Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

TL;DR

This paper explores specific algebraically solvable discrete-time systems with two dependent variables evolving via nonlinear coupled recursions, extending previous models to include higher-degree homogeneous functions.

Contribution

It introduces a broader class of solvable systems by generalizing from quadratic to arbitrary degree homogeneous functions in two coupled discrete-time recursions.

Findings

Identified algebraically solvable two-variable discrete systems.

Extended previous quadratic models to arbitrary degree homogeneous functions.

Clarified properties and structure of the extended models.

Abstract

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,x~n = F^(n)(x1; x2) , n = 1, 2 , which are algebraically solvable. Here l is the "discrete-time" independent variable taking integer values (l = 0, 1, 2,...), xn = xn (l) are 2 dependent variables, and x~n = xn (l + 1) are the corresponding 2 updated variables. In a previous paper the 2 functions F^(n)(x1, x2), n = 1, 2 were defined as follows: F^(n)(x1; x2) = P2 (xn, xn+1), n = 1,2 mod[2]; with P2 (x1; x2) a specific second-degree homogeneous polynomials in the 2 (indistinguishable!) dependent variables x1 (`) and x2 (l). In the present paper we further clarify some aspects of that model and we present its extension to the case when F(n)(x1, x2) = Q^(n)_k(x1, x2), n = 1, 2 mod[2], with Q(n)_k(x1; x2) a specific homogeneous function of arbitrary (integer ) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x1 (l) and x2 (l).

Equations

$${\tilde{x}}_n=F^{\left(n\right)}\left(x_1,x_2\right),n=1,2,$$

$${\tilde{x}}_n=-\left(x_1+x_2\right)\left[a\left(x_1+x_2\right)+Sb\left(x_n-x_{n+1}\right)\right],n=1,2\mathrm{mod}[2].$$

$${\tilde{x}}_n\equiv x_n\left(\ell +1\right)$$

$$p_2\left(z;\ell \right)=z^2+y_1\left(\ell \right)z+y_2\left(\ell \right)=\left[z-x_1\left(\ell \right)\right]\left[z-x_2\left(\ell \right)\right],$$

$${\tilde{x}}_n={\left(B_1{x}_1+B_2{x}_2\right)}^k\left(E_{n1}{x}_1+E_{n2}{x}_2\right),n=1,2,$$

$$E_{11}=d\left(\alpha g_1{B}_1+S\beta B_2{g}_2\right),E_{12}=d{B}_2\left(\alpha g_1+S\beta g_3\right),$$

$$E_{21}=\alpha \left(1-B_{1}d{g}_1\right)\left(B_1/B_2\right)-B_{1}dS\beta g_2,$$

$$E_{22}=\alpha \left(1-B_{1}d{g}_1\right)-B_{1}dS\beta g_3.$$

$$d=\left(1/2\right){\left[{\left(B_1\right)}^2{C}_2+{\left(B_2\right)}^2{C}_1-B_1{B}_2{C}_3\right]}^{-1},$$

$$g_1=2B_1{C}_2-B_2{C}_3,g_2=2B_2{C}_1-B_1{C}_3,g_3=B_2{C}_3-2B_1{C}_2,$$

$${\tilde{x}}_n=\left(B_1{x}_1+B_2{x}_2\right)\left(E_{n1}{x}_1+E_{n2}{x}_2\right),n=1,2;$$

$$E_{11}=d\left(\alpha g_1{B}_1+S\beta B_2{g}_2\right),E_{12}=d\left(\alpha g_1{B}_2+S\beta B_2{g}_3\right),$$

$$E_{21}=\alpha \left(1-B_{1}d{g}_1\right)\left(B_1/B_2\right)-B_{1}dS\beta g_2,$$

$$E_{22}=\alpha \left(1-B_{1}d{g}_1\right)-B_{1}dS\beta g_3.$$

$${\tilde{x}}_n=\frac{D_{n1}{x}_1+D_{n2}{x}_2}{B_1{x}_1+B_2{x}_2},n=1,2;$$

$$D_{11}=d\left(\alpha g_1{B}_1+S\beta B_2{g}_2\right),D_{12}=d\left(\alpha g_1{B}_2+S\beta B_2{g}_3\right),$$

$$D_{21}=\alpha \left(1-B_{1}d{g}_1\right)\left(B_1/B_2\right)-B_{1}dS\beta g_2,$$

$$D_{22}=\alpha \left(1-B_{1}d{g}_1\right)-B_{1}dS\beta g_3.$$

$${\tilde{x}}_n=a_{n1}{\left(x_1\right)}^2+a_{n2}{\left(x_2\right)}^2+a_{n3}{x}_1{x}_2,n=1,2,$$

$${\left(a_{11}{a}_{22}-a_{21}{a}_{12}\right)}^2+\left(a_{13}{a}_{21}-a_{11}{a}_{23}\right)\left(a_{13}{a}_{22}-a_{12}{a}_{23}\right)=0$$

$${\tilde{x}}_n={\left(2x_1+x_2\right)}^k\left[{\left(-1\right)}^{k}a\left(2x_1+x_2\right)+S{\left(-1\right)}^{n}nb\left(x_1-x_2\right)\right],n=1,2.$$

$$x_n\left(\ell +2\right)={\left(-1\right)}^k{3}^{k+1}{a}^k{\left[2x_1\left(\ell \right)+x_2\left(\ell \right)\right]}^{k\left(k+2\right)}\cdot$$

$$\cdot \left\{a^2\left[2x_1\left(\ell \right)+x_2\left(\ell \right)\right]-S\left(\ell \right)S\left(\ell +1\right){\left(-1\right)}^{n}n{b}^2\left[x_1\left(\ell \right)-x_2\left(\ell \right)\right]\right\},$$

$$n=1,2;$$

$${\tilde{y}}_1=\alpha {\left(y_1\right)}^{1+k},{\tilde{y}}_2={\beta }^2{y}_2{\left(y_1\right)}^q+\gamma {\left(y_1\right)}^r,$$

$$y_1\left(\ell \right)={\alpha }^{\left[{\left(1+k\right)}^{\ell }-1\right]/k}{\left[y_1\left(0\right)\right]}^{{\left(1+k\right)}^{\ell }},$$

$$y_2\left(\ell \right)={\beta }^{2\ell }{\alpha }^{\left(q/k^2\right)\left[{\left(1+k\right)}^{\ell }-k\ell -1\right]}{\left[y_1\left(0\right)\right]}^{\left(q/k\right)\left[{\left(1+k\right)}^{\ell }-1\right]}{Y}_2\left(\ell \right),$$

$$Y_2\left(\ell \right)=y_2\left(0\right)$$

$$+\gamma \sum _{s=0}^{\ell -1}\left\{{\beta }^{-2\left(s+1\right)}{\alpha }^{u\left[{\left(1+k\right)}^s-1\right]/k^2+\left(q/k\right)s}{\left[y_1\left(0\right)\right]}^{\left[u{\left(1+k\right)}^s+q\right]/k}\right\},$$

$$u\equiv kr-\left(1+k\right)q.$$

$$q=2k,r=2\left(1+k\right).$$

$$y_1\left(\ell \right)={\alpha }^{\left[{\left(1+k\right)}^{\ell }-1\right]/k}{\left[y_1\left(0\right)\right]}^{{\left(1+k\right)}^{\ell }},$$

$$y_2\left(\ell \right)$$

$$Y_2\left(\ell \right)$$

$$y_1=-\left(x_1+x_2\right),y_2=x_1{x}_2,$$

$$z^2+y_{1}z+y_2=\left(z-x_1\right)\left(z-x_2\right),$$

$${\left(x_n\right)}^2+y_1{x}_n+y_2=0,n=1,2,$$

$$x_n\left(\ell \right)=\left(1/2\right)\left\{-y_1\left(\ell \right)+{\left(-1\right)}^n{\left\{{\left[y_1\left(\ell \right)\right]}^2-4y_2\left(\ell \right)\right\}}^{1/2}\right\},n=1,2.$$

$${\left({\tilde{x}}_n\right)}^2+{\tilde{y}}_1{\tilde{x}}_n+{\tilde{y}}_2=0,n=1,2,$$

$${\tilde{x}}_n=(1/2\left[-{\tilde{y}}_1+{\left(-1\right)}^n{\tilde{\Delta }}_1\right],{\left({\tilde{\Delta }}_1\right)}^2={\left({\tilde{y}}_1\right)}^2-4{\tilde{y}}_2.$$

$${\tilde{x}}_n=\left(1/2\right)\left\{-\alpha {\left[-\left(x_1+x_2\right)\right]}^{k+1}+{\left(-1\right)}^n{\Delta }_1\right\},n=1,2,$$

Full text

465ClassSolvDiscrTimeDynSyst190313

Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

Francesco Calogeroa,b,1 and Farrin Payandeha,c,2

a Physics Department, University of Rome ”La Sapienza”, Rome, Italy

b INFN, Sezione di Roma 1

c Department of Physics, Payame Noor University (PNU), PO BOX 19395-3697 Tehran, Iran

1 francesco.calogero@roma1.infn.it, francesco.calogero@uniroma1.it

2 f_payandeh@pnu.ac.ir, farrinpayandeh@yahoo.com

Abstract

In this paper we identify certain peculiar systems of $2$ discrete-time evolution equations,

[TABLE]

which are algebraically solvable. Here $\ell$ is the ”discrete-time” independent variable taking integer values ($\ell=0,1,2,...$), $x_{n}\equiv x_{n}\left(\ell\right)$ are $2$ dependent variables, and $\tilde{x}_{n}\equiv x_{n}\left(\ell+1\right)$ are the corresponding $2$ updated variables. In a previous paper the $2$ functions $F^{\left(n\right)}\left(x_{1},x_{2}\right),$ $n=1,2,$ were defined as follows: $F^{\left(n\right)}\left(x_{1},x_{2}\right)=P_{2}\left(x_{n},x_{n+1}\right),$ $n=1,2~{}\mathop{\mathrm{m}od}[2],$ with $P_{2}\left(x_{1},x_{2}\right)$ a specific second-degree homogeneous polynomials in the $2$ (indistinguishable!) dependent variables $x_{1}\left(\ell\right)$ and $x_{2}\left(\ell\right)$. In the present paper we further clarify some aspects of that model and we present its extension to the case when $F^{\left(n\right)}\left(x_{1},x_{2}\right)=Q_{k}^{\left(n\right)}\left(x_{1},x_{2}\right),~{}n=1,2,$ with $Q_{k}^{\left(n\right)}\left(x_{1},x_{2}\right)$ a specific homogeneous function of arbitrary (integer) degree $k$ (hence a polynomial of degree $k$ when $k>0$) in the $2$ dependent variables $x_{1}\left(\ell\right)$ and $x_{2}\left(\ell\right)$.

1 Introduction and main results

The results reported in this paper are a nontrivial extension of those reported in [1], to which the interested reader is referred: (i) for a terse overview of an old (see [2] [3])—and recently substantially improved (see [4] [5] [6] [7] [8] [9] [10])—technique to identify solvable dynamical systems in continuous-time $t$ ; (ii) for an introduction to the extension of that approach to the case of discrete-time $\ell$ (see [11] [12] [13]); (iii) for a very terse review of previous results on analogous solvable discrete-time models (see [5]). To make the relevance of the present paper immediately clear we report already in this introductory section what we consider its main findings.

Notation 1-1. Hereafter $\ell=0,1,2,$… denotes the discrete-time independent variable; the dependent variables are $x_{n}\equiv x_{n}\left(\ell\right)$ (with $n=1,2$), and the notation $\tilde{x}_{n}\equiv x_{n}\left(\ell+1\right)$ indicates the once-updated values of these variables. We shall also use other dependent variables, for instance $y_{m}\equiv y_{m}\left(\ell\right)$ (with $m=1,2$) and then of course likewise $\tilde{y}_{m}\equiv y_{m}\left(\ell+1\right)$. Variables such as $x_{n}$ and $y_{m}$ are generally assumed to be complex numbers (this does not exclude that they might in some cases take only *real *values); note that while these quantities generally depend on the discrete-time variable $\ell$, only occasionally this is explicitly indicated. Parameters such as $a$, $b$, $\alpha,$ $\beta,$ $\gamma,$ $B_{n}$, $C_{j}$ (with $n=1,2$; $j=1,2,3$) are generally time-independent complex numbers, while $k,$ $q,$ $r$ are real integers; and indices such as $n$, $m,$ $j$ are of course positive integers (the values they may take shall be explicitly indicated or be quite clear from the context). The quantity $S$ denotes an arbitrarily assigned sign, $S=\pm$: note that generally the assignment of the sign $S$ may depend on the discrete-time $\ell$, $S\equiv S\left(\ell\right)$ (but of course it has the same assigned value $S\left(\ell\right)$ for each value of $\ell$). Finally: the convention is hereafter adopted according to which $\mathop{\textstyle\sum}\limits_{s=s_{-}}^{s_{+}}f\left(s\right)=0$ and $\mathop{\textstyle\prod}\limits_{s=s_{-}}^{s_{+}}f\left(s\right)=1$ whenever $s_{+}<s_{-}$. $\blacksquare$

Remark 1-1. In this paper the term solvable generally characterizes systems of discrete-time evolution equations the initial-values problem of which is explicitly solvable by algebraic operations. $\blacksquare$

The* *main result of [1] is to provide the explicit solution of the initial-values problem for the system of $2$ nonlinearly-coupled discrete-time evolution equations

[TABLE]

(Note here the notational changes with respect to [1]: implying $\alpha=2a,$ $\beta=2b$ and the explicit introduction of the arbitrary $\ell$-dependent sign $S\equiv S\left(\ell\right),$ which is indeed implicit in the formulas (2.11b) and (1.2a) of [1]).

Remark 1.2. The characteristic of the discrete-time evolution of this model is that, if the sign $S\left(\ell\right)$ is positive, $S\left(\ell\right)=+,$ then

[TABLE]

[TABLE]

the $2$ coefficients of which, $y_{1}\left(\ell\right)$ and $y_{2}\left(\ell\right),$ are unambiguously determined and indeed explicitly known in terms of the initial values $x_{1}\left(0\right)$ and $x_{2}\left(0\right)$.

This remark is introduced here as an introduction to the somewhat more peculiar phenomenon associated with the more general models considered in the present paper, see below. $\blacksquare$

The first main result of the present paper is to provide (in the following Section 2) the explicit solution of the initial-values problem for the following, more general, system of $2$ nonlinearly-coupled discrete-time evolution equations

[TABLE]

Remark 1.3. The restriction to integer values of the parameter $k$—and of the analogous exponents $q$ and $r,$ see below and Notation 1.1—is to make sure that the right-hand sides of the main discrete-time evolution equations we introduce and discuss in this paper are analytic, hence unambiguously defined, functions. $\blacksquare$

For $k=1$ the system (4) can of course be re-written as follows:

[TABLE]

For $k=-1$ the system (4) can of course be re-written as follows:

[TABLE]

Remark 1.4. The case with $k=1$ is sufficiently interesting to deserve this additional remark. Its equations of motion (5a) can of course be reformulated as follows:

[TABLE]

with easily obtainable expressions of the $6$ parameters $a_{nj}$ ($n=1,2;$ $j=1,2,3$) in terms of the $7$ arbitrary parameters $\alpha,$ $\beta$ and $B_{n}$, $C_{j}$ (with $n=1,2$; $j=1,2,3$; see (4)). But this does not imply that these $6$ parameters $a_{nj}$ can be arbitrarily assigned: the fact that the right-hand sides of the $2$ recursions (4a) feature a common zero—they both vanish when $B_{1}x_{1}+B_{2}x_{2}$ vanishes—is easily seen to imply that these $6$ parameters are constrained to satisfy (at least!) the following nonlinear relationship:

[TABLE]

(see, if need be, Remark 5.3 of Ref. [8]). $\blacksquare$

The second main result of the present paper is to provide the explicit solution of the initial-values problem for the system of $2$ nonlinearly-coupled discrete-time evolution equations

[TABLE]

Note that in this case—differently from those reported above—the discrete-time evolutions of the $2$ variables $x_{1}\left(\ell\right)$ and $x_{2}\left(\ell\right)$ are different; hence these dependent variables are no more related to each other by just an exchange of their identities. Yet these evolution equations, (9), still have a somewhat analogous property to those discussed above: for any given pair of initial data, $x_{1}\left(0\right)$ and $x_{2}\left(0\right)$, only $2$ different solutions, say the two pairs $x_{1}^{\left(+\right)}\left(\ell\right)$, $x_{2}^{(+)}\left(\ell\right)$ and $x_{1}^{\left(-\right)}\left(\ell\right)$, $x_{2}^{(-)}\left(\ell\right),$ emerge, not $2^{\ell}$ as it might instead be inferred due to the indeterminacy of the signs $S\left(\ell\right)$ appearing in these evolution equations (9) at every step of the discrete-time evolution they yield. This is demonstrated by the explicit solution of the initial-values problem for this model, as reported below; but the interested reader may readily understand the origin of this remarkable phenomenon by noting that a simple iteration of (9) entails the (double-step) formula

[TABLE]

indeed the quantity $S\left(\ell\right)S\left(\ell+1\right)$ is again just a sign, i. e. it can only take the $2$ values $+1$ or $-1$, as implied by the very definition of $S\left(\ell\right),$ see Notation 1.1. Hence this formula clearly shows that—starting from the initial values $x_{n}\left(0\right)$—also at the $\ell=2$ level (as at the $\ell=1$ level)—this evolution yields only two (not four!) alternative values for the pair $x_{1}\left(2\right),$ $x_{2}\left(2\right)$, say $x_{1}^{\left(+\right)}\left(2\right),$ $x_{2}^{\left(+\right)}\left(2\right)$ (corresponding to $S\left(0\right)S\left(1\right)=+$) respectively $x_{1}^{\left(-\right)}\left(2\right),$ $x_{2}^{\left(-\right)}\left(2\right)$ (corresponding to $S\left(0\right)S\left(1\right)=-$); and this phenomenology of course prevails—see below—at every subsequent level $\ell>2$ of the discrete-time evolution.

Additional results and proofs—including the explicit solutions of the initial-values problems for the $2$ systems of $2$ discrete-time evolution equations (4) respectively (9)—are provided in Section 2. A concluding Section 3 outlines additional developments.

2 Additional results and proofs

The starting point of our treatment in this Section 2 is the following algebraically solvable system of $2$ discrete-time evolution equations in the $2$ dependent variables $y_{1}\equiv y_{1}\left(\ell\right)$ and $y_{2}\equiv y_{2}\left(\ell\right)$:

[TABLE]

where the $6$ parameters $\alpha,\beta,$ $\gamma,$ $k,$ $q$, $r$ can be* a priori arbitrarily assigned* (see Notation 1.1; but see also below for eventual restrictions on these parameters).

The fact that the initial-values problem for this system of $2$ discrete-time evolution equations is solvable is demonstrated by exhibiting its solution:

[TABLE]

For reasons that shall be clear in the following, we are also interested in the solution of the system (11) in the particular case when the $2$ parameters $q$ and $r$ are expressed in terms of $k$ as follows:

[TABLE]

Note that this assignment implies $u=0$ (see (12d)). In this case the sum in the right-hand side of (12c) becomes a geometric sum, hence it can be performed explicitly; therefore in this special case the formulas (12) are replaced by the following, more explicit, versions:

[TABLE]

Our next task is to identify various systems—satisfied by $2$ new dependent variables $x_{1}\equiv x_{1}\left(\ell\right)$ and $x_{2}\equiv x_{2}\left(\ell\right)$—the solution of which can be identified via the solution of the system (11). To this end we set, to begin with,

[TABLE]

[TABLE]

[TABLE]

Remark 2.1. The $\pm$ sign $S\equiv S\left(\ell\right)$ in this definition (17b) of $\Delta_{1}$ might be considered pleonastic in view of the sign indeterminacy of the square-root in the right-hand side of this formula (17b). We did put it there as a reminder of the fact that, for every value of the discrete time $\ell$, the assignment of the labels $1$ or $2$ to the solutions of the second-degree evolution equation (17) is optional. Indeed the two variables $x_{1}\left(\ell\right)$ and $x_{2}\left(\ell\right)$—the discrete-time evolution of which is identified with the evolution of the $2$ zeros of the second-degree $\ell$-dependent (monic) polynomial (15b) the $2$ coefficients $y_{1}\left(\ell\right)$ and $y_{2}\left(\ell\right)$ of which evolve according to the solvable system (11)—should be considered indistinguishable. Note that this implies that this evolution equation is actually not quite deterministic; it is only deterministic for the couple of *indistinguishable *dependent variables $x_{1}\equiv x_{1}\left(\ell\right)$, $x_{2}\equiv x_{2}\left(\ell\right)$: a well-known phenomenon for this kind of evolution equations, as discussed above and in the past—see for instance [13] and Chapter 7 (”Discrete Time”) of the book [5] (in particular Remark 7.1.2 there). $\blacksquare$

Remark 2.2. Of course when $s$ is an integer the power $\left(-z\right)^{s}$ can be replaced by $z$ respectively $-z$ for $s$ even respectively odd. $\blacksquare$

The results obtained so far allow to formulate the following

Proposition 2.1. The solution of the initial-values problem for the system of discrete-time evolution equations (17) is provided—up to the limitations implied by Remark 2.1—by the $2$ zeros $x_{1}\equiv x_{1}\left(\ell\right)$ and $x_{2}\equiv x_{2}\left(\ell\right)$ of the polynomial (15b) (see (15d)), with its coefficients $y_{1}\equiv y_{1}\left(\ell\right)$ and $y_{2}\equiv y_{2}\left(\ell\right)$ given by the formulas (12) where of course (see (15a)) $y_{1}\left(0\right)=-x_{1}\left(0\right)-x_{2}\left(0\right)$ and $y_{2}\left(0\right)=x_{1}\left(0\right)x_{2}\left(0\right)$. $\blacksquare$

We believe that the interest—both theoretical and applicative—of the system (17) is modest, due to the appearance of a square root in the right-hand side of its equations of motion. Hence our next step is to restrict attention to the values identified by eq. (13). Indeed these assignments—beside allowing the more explicit solution of the system of recursions (11) characterizing the discrete-time evolution of the $2$ dependent variables $y_{1}\left(\ell\right)$ and $y_{2}\left(\ell\right),$ see (14)—also allow (remarkably!) to get rid of the square-root in the right-hand side of (17b), provided we moreover make the assignments

[TABLE]

obtaining thereby just the system of evolution equations (4). This allows us to prove our first main result, in the guise of the following

Proposition 2.2. The solution of the initial-values problem for the system of discrete-time evolution equations (4) is provided by the $2$ zeros $x_{1}\equiv x_{1}\left(\ell\right)$ and $x_{2}\equiv x_{2}\left(\ell\right)$ of the polynomial (15b) (see (15d)), with its coefficients $y_{1}\equiv y_{1}\left(\ell\right)$ and $y_{2}\equiv y_{2}\left(\ell\right)$ given by the formulas (14) (with the assignments (18)), where of course (see (15a)) $y_{1}\left(0\right)=-x_{1}\left(0\right)-x_{2}\left(0\right)$ and $y_{2}\left(0\right)=x_{1}\left(0\right)x_{2}\left(0\right)$.

Let us again emphasize that, for each value of $\ell$, the assignment of the labels $1$ or $2$ to the two zeros of the polynomial (15b) is optional. $\blacksquare$** **

This **Proposition 2.2 **corresponds to the first main result reported in Section 1.

Our next step is to modify the relationship among the variables $x_{n}\left(\ell\right)$ and $y_{n}\left(\ell\right)$ by replacing the second-degree (monic) polynomial (15b) with the following (monic) third-degree polynomial:

[TABLE]

Remark 2.3. It is for instance easy to check that the coefficient $y_{3}$ of the polynomial (19a) is given by the following formula in terms of the other $2$ coefficients $y_{1}$ and $y_{2}$:

[TABLE]

It is now convenient to write again these expressions of the $2$ zeros $x_{n}\equiv x_{n}\left(\ell\right),$ but with $\ell$ replaced by $\ell+1$:

[TABLE]

Our next step is to then assume again that the two coefficients $\tilde{y}_{1}\equiv y_{1}\left(\ell+1\right)$ and $\tilde{y}_{2}\equiv y_{2}\left(\ell+1\right)$ evolve according to the solvable discrete-time system (11). By proceeding in close analogy with the previous treatment—i. e., by replacing in the right-hand sides of (21) the variables $\tilde{y}_{1}$ and $\tilde{y}_{2}$ via the evolution equations (11) and then in the right-hand sides of the resulting equations $y_{1}$ and $y_{2}$ via (19b)—we thereby obtain the following

Proposition 2.3. The solution of the initial-values problem for the following system of discrete-time evolution equations

[TABLE]

Note that this system features the same kind of $2$-fold ambiguity as discussed in the previous Section 1 (see after eq. (9)). $\blacksquare$

However, a ”defect” of this solvable system is the appearance in the right-hand side of its discrete-time equations of motion (22) of a square root; but this ”defect” can now be eliminated by restricting the parameters $q$ and $r$ to satisfy the condition (13)—just the same condition that allows to replace the solution (12) with the more explicit solution (14)—and by moreover replacing the assignments (18) with the following assignments

[TABLE]

It is indeed easily seen that there thereby holds the following

Proposition 2.4. The solution of the initial-values problem for the system of discrete-time evolution equations (9) is provided by the $2$ distinct zeros $x_{1}\equiv x_{1}\left(\ell\right)$ and $x_{2}\equiv x_{2}\left(\ell\right)$ of the polynomial (19)—i. e., by the formulas (20b)—with the coefficients $y_{1}\equiv y_{1}\left(\ell\right)$ and $y_{2}\equiv y_{2}\left(\ell\right)$ given by the formulas (14) where of course now (see (19b)) $y_{1}\left(0\right)=-2x_{1}\left(0\right)-x_{2}\left(0\right)$ and $y_{2}\left(0\right)=x_{1}\left(0\right)\left[x_{1}\left(0\right)+2x_{2}\left(0\right)\right]$.

Let us again emphasize that, for each value of the discrete-time $\ell$, this prescription yields $2$ different solutions, say the $2$ different pairs $x_{1}^{\left(+\right)}\left(\ell\right),$ $x_{2}^{\left(+\right)}\left(\ell\right)$ and $x_{1}^{\left(-\right)}\left(\ell\right),$ $x_{2}^{\left(-\right)}\left(\ell\right)$. $\blacksquare$

This **Proposition 2.4 **corresponds to the second main result reported in Section 1.

3 Additional developments

An important issue is the possibility to generalize the algebraically solvable systems treated in the previous Section 2—which feature the $2$ arbitrary (possibly *complex) *parameters $a$ and $b$—to more general analogous models involving more free parameters. Following the treatment given in [1], let us outline how this can be done for the system (9).

The procedure is to introduce the simple invertible change of dependent variables

[TABLE]

It is then a matter of simple algebra to obtain the—of course algebraically solvable—evolution equations satisfied by the dependent variables $z_{1}\equiv z_{1}\left(\ell\right)$ and $z_{2}\equiv z_{2}\left(\ell\right)$:

[TABLE]

Let us also display these equations in the—possibly more relevant to applicative contexts—special cases with $k=\pm 1$.

For $k=1$:

[TABLE]

For $k=-1$:

[TABLE]

An alternative generalization of this approach is based on the replacement of the relations (15a) and (19b) and their generalization via (24) with the following more general relations:

[TABLE]

Note that these relations involve the $5$ a priori arbitrary parameters $B_{n}$ and $C_{j}$ ($n=1,2$; $j=1,2,3$), and that they are easily inverted:

[TABLE]

[TABLE]

Starting from these formulas, and proceeding in close analogy with the treatment provided above—which involve of course the assumption that the quantities $z_{n}$ and $y_{n}$ ($n=1,2$) are $\ell$-dependent while the parameters $B_{n}$ and $C_{j}$ ($n=1,2$; $j=1,2,3$) are $\ell$-independent, and moreover that the quantities $y_{n}$ evolve according to the solvable discrete-time evolution equations (11)—one arrives at the following—of course, also *solvable—*evolution equations for the quantities $z_{n}\equiv z_{n}\left(\ell\right)$:

[TABLE]

The special cases of these equations corresponding to the assignments $k=1$ and $k=-1$ are also worth explicit display:

[TABLE]

Assigning solvable evolutions to $y_{1}$ and $y_{3}$ or $y_{2}$ and $y_{3}$ (rather than to $y_{1}$ and $y_{2}$; in the case of the third-degree polynomial (19a)) are possible further developments, but we postpone the relevant treatments to future papers.

4 Acknowledgements

Both authors like to thank Piotr Grinevich, Paolo Santini and Nadezda Zolnikova for very useful suggestions. FP likes to thank the Physics Department of the University of Rome ”La Sapienza” for the hospitality from February 2018 to April 2019 (during her sabbatical), when the results reported in this paper were obtained.