463SolvDiscrTimeDynSyst190203

Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via
2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-
Hand Sides

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<EMAIL_ADDRESS><EMAIL_ADDRESS>
2<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

The evolution equations mentioned in the title of this paper read as follows:

$\tilde{x}_{n}=P^{\left(n\right)}\left(x_{1},x_{2}\right)~{},~{}~{}~{}n=1,2~{},$

where $\ell$ is the ”discrete-time” independent variable taking integer values
($\ell=0,1,2,...$), $x_{n}\equiv x_{n}\left(\ell\right)$ are the $2$ dependent
variables, $\tilde{x}_{n}\equiv x_{n}\left(\ell+1\right)$, and the $2$
functions $P^{\left(n\right)}\left(x_{1},x_{2}\right),~{}n=1,2,$ are $2$
polynomials in the $2$ dependent variables $x_{1}\left(\ell\right)$ and
$x_{2}\left(\ell\right)$. The results reported in this paper have been
obtained by an appropriate modification of a recently introduced technique to
obtain analogous results in continuous-time $t$—in which case $x_{n}\equiv
x_{n}\left(t\right)$ and the above recursion relations are replaced by first-
order ODEs. Their potential interest is due to the relevance of this kind of
evolution equations in various applicative contexts.

##  1 Introduction

In this introductory Section 1, after providing some notational prescriptions,
we tersely review previous relevant findings.

Notation 1-1. Hereafter $\ell=0,1,2,...$denotes the discrete-time independent
variable; the dependent variables are $x_{n}\equiv x_{n}\left(\ell\right)$
(generally 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
$y_{m}\equiv y_{m}\left(\ell\right),$ and then of course likewise
$\tilde{y}_{m}\equiv y_{m}\left(\ell+1\right)$. All variables such as $x$,
$y$, $z$, (generally equipped with indices) are assumed to be complex numbers,
unless otherwise indicated; it shall generally be clear from the context which
of these and other quantities depend on time (as occasionally—but not
always—explicitly indicated); parameters such as $a$, $\alpha,$ $\beta,$
$\gamma,$ $A,$ etc. (often equipped with indices) are generally time-
independent complex numbers; and indices such as $n$, $m$, $j$ are generally
positive integers (the values they may take shall be explicitly indicated or
quite clear from the context). $\blacksquare$

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

In the following Subsection 1.1 we tersely review—mainly via quotations (with
minor adjustments) from a recent paper of ours [1]—a recent approach to
identify solvable dynamical systems in continuous-time $t$, as introduction to
the extension of (some of) these results to the case of discrete-time $\ell$,
which is the topic of the present paper. Previous results on solvable
discrete-time models are tersely reviewed in the subsequent Subsection 1.2.
Our main findings are reported in Section 2 (also based on the results
reported in Appendix A). A concluding Section 3 outlines tersely possible
additional developments.

###  1.1 Review of an analogous approach in the continuous-time context

”Long time ago the idea has been introduced to identify dynamical systems
(evolving in continuous-time $t$) which are solvable by using as a tool the
relations between the time evolutions of the coefficients and the zeros of a
generic time-dependent polynomial [2]. The basic idea of this approach is to
relate the time-evolution of the $N$ zeros $x_{n}\left(t\right)$ of a generic
time-dependent polynomial $p_{N}\left(z;t\right)$ of degree $N$ in its
argument $z,$

$p_{N}\left(z;t\right)=z^{N}+\sum_{m=1}^{N}\left[y_{m}\left(t\right)z^{N-m}\right]=\prod_{n=1}^{N}\left[z-x_{n}\left(t\right)\right]~{},$
(1a) to the time-evolution of its $N$ coefficients $y_{m}\left(t\right)$.
Indeed, if the time evolution of the $N$ coefficients $y_{m}\left(t\right)$ is
determined by a system of ODEs which is itself solvable, then the
corresponding time-evolution of the $N$ zeros $x_{n}\left(t\right)$ is also
solvable, via the following $3$ steps: (i) given the initial values
$x_{n}\left(0\right),$ the corresponding initial values $y_{m}\left(0\right)$
can be obtained from the explicit formulas—expressing the $N$ coefficients
$y_{m}\left(t\right)$ of the polynomial (1a) in terms of its $N$ zeros
$x_{n}\left(t\right)$—reading (for all time, hence in particular at $t=0$)
$y_{m}\left(t\right)=\left(-1\right)^{m}\sum_{1\leq n_{1}<n_{2}<...<n_{m}\leq
N}^{N}\left\\{\prod_{\ell=1}^{M}\left[x_{n_{\ell}}\left(t\right)\right]\right\\}~{},~{}~{}~{}m=1,2,...,N~{};$
(1b) (ii) from the $N$ values $y_{m}\left(0\right)$ thereby obtained, the $N$
values $y_{m}\left(t\right)$ are then evaluated via the—assumedly
solvable—system of ODEs satisfied by the $N$ coefficients
$y_{m}\left(t\right)$; (iii) the $N$ values $x_{n}\left(t\right)$—i. e., the
$N$ solutions of the dynamical system satisfied by the $N$ variables
$x_{n}\left(t\right)$—are then determined as the $N$ zeros of the polynomial,
see (1a), itself known at time $t$ in terms of its $N$ coefficients
$y_{m}\left(t\right)$ (the computation of the zeros of a known polynomial
being an algebraic operation; of course generally explicitly performable only
for polynomials of degree $N\leq 4$)…

The viability of this technique to identify solvable dynamical systems depends
of course on the availability of an explicit method to relate the time-
evolution of the $N$ zeros of a polynomial to the corresponding time-evolution
of its $N$ coefficients. Such a method was indeed provided in [2], opening the
way to the identification of a vast class of algebraically solvable dynamical
systems (see also, for instance, [3] and references therein); but that
approach was essentially restricted to the consideration of linear time
evolutions of the coefficients $y_{m}\left(t\right)$.

A development allowing to lift this quite strong restriction emerged
relatively recently [4], by noticing the validity of the identity

$\dot{x}_{n}=-\left[\prod_{\ell=1,~{}\ell\neq
n}^{N}\left(x_{n}-x_{\ell}\right)\right]^{-1}\sum_{m=1}^{N}\left[\dot{y}_{m}\left(x_{n}\right)^{N-m}\right]$
(2)

which provides a convenient explicit relationship among the time evolutions of
the $N$ zeros $x_{n}\left(t\right)$ and the $N$ coefficients
$y_{m}\left(t\right)$ of the generic polynomial (1a). This allowed a major
enlargement of the class of algebraically solvable dynamical systems
identifiable via this approach: for many examples see [5] and references
therein…

A new twist of this approach was then provided by its extension to nongeneric
polynomials featuring—for all time—multiple zeros. The first step in this
direction focussed on time-dependent polynomials featuring for all time a
single double zero [6]; and subsequently significant progress has been made to
treat the case of polynomials featuring a single zero of arbitrary
multiplicity [7]. A convenient method was then provided which is suitable to
treat the most general case of polynomials featuring an arbitrary number of
zeros each of which features an arbitrary multiplicity. While all these
developments might appear to mimic scholastic exercises analogous to the
discussion among medieval scholars of how many angels might dance
simultaneously on the tip of a needle, they do indeed provide new tools to
identify new dynamical systems featuring interesting time evolutions
(including systems displaying remarkable behaviors such as isochrony or
asymptotic isochrony: see for instance [6] [7]); dynamical systems
which—besides their intrinsic mathematical interest—are quite likely to play
significant roles in applicative contexts…

We then focused on another twist of this approach to identify new solvable
dynamical systems which was introduced quite recently [8]. It is again based
on the relations among the time-evolution of the coefficients and the zeros of
time-dependent polynomials [4] [5] with multiple roots (see [6], [7] and
above); restricting moreover attention to such polynomials featuring only $2$
zeros. Again, this might seem such a strong limitation to justify the doubt
that the results thereby obtained be of much interest. But the effect of this
restriction is to open the possibility to identify algebraically solvable
dynamical models characterized by the following systems of $2$ ODEs,

$\dot{x}_{n}=P^{\left(n\right)}\left(x_{1},x_{2}\right)~{},~{}~{}~{}n=1,2~{},$
(3)

with $P^{\left(n\right)}\left(x_{1},x_{2}\right)$ $2$ polynomials in the $2$
dependent variables $x_{1}\left(t\right)$ and $x_{2}\left(t\right)$; hence
systems of considerable interest, both from a theoretical and an applicative
point of view (see [8] and references quoted there).” [1]

This completes our review—via a long quotation from a previous paper—of recent
developments concerning certain classes of standard dynamical systems in
continuous time. In the present paper—after tersely reviewing, in the
following Subsection 1.2, some past results in the discrete-time context—we
focus on the derivation in such a context of analogous results to some of
those reported in the continuous-time context in [1].

###  1.2 Review of somewhat analogous past findings in the discrete-time
context

Somewhat analogous results to those reviewed in the first part of the previous
Subsection 1.1 have been developed over time in the context of discrete-time
evolutions, by focussing on the evolution of the zeros of generic monic
polynomials the coefficients of which evolve in a solvable manner in discrete
time.

The new results reported below consists essential of extensions to the
discrete-time context of the results outlined in the second part of the
preceding Subsection 1.1. Note however that here and below we actually
dispense from a general discussion of the evolution of the zeros of a
polynomial the coefficients of which evolve in discrete time in a solvable
manner, both in the case of generic monic polynomials (as treated in Chapter 7
of [5]) and in the case of the special polynomials of higher degree than $2$
which nevertheless feature for all time only $2$ (of course multiple) zeros
(as treated in [8] [1]); below we rather employ the simpler
technique—described in the following Section 2—to identify solvable nonlinear
evolution equations that emerged from that approach and which actually
subtends most of the explicit findings reported in [1]. Hence from the
previous findings for discrete-time evolutions—see [9] and Chapter 7
(”Discrete time”) of [5]—we only use below the following discrete-time
equivalent of the identity (2) (originating from the polynomial (1) with $t$
replaced by $\ell$),

$\prod\limits_{j=1}^{N}\left(\tilde{x}_{n}-x_{j}\right)+\sum_{m=1}^{N}\left[\left(\tilde{y}_{m}-y_{m}\right)\left(\tilde{x}_{n}\right)^{N-m}\right]=0~{},$
(4)

hence, for the $N=2$ case,

$\left(\tilde{x}_{n}-x_{1}\right)\left(\tilde{x}_{n}-x_{2}\right)+\left(\tilde{y}_{1}-y_{1}\right)\tilde{x}_{n}+\tilde{y}_{2}-y_{2}=0~{},~{}~{}~{}n=1,2$
(5a) of course with (see (1b))
$y_{1}\left(\ell\right)=-\left[x_{1}\left(\ell\right)+x_{2}\left(\ell\right)\right]~{},~{}~{}~{}y_{2}\left(\ell\right)=x_{1}\left(\ell\right)x_{2}\left(\ell\right)~{}.$
(5b)

##  2 A solvable system of 2 nonlinearly coupled evolution equations in
discrete-time satisfied by 2 dependent variables

In this Section 2 we present our main results, consisting in the
identification of a solvable systems of $2$ nonlinearly-coupled discrete-time
evolution equations belonging to the class

$\tilde{x}_{n}=P^{\left(n\right)}\left(x_{1},x_{2}\right)~{},~{}~{}~{}n=1,2~{},$
(6)

where the $2$ functions $P^{\left(n\right)}\left(x_{1},x_{2}\right)$ are $2$
appropriately identified polynomials in the variables $x_{1}\left(\ell\right)$
and $x_{2}\left(\ell\right)$; a system which is clearly the natural
generalization to discrete time of the continuous-time system (3). In
particular we demonstrate the solvable character of the following dynamical
system:

$\tilde{z}_{n}=a_{n1}\left(z_{1}\right)^{2}+a_{n2}\left(z_{2}\right)^{2}+a_{n3}z_{1}z_{2}~{},~{}~{}~{}n=1,2~{},$
(7)

with the $6$ parameters $a_{nj}$ ($n=1,2,$ $j=1,2,3$) explicitly given by $6$
algebraic expressions in terms of $6$ arbitrary parameters.

Let the $2$ dependent variables $y_{1}\left(\ell\right)$ and
$y_{2}\left(\ell\right)$ evolve in discrete time according to the following
discrete-time evolution equations (the solvability of which is demonstrated in
Appendix A):

$\tilde{y}_{1}=\alpha\left(y_{1}\right)^{2}~{},~{}~{}~{}\tilde{y}_{2}=\beta^{2}\left(y_{1}\right)^{2}y_{2}+\gamma\left(y_{1}\right)^{4}~{};$
(8)

and assume again that the $2$ variables $y_{1}\left(\ell\right)$ and
$y_{2}\left(\ell\right)$ are related to the $2$ variables
$x_{1}\left(\ell\right)$ and $x_{2}\left(\ell\right)$ as follows (see (5b)):

$y_{1}\left(\ell\right)=-\left[x_{1}\left(\ell\right)+x_{2}\left(\ell\right)\right]~{},~{}~{}~{}y_{2}\left(\ell\right)=x_{1}\left(\ell\right)x_{2}\left(\ell\right)~{}.$
(9)

Remark 2-1. Let us re-emphasize that, if the discrete-time evolution of the
$2$ variables $y_{1}\left(\ell\right)$ and $y_{2}\left(\ell\right)$ is
solvable, then the discrete-time evolution of the $2$ variables
$x_{1}\left(\ell\right)$ and $x_{2}\left(\ell\right)$ is as well solvable,
because the ansatz (9) can be inverted via an algebraic operation, indeed
quite explicitly, since it clearly implies that $x_{1}\left(\ell\right)$ and
$x_{2}\left(\ell\right)$ are the $2$ roots of the following monic polynomial
of degree $2$:

$p_{2}\left(z\right)=z^{2}+y_{1}z+y_{2}=\left(z-x_{1}\right)\left(z-x_{2}\right)~{}.~{}~{}~{}\blacksquare$
(10)

It is then a matter of trivial algebra—either via the formulas (5) or directly
from (8) and (9)—to derive the following system of two discrete-time evolution
equations satisfied by the $2$ dependent variables $x_{1}\left(\ell\right)$
and $x_{2}\left(\ell\right)$:

$\left(\tilde{x}_{n}\right)^{2}+\alpha\left(x_{1}+x_{2}\right)^{2}\tilde{x}_{n}+\left(x_{1}+x_{2}\right)^{2}\left[\beta^{2}x_{1}x_{2}+\gamma\left(x_{1}+x_{2}\right)^{2}\right]=0~{},$
(11a) implying
$\tilde{x}_{n}=-\frac{\alpha\left(x_{1}+x_{2}\right)^{2}+\left(-1\right)^{n}\Delta}{2}~{},$
(11b) with
$\Delta^{2}=\left(x_{1}+x_{2}\right)^{2}\left[\left(\alpha^{2}-4\gamma\right)\left(x_{1}+x_{2}\right)^{2}-4\beta^{2}x_{1}x_{2}\right]~{}.$
(11c)

Assume now that

$\gamma=\frac{\alpha^{2}-\beta^{2}}{4}~{},$ (12a) so that the right-hand side
of (11c) become an exact square implying
$\Delta=\pm\beta\left[\left(x_{1}\right)^{2}-\left(x_{2}\right)^{2}\right]~{}.$
(12b) Then clearly
$\tilde{x}_{n}=-\frac{\alpha\left(x_{1}+x_{2}\right)^{2}+\left(-1\right)^{n}\beta\left[\left(x_{1}\right)^{2}-\left(x_{2}\right)^{2}\right]}{2}~{},~{}~{}~{}n=1,2~{},$
(13a) namely
$\tilde{x}_{n}=a_{1}^{\left(n\right)}\left(x_{1}\right)^{2}+a_{2}^{\left(n\right)}\left(x_{2}\right)^{2}+a_{3}^{\left(n\right)}x_{1}x_{2}~{},~{}~{}~{}n=1,2~{},$
(13b) with
$a_{1}^{\left(n\right)}=-\frac{\alpha+\left(-1\right)^{n}\beta}{2}~{},~{}~{}~{}a_{2}^{\left(n\right)}=-\frac{\alpha+\left(-1\right)^{n}\beta}{2}~{},~{}~{}~{}a_{3}^{\left(n\right)}=-\alpha~{}.$
(13c)

We have thereby identified a simple solvable system of type (6), featuring in
its right-hand side $2$ homogeneous polynomials of second degree in the $2$
variables $x_{1}\left(\ell\right)$ and $x_{2}\left(\ell\right)$, the $6$
coefficients $a_{j}^{\left(n\right)}$ ($n=1,2,~{}~{}j=1,2,3$) of which depend
on the $2$ a priori arbitrary parameters $\alpha$ and $\beta$, see (13c).

From this system an analogous system featuring more arbitrary parameters can
be identified via the simple trick of introducing $2$ new dependent variables,
$z_{1}\left(\ell\right)$ and $z_{2}\left(\ell\right),$ related linearly to the
$2$ variables $x_{1}\left(\ell\right)$ and $x_{2}\left(\ell\right)$:

$z_{1}=A_{11}x_{1}+A_{12}x_{2}~{},~{}~{}~{}z_{2}=A_{21}x_{1}+A_{22}x_{2}~{},$
(14a)
$x_{1}=\left(A_{22}z_{1}-A_{12}z_{2}\right)/D~{},~{}~{}~{}x_{2}=\left(-A_{21}z_{1}+A_{11}z_{2}\right)/D~{},$
(14b) $D=A_{11}A_{22}-A_{12}A_{21}~{}.$ (14c) It is easily seen that the new
system is then just the system (7), with the $6$ parameters $a_{nj}$ ($n=1,2,$
$j=1,2,3$) explicitly expressed as follows in terms of the $4$ arbitrary
parameters $A_{nm}$ ($n=1,2,$ $m=1,2$) and the $2$ arbitrary parameters
$\alpha$ and $\beta$ (see (13c)):  $\displaystyle a_{n1}$ $\displaystyle=$
$\displaystyle
D^{-2}\left[\left(A_{22}\right)^{2}\left(A_{n1}a_{1}^{\left(1\right)}+A_{n2}a_{1}^{\left(2\right)}\right)+\left(A_{21}\right)^{2}\left(A_{n1}a_{2}^{\left(1\right)}+A_{n2}a_{2}^{\left(2\right)}\right)\right.$
(15a)
$\displaystyle\left.-A_{22}A_{21}\left(A_{n1}a_{3}^{\left(1\right)}+A_{n2}a_{3}^{\left(2\right)}\right)\right]~{},~{}~{}~{}n=1,2~{},$

$\displaystyle a_{n2}$ $\displaystyle=$ $\displaystyle
D^{-2}\left[\left(A_{12}\right)^{2}\left(A_{n1}a_{1}^{\left(1\right)}+A_{n2}a_{1}^{\left(2\right)}\right)+\left(A_{11}\right)^{2}\left(A_{n1}a_{2}^{\left(1\right)}+A_{n2}a_{2}^{\left(2\right)}\right)\right.$
(15b)
$\displaystyle\left.-A_{11}A_{12}\left(A_{n1}a_{3}^{\left(1\right)}+A_{n2}a_{3}^{\left(2\right)}\right)\right]~{},~{}~{}~{}n=1,2~{},$
$\displaystyle a_{n3}$ $\displaystyle=$ $\displaystyle
D^{-2}\left[-2A_{12}A_{22}\left(A_{n1}a_{1}^{\left(1\right)}+A_{n2}a_{1}^{\left(2\right)}\right)-2A_{21}A_{11}\left(A_{n1}a_{2}^{\left(1\right)}+A_{n2}a_{2}^{\left(2\right)}\right)\right.$
(15c)
$\displaystyle\left.+\left(A_{11}A_{22}+A_{12}A_{21}\right)\left(A_{n1}a_{3}^{\left(1\right)}+A_{n2}a_{3}^{\left(2\right)}\right)\right]~{},~{}~{}~{}n=1,2~{}.$

For the inversion of these transformations—i. e., the issue of expressing the
$4$ parameters $A_{nm}$ ($n=1,2;$ $m=1,2$) and the $2$ parameters $\alpha,$
$\beta$ (see (13c)) in terms of the $6$ parameters $a_{n\ell}$ ($n=1,2;$
$\ell=1,2,3$)—we refer to the analogous discussion in [1].

##  3 Outlook

In this final Section 3 we outline tersely possible future developments of the
findings reported above.

The findings reported in this paper extend to evolutions in discrete-time only
some of the findings for evolutions in continuous-time reported in [8] and [1]
and tersely reviewed above (in Section 1). It is therefore quite natural to
envisage an extension from the continuous-time context to the discrete-time
context of other results reported in [8] and [1].

A different research line might be directed towards applications of the
solvable discrete-time evolution equation (7), including its generalization
via an assigned shift of the dependent variables $z_{n}\left(\ell\right)$, say

$z_{n}\left(\ell\right)=w_{n}\left(\ell\right)+f_{n}\left(\ell\right)~{},~{}~{}~{}n=1,2~{},$
(16a) with $f_{n}\left(\ell\right)$ two arbitrarily assigned functions of the
discrete-time $\ell,$ implying that the new dependent-variable
$w_{n}\left(\ell\right)$ satisfies the (of course still solvable) discrete-
time evolution equation $\displaystyle\tilde{w}_{n}$ $\displaystyle=$
$\displaystyle
a_{n1}\left(w_{1}\right)^{2}+a_{n2}\left(w_{2}\right)^{2}+a_{n3}w_{1}w_{2}+g_{n1}w_{1}+g_{n2}w_{2}+h_{n}~{},$
$\displaystyle g_{n1}\left(\ell\right)$ $\displaystyle\equiv$ $\displaystyle
2a_{n1}f_{1}\left(\ell\right)+a_{n3}f_{2}\left(\ell\right)~{},~{}~{}~{}g_{n2}\left(\ell\right)\equiv
2a_{n2}f_{2}\left(\ell\right)+a_{n3}f_{1}\left(\ell\right)~{},$ $\displaystyle
h_{n}\left(\ell\right)$ $\displaystyle\equiv$ $\displaystyle
a_{n1}\left[f_{1}\left(\ell\right)\right]^{2}+a_{n2}\left[f_{2}\left(\ell\right)\right]^{2}+a_{n3}f_{1}\left(\ell\right)f_{2}\left(\ell\right)-f_{n}\left(\ell+1\right)~{},$
$\displaystyle n$ $\displaystyle=$ $\displaystyle 1,2~{}.$ (16b)

##  4 Acknowledgements

FP likes to thank the Physics Department of the University of Rome ”La
Sapienza” for the hospitality from April to November 2018 (during her
sabbatical), when the results reported in this paper were obtained.

##  5 Appendix A: A useful class of solvable systems of $2$ nonlinear
discrete-time evolution equations for the $2$ variables
$y_{m}\left(\ell\right)$

The system (8) of discrete-time evolution equations discussed in this Appendix
A is too simple to justify considering its solution as a new finding; its
solution is reported here because of its role in solving the novel, more
interesting, discrete-time model discussed above. The solution of the initial-
value problem of the first of the $2$ discrete-time evolution equations (8),

$\tilde{y}_{1}=\alpha\left(y_{1}\right)^{2}~{},$ (17a) is an easy task:
$y_{1}\left(\ell\right)=\alpha^{-1}\left[\alpha
y_{1}\left(0\right)\right]^{2^{\ell}}~{},~{}~{}~{}\ell=0,1,2,...~{}.$ (17b)

To solve the initial-value problem for the second of the $2$ discrete-time
evolution equations (8),

$\tilde{y}_{2}=\beta^{2}\left(y_{1}\right)^{2}y_{2}+\gamma\left(y_{1}\right)^{4}~{},$
(18)

it is convenient to introduce the ansatz

$y_{2}\left(\ell\right)=\left\\{\beta^{2\ell}\mathop{\textstyle\prod}\limits_{s=0}^{\ell-1}\left[y_{1}\left(s\right)\right]^{2}\right\\}Y\left(\ell\right)~{},$
(19a) implying $Y\left(0\right)=y_{2}\left(0\right)~{}.$ (19b)

Remark A-1. We always use the standard convention according to which, if
$s_{1}>s_{2}$,

$\sum_{s=s_{1}}^{s_{2}}f\left(s\right)=0~{},~{}~{}~{}\mathop{\textstyle\prod}\limits_{s=s_{1}}^{s_{2}}f\left(s\right)=1$
(20)

for any arbitrary function $f\left(s\right)$ of the discrete-time variable
$s$. $\blacksquare$

It is then easily seen that (18) implies

$Y\left(\ell+1\right)=Y\left(\ell\right)+F\left(\ell\right)~{},$ (21a) with
$F\left(\ell\right)=\gamma\left[y_{1}\left(\ell\right)\right]^{4}\beta^{-2\left(\ell+1\right)}\mathop{\textstyle\prod}\limits_{s=0}^{\ell}\left[y_{1}\left(s\right)\right]^{-2}~{},$
(21b) implying, via (17b),
$F\left(\ell\right)=\gamma\alpha^{-4}\left(\alpha/\beta\right)^{2\left(\ell+1\right)}\left[\alpha
y_{1}\left(0\right)\right]^{2}~{}.$ (21c)

Hence in conclusion, from (21) with (19b),

$Y\left(\ell\right)=y_{2}\left(0\right)+\gamma\left(\alpha\beta\right)^{-2}\left[\frac{\left(\alpha/\beta\right)^{2\ell}-1}{\left(\alpha/\beta\right)^{2}-1}\right]\left[\alpha
y_{1}\left(0\right)\right]^{2}~{},$ (22a) hence, via (19a), $\displaystyle
y_{2}\left(\ell\right)=\left(\beta/\alpha\right)^{2\ell}\left[\alpha
y_{1}\left(0\right)\right]^{2^{\ell+1}-2}\left\\{y_{2}\left(0\right)\right.$
$\displaystyle\left.+\gamma\left(\alpha\beta\right)^{-2}\left[\frac{\left(\alpha/\beta\right)^{2\ell}-1}{\left(\alpha/\beta\right)^{2}-1}\right]\left[\alpha
y_{1}\left(0\right)\right]^{2}\right\\}~{}.$ (22b)

The $2$ formulas (17b) and (22b) provide the explicit solution of the initial-
values problem of the discrete-time evolution (8). Of course in (17b), to make
this solution applicable to the final findings reported in Section 2, the
assignment $\gamma=\left(\alpha^{2}-\beta^{2}\right)/4$ must be made, see
(12a).

## References

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