j = m (cid:20) z m (1) − a + z j (0) z m (1) − z j (1) (cid:21) , m = 1 , . . . , N, (3.30b)so that M nm (0) = z m (1) − a + z m (0) z m (1) − a + z n (0) N Y j =1 , j = m (cid:20) z m (1) − a + z j (0) z m (1) − z j (1) (cid:21) , n, m = 1 , . . . , N. (3.30c)Next, we note that the expression (3.30a) of the matrix M (0) entails – via Lemma A.5 (with f n = 1, g m = g m (0), ξ m = z m (1), η n = a + z n (0)) and the expression of g m (0) given above – thatits inverse, appearing in the right-hand-side of (3.29b), is explicitly given, componentwise, asfollows: (cid:8) [ M (0)] − (cid:9) nm = a − N + (cid:20) a + z m (0) − z m (1) a + z m (0) − z n (1) (cid:21) N Y j =1 , j = m (cid:20) a + z m (0) − z j (1) z m (0) − z j (0) (cid:21) ,n, m = 1 , . . . , N. (3.30d)The insertion of this formula and (3.30c) in (3.29b) entails that the matrix U (1) reads compo-nentwise as follows: U nm (1) = a − N + N Y j =1 , j = m (cid:20) a + z m (0) − z j (1) z m (0) − z j (0) (cid:21) × N X k =1 z k (1) Q j =1 , j = n [ z k (1) − a + z j (0)] Q j =1 , j = k [ z k (1) − z j (1)] , n, m = 1 , . . . , N. And it is then immediately seen that this formula yields, via the identity (A.12) (with η k = z k (1), k = 1 , . . . , N ; ζ j = a + z j (0), j = 1 , . . . , n − , n + 1 , . . . , N ), the formula (2.15d).6 F. Calogero The treatment in this subsection is rather terse, since it is analogous to that of the precedingsubsections; and the notation is of course analogous. But now the starting point is the followingnonlinear system of two discrete-time matrix evolution equations:˜ U = αU + βV + ηU V, ˜ V = ρ + γV, (3.31)featuring the 5 arbitrary constants α , β , η , ρ , γ (which, as noted in Subsection 2.4, can bereduced to 4 by taking advantage of the freedom to rescale V ).It is a standard task to see that the solution of the initial-value problem for this matrixsystem reads as follows: V ( ℓ ) = γ ℓ V (0) + ρ γ ℓ − γ − I, with U ( ℓ ) given by (2.21).We now proceed again as in the first part of Section 3. Via (3.3a) and (3.3c) we get from (3.31)the two matrix equations M ˜ Z = αZM + ( β + ηZ ) Y, (3.32a) M ˜ Y = ρM ˜ M + γY ˜ M . (3.32b)The first, (3.32a), of these two matrix equations entails, componentwise, Y nm = (cid:18) ˜ z m − αz n ηz n + β (cid:19) M nm , n, m = 1 , . . . , N ;hence the second, (3.32b), of these two matrix equations yields (when written componentwise)the following N equations: N X k =1 " M nk ˜ M km e ˜ z m − α ˜ z k η ˜ z k + β − ρ − γ ˜ z k − αz n ηz n + β ! = 0 , n, m = 1 , . . . , N, (3.33a)which, as can be easily verified, can be conveniently rewritten as follows: N X k =1 " M nk ˜ M km e ˜ z m − a ˜ z k − bη ˜ z k + β − γ ˜ z k − az n − bηz n + β ! = 0 , n, m = 1 , . . . , N, (3.33b)with the two constants a and b defined by (2.17c).Next, we make the following ansatz for the matrix M nm : M nm = g m ˜ z m − az n − b , n, m = 1 , . . . , N. (3.34)Here the N quantities g m are a priori arbitrary; they shall be determined as functions of the2 N un-updated and once-updated coordinates z n ≡ z n ( ℓ ) and ˜ z n ≡ z n ( ℓ + 1), or of the once andtwice updated coordinates ˜ z n ≡ z n ( ℓ + 1) and e ˜ z n ≡ z n ( ℓ + 2), see below. It is indeed immediatelyseen that via this ansatz (3.34) the system of N equations (3.33b) can be rewritten as follows: N X k =1 " g k ( ηz n + β )( η ˜ z k + β )(˜ z k − az n − b ) − γ g k (cid:0)e ˜ z m − a ˜ z k − b (cid:1) = 0 , n, m = 1 , . . . , N ; (3.35a)iscrete-Time Goldfishing 27hence it can be replaced by the following two separated systems of only N equations: N X k =1 (cid:20) g k ( η ˜ z k + β )(˜ z k − az n − b ) (cid:21) = 1( ηz n + β ) , n = 1 , . . . , N, (3.35b) N X k =1 (cid:18) g k e ˜ z n − a ˜ z k − b (cid:19) = 1 γ , n = 1 , . . . , N. (3.35c)The first of these two systems defines uniquely the N quantities g k ≡ ˆ g k ( z, ˜ z ), yielding theirexpression (2.17b) (see Appendix B for a proof). The second equation then yields the evolutionequation (2.17a).The alternative possibility is to determine (via Lemma A.1, with ξ k = a ˜ z k , η n = e ˜ z n − b, c = − /γ ) the quantities g k ≡ ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) as solutions of the second system, yielding the formula (2.18b);and to then insert these expressions of g k ≡ ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) in the first system of equations. Clearly inthis manner one arrives at the equations of motion (2.19a).And a third possibility is of course to equate (2.17b) to (2.18b), see (2.20).The identification of the three variants of the equations of motion of the fourth discrete-time dynamical system of goldfish type, see Subsection 2.4, is thereby accomplished.The proof of Proposition 2.4 follows immediately from the above treatment; and we trustthat the identification in terms of the 2 N initial data z n (0) and z n (1) of the two matrices U (0)and V (0), see (2.4) and (2.4), is sufficiently obvious (also in the light of the analogous treatmentin the preceding subsections of this section) not to require an explicit justification here.We end this subsection with a terse mention of the continuous-time model that obtains fromthat treated above (in this subsection) via the limiting transition from discrete to continuous time. The point of departure for the treatment of the continuous-time dynamical system ofgoldfish type obtained in this manner is the following continuous-time system of two first-ordermatrix evolution equations˙ U = a U + a V + a U V, ˙ V = a + a V, that obtains from (3.31) via the assignments t ⇒ εℓ , U ( ℓ ) ⇒ U ( t ), V ( ℓ ) ⇒ V ( t ), α = 1 + εa , β = εa , η = εa , ρ = εa , γ = 1 + εa , with ε infinitesimal. The resulting model of goldfishtype was, to the best of our knowledge, new ; a detailed treatment of it is provided in [9]. In this paper we have introduced and tersely analyzed 4 different discrete-time dynamicalsystems of goldfish type. The possibility to identify other discrete-time evolution equationsamenable to exact treatment by variations of the methodology used in this paper is open: letus outline here an avenue to such generalizations.Consider the system of two N × N matrix discrete-time first-order evolution equations S X s =1 (cid:2) F ,s ( U ) F ,s ( ˜ U ) (cid:3) = S X s =1 (cid:2) Φ ,s ( U ) V Φ ,s (cid:0) ˜ U (cid:1)(cid:3) , (4.1a) S X s =1 (cid:2) F ,s ( U ) F ,s (cid:0) ˜ U (cid:1)(cid:3) ˜ V = Φ ( U ) V Φ (cid:0) ˜ U (cid:1) , (4.1b)where the two N × N matrices U ≡ U ( ℓ ) and V ( ℓ ) are the dependent variables, ℓ = 0 , , , . . . is the independent discrete-time variable, S , S , S are 3 arbitrary positive integers, F ,s ( u ), F ,s ( u ), F ,s ( u ), F ,s ( u ) and Φ ,s ( u ), Φ ,s ( u ), Φ ( u ), Φ ( u ) are 2( S + S + S + 1) a priori u (of course becoming N × N matriceswhen the scalar u is replaced by an N × N matrix). Then introduce the eigenvalues z n ( ℓ ) of thematrix U ( ℓ ), as well as the matrices Z ≡ Z ( ℓ ), R ≡ R ( ℓ ), M ≡ M ( ℓ ) and Y ≡ Y ( ℓ ), as above(see (3.2) and (3.3)). It is then plain that the matrix evolution equation (4.1a) becomes S X s =1 (cid:2) F ,s ( Z ) M F ,s (cid:0) ˜ Z (cid:1)(cid:3) = S X s =1 (cid:2) Φ ,s ( Z ) Y Φ ,s (cid:0) ˜ Z (cid:1)(cid:3) (4.2a)entailing componentwise M nm = S P s =1 (cid:2) Φ ,s ( z n )Φ ,s (˜ z m ) (cid:3) S P s =1 (cid:2) F ,s ( z n ) F ,s (cid:0) ˜ z m (cid:1)(cid:3) Y nm , n, m = 1 , . . . , N. (4.2b)Likewise (4.1b) becomes S X s =1 (cid:2) F ,s ( Z ) M F ,s ( ˜ Z ) (cid:3) ˜ Y = Φ ( Z ) Y Φ (cid:0) ˜ Z (cid:1) ˜ M , entailing componentwise (via (4.2b)) N X k =1 Y nk ˜ Y km " S X s =1 F ,s ( z n ) F ,s (˜ z k ) S P σ =1 (cid:2) Φ ,σ ( z n )Φ ,σ (˜ z k ) (cid:3) S P σ =1 (cid:2) F ,σ ( z n ) F ,σ (˜ z k ) (cid:3) = Φ ( z n ) N X k =1 Y nk ˜ Y km Φ (˜ z k ) S P s =1 (cid:2) Φ ,s (˜ z k )Φ ,s (cid:0)e ˜ z m (cid:1)(cid:3) S P s =1 (cid:2) F ,s (˜ z k ) F ,s (cid:0)e ˜ z m (cid:1)(cid:3) , n, m = 1 , . . . , N. And via the ansatz Y nm = g m (see (3.9)) this system of N equations can clearly be replacedby the following two systems of N linear algebraic equations for the N quantities g k : N X k =1 g k " S X s =1 F ,s ( z n ) F ,s (˜ z k ) S P σ =1 (cid:2) Φ ,σ ( z n )Φ ,σ (˜ z k ) (cid:3) S P σ =1 (cid:2) F ,σ ( z n ) F ,σ (˜ z k ) (cid:3) = Φ ( z n ) , n = 1 , . . . , N, (4.3a) N X k =1 g k Φ (˜ z k ) S P s =1 (cid:2) Φ ,s (˜ z k )Φ ,s (cid:0)e ˜ z n (cid:1)(cid:3) S P s =1 (cid:2) F ,s (˜ z k ) F ,s (cid:0)e ˜ z n (cid:1)(cid:3) = 1 , n = 1 , . . . , N. (4.3b)One can then solve the first, (4.3a), respectively the second, (4.3b), of these two linear systemsfor the N quantities g k , getting the expressions g k = ˆ g k ( z, ˜ z ) respectively g k = ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) . Onearrives thereby at three equivalent systems of N discrete-time second-order evolution equationsfor the N coordinates z n ( ℓ ): ( i ) by inserting the expression ˆ g k ( z, ˜ z ) in (4.3b); ( ii ) by inserting theexpression ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) in (4.3a); ( iii ) by setting ˆ g n ( z, ˜ z ) = ˇ g n (cid:0) ˜ z, e ˜ z (cid:1) . While of course the evolutionin the discrete time ℓ entailed by these equations of motions corresponds to the evolution ofiscrete-Time Goldfishing 29the eigenvalues of the matrix U ( ℓ ) solution of the matrix evolution equation (4.1a) (with the ansatz (3.9) properly taken into account). Hence if that matrix evolution equation, (4.1a), is solvable , the discrete-time dynamical system described by these three equivalent sets of second-order evolution equations is as well solvable .One has thereby identified a discrete-time solvable dynamical system. A remaining openquestion is the extent to which its equations of motion can be exhibited in reasonably neatform: this depends on the extent that the two quantities g k ≡ ˆ g k ( z, ˜ z ) respectively g k ≡ ˇ g k (cid:0) ˜ z, e ˜ z (cid:1) defined as solutions of the two linear systems (4.3a) respectively (4.3b) can be expressed moreexplicitly than via the standard Cramer formula (ratio of two determinants).Clearly the 4 models treated in this paper belong to this class (4.1) (the last one, however,only if ρ = 0): see (3.6a), (3.18), (3.22a) and (3.31). The interest of additional models treatablevia this approach depends on the neatness of the corresponding equations of motion, which canonly be investigated on a case-by-case basis. A Appendix
In this appendix we collect various mathematical developments whose treatment in the body ofthe paper would interrupt the flow of the presentation.First of all we report several mathematical identities. We consider all of them well-known,but for completeness we either prove them below, or indicate where proofs can be found. Theseformulas feature sets of N numbers such as ξ n or η n or ζ n ; these numbers are arbitrary butfor simplicity we assume them to be distinct . The formulas of course remain valid when thesenumbers are not distinct, but possibly only by taking appropriate limits. Sometimes an arbitrary number z also appears. N X k =1 N Y j =1 , j = k (cid:18) ζ j − zζ j − ζ k (cid:19) = 1 , (A.1) N X k =1 ζ k N Y j =1 , j = k (cid:18) ζ j − ζ k (cid:19) = N Y j =1 (cid:18) ζ j (cid:19) , (A.2) N X k =1 ζ n − k N Y j =1 , j = k (cid:18) ζ j − zζ j − ζ k (cid:19) = z n − , n = 1 , , . . . , N, (A.3) N X k =1 ζ n − k N Y j =1 ,j = k (cid:18) ζ k − ζ j (cid:19) = δ nN , n = 1 , , . . . , N, (A.4) N Y j =1 , j = m (cid:18) ζ j − ζ n ζ j − ζ m (cid:19) = δ nm , n, m = 1 , , . . . , N, (A.5) N X k =1 N Y j =1 , j = k (cid:18) ξ j − η n ξ j − ξ k (cid:19) N Y j =1 , j = m (cid:18) η j − ξ k η j − η m (cid:19) = δ nm , n, m = 1 , , . . . , N, (A.6) N X k =1 (cid:18) z − η k (cid:19) N Q j =1 , j = n ( ζ j − η k ) N Q j =1 , j = k ( η j − η k ) = N Q j =1 , j = n ( ζ j − z ) N Q j =1 ( η j − z ) ≡ z − ζ n N Y j =1 (cid:18) ζ j − zη j − z (cid:19) ,n = 1 , , . . . , N, (A.7)0 F. Calogero N X k =1 N Q j =1 , j = n ( ζ j − η k ) N Q j =1 , j = k ( η j − η k ) ≡ N X k =1 (cid:18) η k − ζ k η k − ζ n (cid:19) N Y j =1 , j = k (cid:18) ζ j − η k η j − η k (cid:19) = 1 ,n = 1 , , . . . , N, (A.8) N X k =1 (cid:18) η k − ξ n (cid:19) N Y j =1 , j = k (cid:18) η j − η k (cid:19) = N Y j =1 (cid:18) η j − ξ n (cid:19) , n = 1 , , . . . , N, (A.9) N X k =1 (cid:18) η k − ζ k η k − z (cid:19) N Y j =1 , j = k (cid:18) ζ j − η k η j − η k (cid:19) ≡ N X k =1 N Q j =1 ( ζ j − η k )( z − η k ) N Q j =1 , j = k ( η j − η k ) = 1 − N Y j =1 (cid:18) ζ j − zη j − z (cid:19) , (A.10) N X k =1 N Q j =1 ( ζ j − η k ) N Q j =1 , j = k ( η j − η k ) = N X k =1 ( ζ k − η k ) , (A.11) N X k =1 η k N Q j =1 , j = n ( ζ j − η k ) N Q j =1 , j = k ( η j − η k ) = N X k =1 , k = n ( ζ k ) − N X k =1 ( η k ) , n = 1 , . . . , N. (A.12)The identity (A.1) (with z an arbitrary number) is implied by the fact that its left-hand sideis a polynomial in z of degree less than N (in fact, of degree at most N −
1) which clearly hasthe value unity at the N points ζ n , and the right-hand side, i.e. unity , is the unique polynomialof degree less than N in z that has the value unity in N distinct points. The identity (A.2)is the special case of (A.1) with z = 0. The identity (A.3) coincides with equation (2.4.2-32)of [4] (or, as above, it is implied by the observation that its left-hand side is a polynomial in z of degree less than N the values of which at the N points ζ k coincide with the values of thepolynomial z n at z = ζ k ). The identity (A.4) coincides with equations (2.4.3-12) and (2.4.3-21)of [4]. The identity (A.5) is obvious. The identities (A.6) respectively (A.7) coincide withequations (2.4.2-26) respectively (2.4.2-27) of [4] (via the definition (2.4.2-24), with x n = ξ n ,y n = η n , respectively x n = z , y n = η n , z n = ζ n ). The identities (A.8) respectively (A.9) followfrom (A.7) in the limit z → ∞ respectively ζ j → ∞ . The identity (A.10) follows from (A.8)and (A.7) via the trivial identity ζ n − η k z − η k ≡ − ζ n − zη k − z . Finally, the identity (A.11) follows from (A.10) in the limit z → ∞ , and the identity (A.12) isjust the special case of the preceding identity (A.11) with ζ n = 0.Next we report a simple lemma (for a neat proof see for instance [18], or below, after theproof of the following Lemma A.2).iscrete-Time Goldfishing 31 Lemma A.1.
The solution of the set of N linear algebraic equations for the N variables g k reading N X k =1 g k ξ k − η n = c, n = 1 , . . . , N (A.13a) is provided by the formula g k = c ( ξ k − η k ) N Y j =1 , j = k (cid:18) ξ k − η j ξ k − ξ j (cid:19) , k = 1 , . . . , N. (A.13b)A generalization of this lemma reads as follows: Lemma A.2.
The solution of the set of N linear algebraic equations for the N variables g k reading N X k =1 g k ξ k − η n = c n , n = 1 , . . . , N (A.14a) is provided by the formula g k = N X s =1 c s ( ξ k − η s ) N Y j =1 , j = k (cid:18) ξ j − η s ξ j − ξ k (cid:19) N Y j =1 , j = s (cid:18) η j − ξ k η j − η s (cid:19) ,n = 1 , . . . , N, (A.14b) or equivalently g k = ( ξ k − η k ) N Y j =1 , j = k (cid:18) η j − ξ k ξ j − ξ k (cid:19) N X s =1 c s N Q j =1 , j = k ( ξ j − η s ) N Q j =1 , j = s ( η j − η s ) ,n = 1 , . . . , N. (A.14c)To prove this formula one inserts this expression, (A.14b), of g k in (A.14a), and notes thatone obtains thereby an equality provided there holds the formula N X k =1 ξ k − η s ξ k − η n N Y j =1 , j = k (cid:18) ξ j − η s ξ j − ξ k (cid:19) N Y j =1 , j = s (cid:18) η j − ξ k η j − η s (cid:19) = δ sn or, equivalently, N X k =1 N Y j =1 , j = k (cid:18) ξ j − η n ξ j − ξ k (cid:19) N Y j =1 , j = s (cid:18) η j − ξ k η j − η s (cid:19) = δ sn N Y j =1 (cid:18) ξ j − η n ξ j − η s (cid:19) . Clearly this formula is implied by the identity (A.6). Lemma A.2 is thus proven.Note that, by setting c n = c in (A.14c) and using the identity (A.8) (with the dummy index k replaced by s and the index n replaced by k ), one reobtains (A.13b), thereby proving Lemma A.1.We now report, and prove, 3 other lemmata.2 F. Calogero Lemma A.3.
There holds the formula det[ I + X ] = 1 + N X k =1 x k , (A.15a) provided I is the N × N unit matrix and the N × N matrix X is defined componentwise asfollows: X nm = x m , n, m = 1 , . . . , N. (A.15b)This (presumably well-known) formula is easily proven by recursion. Lemma A.4.
The N eigenvalues of the N × N matrix U nm = δ nm ζ n + η m , n, m = 1 , . . . , N, (A.16a) coincide with the N solutions of the following algebraic equation in z : N X k =1 (cid:18) η k z − ζ k (cid:19) − , (A.16b) i.e. they are the N roots of the polynomial of degree N in z that obtains by multiplying theleft-hand side of this equation by N Q j =1 ( z − ζ k ) . This lemma is an immediate consequence of the preceding Lemma A.3, because the secularequation associated with the matrix (A.16a) (whose roots provide the eigenvalues) is easily seento coincide (up to an overall, hence irrelevant, multiplicative constant) with the vanishing of thedeterminant in the left-hand side of (A.15a) with (A.15b) and x k = η k / ( z − ζ k ). Lemma A.5.
The inverse of the matrix defined componentwise as follows, M nm = f n g m ξ m − η n , n, m = 1 , . . . , N, (A.17a) is defined componentwise as follows: (cid:2) M − (cid:3) nm = (cid:18) ξ n − η m g n f m (cid:19) N Y j =1 , j = n (cid:18) ξ j − η m ξ j − ξ n (cid:19) N Y j =1 , j = m (cid:18) η j − ξ n η j − η m (cid:19) ,n, m = 1 , . . . , N. (A.17b)The proof of this formula goes as follows. The matrix formula M M − = I , written compo-nentwise, reads, via (A.17a), N X k =1 g k (cid:2) M − (cid:3) km ξ k − η n ! = δ nm f n , n, m = 1 , . . . , N. Then, for fixed m , apply Lemma A.2 with g k replaced by g k (cid:2) M − (cid:3) km , and c n replaced by δ nm /f n . This yields, rather immediately, the formula (A.17b), which is thereby proven.iscrete-Time Goldfishing 33 B Appendix
In this appendix we detail the derivation of some findings for the fourth model, firstly theexpression (2.17b) of ˆ g n ( z, ˜ z ) as solution of the linear system (3.35b), and secondly the derivationof (2.19b) from (2.19a).Via the identity1( η ˜ z k + β )(˜ z k − az n − b ) = 1 η ( az n + b ) + β (cid:18) z k − az n − b − ηη ˜ z k + β (cid:19) , (B.1)and the definition σ (0) = N X k =1 (cid:18) ηg k η ˜ z k + β (cid:19) , (B.2)the linear system (3.35b) can be conveniently reformulated as follows: N X k =1 (cid:18) g k ˜ z k − az n − b (cid:19) = c n ,c n = σ (0) + η ( az n + b ) + βηz n + β = σ (0) + a + aη (cid:20) β (1 − α ) az n + aβ/η (cid:21) . We now use Lemma A.2 (with ξ k = ˜ z k , η n = a z n + b and c n defined as above). We thus getˆ g n ( z, ˜ z ) = (˜ z n − az n − b ) N Y j =1 , j = n (cid:18) ˜ z n − az j − b ˜ z n − ˜ z j (cid:19) (cid:20) ( σ + a ) σ (1) n + aβ (1 − α ) η σ (2) n (cid:21) ,σ (1) n = N X k =1 N Q j =1 , j = n (˜ z j − az k − b ) N Q j =1 , j = k ( az j − az k ) , n = 1 , . . . , N,σ (2) n = N X k =1 az k + aβ/η N Q j =1 , j = n (˜ z j − az k − b ) N Q j =1 , j = k ( az j − az k ) , n = 1 , . . . , N. It is now plain, via the identity (A.8) (with η k = az k + b, ζ j = ˜ z j ) that σ (1) n = 1. As forthe sum σ (2) n , it is also easily evaluated via the identity (A.7) (now with η k = az k + b , ζ j = ˜ z j , z = b − aβ/η ): σ (2) n = a − N ηη ˜ z n + aβ − bη N Y j =1 (cid:18) η ˜ z j + aβ − bηηz j + β (cid:19) . Henceˆ g n ( z, ˜ z ) = (˜ z n − az n − b ) N Y j =1 , j = n (cid:18) az j + b − ˜ z n ˜ z j − ˜ z n (cid:19) σ (0) + a + σ (3) η ˜ z n + aβ − bη ! , (B.3a) σ (3) = a − N β (1 − α ) N Y j =1 (cid:18) η ˜ z j + αβηz j + β (cid:19) . (B.3b)4 F. CalogeroNow, using this expression of ˆ g n ( z, ˜ z ), we can evaluate σ from its definition (B.2), therebyobtaining σ (0) = aσ (4) + σ (3) σ (5) − σ (4) = − a + a + σ (3) σ (5) − σ (4) ,σ (4) = N X k =1 (cid:18) ˜ z k − az k − b ˜ z k + β/η (cid:19) N Y j =1 , j = k (cid:18) ˜ z k − az j − b ˜ z k − ˜ z j (cid:19) ,σ (5) = N X k =1 ˜ z k − az k − b ( η ˜ z k + β )(˜ z k + αβ/η ) N Y j =1 , j = k (cid:18) az j + b − ˜ z k ˜ z j − ˜ z k (cid:19) . It is now plain (via (A.10) with η j = ˜ z j , ζ j = a z j + b, z = − β/η ) that σ (4) = 1 − N Y j =1 (cid:18) ηaz j + ηb + βη ˜ z j + β (cid:19) . (B.4)To evaluate σ , we use again an identity analogous to that used above:1( η ˜ z k + β )(˜ z k + αβ/η ) = 1 β (1 − α ) (cid:18) z k + αβ/η − ηη ˜ z k + β (cid:19) . Thereby σ (5) = σ (6) − σ (7) β (1 − α ) , (B.5a) σ (6) = N X k =1 ˜ z k − az k − b ˜ z k + αβ/η N Y j =1 , j = k (cid:18) az j + b − ˜ z k ˜ z j − ˜ z k (cid:19) , (B.5b) σ (7) = N X k =1 ˜ z k − az k − b ˜ z k + β/η N Y j =1 , j = k (cid:18) az j + b − ˜ z k ˜ z j − ˜ z k (cid:19) . (B.5c)Both these sums can be evaluated via the identity (A.10), with η k = ˜ z k , ζ j = az j + b, and with z = αβ/η respectively with z = − β/η , obtaining σ (6) = 1 − a N N Y j =1 (cid:20) ηz j + βη ˜ z j + αβ (cid:21) , σ (7) = 1 − N Y j =1 (cid:20) η ( az j + b ) + βη ˜ z j + β (cid:21) , hence (via (B.5a)) σ (5) = 1 β (1 − a ) + bη N Y j =1 (cid:20) η ( az j + b ) + βη ˜ z j + β (cid:21) − a N N Y j =1 (cid:20) ηz j + βη ˜ z j + αβ (cid:21) , and via this formula together with (B.4) and (B.3b) we finally get σ (0) + a = a − N N Y j =1 (cid:18) η ˜ z j + αβηz j + β (cid:19) . The insertion of this expression of σ (0) + a and of the expression (B.3b) of σ (3) in (B.3a)completes the derivation of the expression (2.17b) of ˆ g n ( z, ˜ z ).Finally let us tersely outline the derivation of (2.19b) from (2.19a). Firstly one uses in (2.19a)the identity (B.1); then one uses twice the identity (A.10), with ζ k = e ˜ z k − b , η k = a ˜ z k and with z = a ( az n + b ) respectively with z = − aβ/η . And the rest is trivial algebra.iscrete-Time Goldfishing 35 Acknowledgements
It is a pleasure to thank my colleague and friend Orlando Ragnisco for pointing out relevantreferences.
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