The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations

1.   Introduction

In the Classroom Notes section of American Mathematical Monthly, Louis Brand ^[5] in 1955 presented an analysis of the so called Riccati real difference equation

where $c\neq 0$ and $ad-bc\neq 0.$ By setting

and $\beta: = \frac{d}{c^2}$, Brand transformed Eq (1.1) into

where

Next, putting

from Eq (1.2) the linear difference equation

is obtained. If $k_1, k_2$ are the roots of its characteristic quadratic $k^2-\alpha k+\beta = 0$, the general solution is determined and, finally, among others, it is shown the following results:

ⅰ) If $\alpha\neq 0$, $\alpha^2 > 4\beta$ and $y_1\neq k_1$, then $y_{n+1}\to k_2,$ when $|k_2| > |k_1|.$

ⅱ) If $\alpha^2 = 4\beta$ and $k_1 = k_2( = \frac{1}{2}\alpha)$, then $y_n\to k_1$.

ⅲ) If $\alpha^2 < 4\beta,$ then $k_1, k_2$ are complex conjugate and let $\theta: = \cos^{-1}\big(\frac{\alpha}{2\sqrt{\beta}}\big)$. If $\theta/\pi$ is rational, then the sequence $(y_n)$ is periodic. If $\theta/\pi$ is irrational, then the set $\{y_n:\; n = 1, 2, \cdots\}$ is dense in the real line.

In the complex case, Eq (1.1) can be written in the form

where $f(x): = \frac{ax+b}{cx+d}.$ This is the Möbius transformation in the complex plane which include dilations, rotations, translations, and complex inversion.

We would, also, like to refer to the work done by Elaydi and Sacker ^[10], where for Eq (1.1), they give the same convergence results as by Brand. Moreover, discuss the equation where the coefficients are periodic having the same period. A special case which they discuss is the so called Beverton-Holt stock-recruitment equation

which has wide applications in population dynamics ^[4]. For $\mu\in(0, 1)$, the point $0$ is globally asymptotically stable, whereas, for $\mu > 1$, the fixed point $K$ is globally asymptotically stable. Furthermore, they investigated the existence of a periodic solution of (1.3), when the parameter $K$ is replaced by the term $K_n$ of a periodic sequence. Notice that this form of equation has been investigated by Cushing and Henson in ^[9]. See, also, ^[8,25].

For $b = 0$ and $c = 1$, Eq (1.1) describes a population model and it is refereed in ^[12], p. 43.

In ^[1], the global asymptotic stability of the difference equation

where $A, B$ are nonnegative real numbers and $C, D > 0.$ In ^[15] Kocić, Ladas and Rondrigues studied the $k+1$-order rational difference equation of the form

where $a, A, b$ are nonnegative real numbers and $k$ is a positive integer. They proved that the positive equilibrium point of Eq (1.4) is globally asymptotically stable. In addition, they showed that all positive solutions oscillate about the positive equilibrium point.

The first thing we shall discuss is the forbidden set $\mathbb{F}.$ For Eq (1.1) in the real case, the forbidden set consists of all initial values $x_0$ which lead to a solution $(x_n)$ such that, for some index $n$, the denominator $cx_n+d$ becomes zero. This means that the term $x_{n+1}$ becomes infinity and so it is not a finite number. An interesting study of the forbidden set of Eq (1.1), in the real case, is given by Kulenović and Ladas in their informative book ^[16], pp. 17–24. Also, the authors describe in detail the long and short term behavior of the solutions when the initial value does not belong to the forbidden set. The book provides a very rich reference list. The same equation, or some of its variants, is studied by many authors, e.g. ^[2], p. 122, ^{[20,21,22,26]}. See, also the references therein.

In this paper we are dealing with the circular system of Riccati difference equations of the form

with complex coefficients. Circular systems of rational real difference equations are investigated in a great number of papers, such as ^{[3,6,11,14,17,23,24]} and in the references therein. Notice that such systems were first presented by Laplace in 1773 (^[18], p. 140), in his attempt to present the solution of a differential equation in a series form.

By the term solution of Eq (1.5) we mean a (finite, or infinite) sequence of complex numbers satisfying it by choosing suitable initial values. As we shall see, system (1.5) can be written in the form

where the complex numbers $R_j, \; j = 1, 2, \cdots, k$ are the so called Riccati numbers for the system, defined by

In this new form the forbidden set consists of $k$-tuples, $\mathbb{F}_j$, $j = 1, 2, \cdots, k$ where $\mathbb{F}_j$ corresponds to the $j$-th coordinate. More facts about them we will see in the text. Next we show that for each index $j$ and some values of $R_j$ we get properties analogous to those described by Brand, as above. But notice that we shall work on the complex plane.

The paper is organized as follows: In Section 2, we describe the collection of the forbidden sets $\mathbb{F}_j, \; j = 1, 2, \cdots, k$ of system (1.5) and then we proceed to the solvability of the system. We do that by leaving all coefficients and the initial values to be complex numbers, thus the solutions are complex, too. We work further to express the solutions in terms of their initial values. Then, by using the expression of the solutions, we discuss the asymptotic behavior of them, and show that the solutions either converge to a fixed point, are periodic, or their range is dense in a line or in a circle of the complex plane. Periodicity of nonlinear difference equations is a very significant subject studied elsewhere, e.g. ^[13] and the references therein. In Section 3, we consider the case of negative Riccati real numbers and we derive the set where the unique positive fixed point belongs, which attracts all solutions starting from positive states.

2.   The complex case

In this section, we shall study the system when all coefficients are complex numbers. Due to the fact that we have to deal with a circular system, in order to simplify our presentation, we start with the following convention: For any $m\in\mathbb{Z}$ and $l, k\in\mathbb{N},$ we shall use the symbol mod$^*(k)$ as in the type

2.1.   The forbidden sets

To better understand the forbidden set we can use the Riemann sphere $S$ and the stereographic projection $P$ of the extended complex plane onto $S$, when the primitive circle of the plane is the equator of the sphere. As it is known, this function is given by

Obviously we have

and

Now, consider Eq (2.2) and assume^* that

^*The special case $c_j(a_j+b_j) = 0,$ for some or for all indices $j$, will be discussed in a forthcoming work.

By defining the sequence $y_n^{(j)}: = \frac{1}{a_j+d_j}(c_ju_n^{(j)}+d_j), \; j\in\overline{1, k},$ system (1.5) is written in the form

where $y_n^{(k+1)}: = y_n^{(1)}.$ The first thing we must do is to obtain the forbidden set $\mathbb{F}_j$ of each coordinate $j$ of system (1.5), namely the set of all initial values $u_0^{(j)},$ which lead to the solution with a zero denominator $c_ju_n^{(j)}+d_j.$ Actually, we are interested in the complement of the forbidden set, namely in the set of all values $u_0^{(j)}$ of the complex plane, which lead to solutions defined on the whole set of natural numbers. We call this set the active set of the system. Making the transformation from $u_n^{(j)}$ to $y_n^{(j)}$, it is enough to seek for the active sets of the new sequences, namely for the vectors $(y_0^{(1)}, y_0^{(2)}, \cdots, y_0^{(k)}),$ which produce solutions of the system (2.2) with no zero terms. Taking the stereographic projection $P$, we see that a term $y_n^{(j)}$ vanishes when $P(y_n^{(j)})$ is the south pole of the Riemann sphere. Then $P(y_{n+1}^{(j+1)})$ reaches the north pole, $P(y_{n+2}^{(j+2)})$ reaches the south pole, etc.

To continue, for any $j\in\overline{1, k}$, consider the sequence of complex numbers $q_n^{(j)}, \; n = 1, 2, \cdots$ defined inductively as

where $q_{n}^{(k+1)}: = q_n^{(1)}$ and, for all $n\geq 0$ for which the terms of the sequence are defined. Our first result is the following.

Theorem 2.1. Let $j\in\{1, 2, \cdots, k\}.$ The forbidden set for the $j_{th}$ coordinate of the system (2.2) is the set $\mathbb{F}_j = \{q_n^{(j)}, \; n = 0, 1, 2, \cdots, {N}\},$ where $N\in\mathbb{N}\cup\{+\infty\}.$

Proof. To prove the theorem, assume that $y_0^{(j)} = q_n^{(j)},$ for some $j\in\overline{1, k}$ and $n\geq1.$ Then, we have

Hence

This relation gives

Continue in this way, inductively, to obtain

Therefore,

If $\hat{j}$ is the (unique) index in the interval of integers $\{1, 2, \cdots, k\}$ such that ${j+n} = \hat{j}(\text{mod}^*(k)),$ then $y_n^{(\hat{j})} = 0.$ Actually, this is the pre-image of $q_n^{(j)}$ through the Möbius transformation.

Conversely, assume that for some indices $j, n$ it holds $y_n^{(j)} = 0.$ Then, we have

This implies that

which gives

The last step of this procedure is the relation $y_0^{(j-n)} = q_{n}^{(j-n)}.$ If $\tilde{j}$ is the index in the set $\{1, 2, \cdots, k\}$ such that $j-n = \tilde{j}(\text{mod}^*(k))$, then we obtain $y_0^{(\tilde{j})} = q_{n}^{(\tilde{j})}\in\mathbb{F}_{\tilde{j}}.$ The proof of the theorem is complete. □

Since the system is circular, if the initial value of the $j_{th}$- coordinate of the solution is taken from the forbidden set $\mathbb{F}_j$, then the evolution of the whole vector solution $(y_n^{(1)}, y_n^{(2)}, \cdots, y_n^{(k)})$ is completely determined.

Example 1. Consider the circular system

The first five elements of the forbidden sets for this system are the following:

If we consider the point $\frac{1+i}{2}$ as the initial value $y_0^{(2)}$ for the 2-coordinate, we obtain $y_1^{(1)} = -1, \; y_2^{(2)} = 1+i, \; y_3^{(1)} = 0.$

Now the question which arises is what happens in case $1-q_{n_0}^{(j_0)} = 0$, for some indices $j_0, n_0$. Our answer is that the result of the theorem remains in force. Indeed, all steps in the proof of the theorem are made between finite numbers among those quantities $q_n^{(j)}$ which exist. Moreover, it is easy to see that if for some indices $j_0, n_0$ it holds $q_{n_0}^{(j_0)} = 1,$ then the quantity $q_{n_0+1}^{(j_0-1)}$ does not exist, and therefore the forbidden sets are finite.

Example 2. Consider the system

Here, we observe that $q_1^{(2)} = 1.$ Therefore the forbidden sets are finite. Indeed, they are defined by $\mathbb{F}_1 = \{0, 2\}$ and $\mathbb{F}_2 = \{0, 1, -1\}$.

2.2.   Solvability

Solvability of a difference equation, in general, is a not easy problem, and it has attracted the interest of many authors. Actually, Brand ^[5] was the first to show how to solve a scalar difference equation, where the map is a Möbius transformation. See, also, ^{[3,11,14,19,21,23,26]} and the references therein. In this subsection, we shall present a way of expressing, in a closed form, the solution of the circular system of difference equations (1.5), when the initial values do no belong to the forbidden sets. An advantage of the expression of the solutions is the fact that it helps us to describe its asymptotic behaviour.

In the sequel, we assume that assumption (2.1) holds which guarantees that circular system (1.5) is transformed into (2.2).

First, our plan is to give the general expression of each coordinate of the solution, in terms of general parameters. This is the easy part of the problem. The second step is to express the general parameters in terms of the initial values $(y_0^{(j)}),$ $j = 1, 2, \cdots, k,$ so that the expression of the solutions are in terms of their initial values.

The central role in all the sequel will be played by the $k$- triples of parameters $(a_m^{(j)}, b_m^{(j)}, c_m^{(j)})$ defined inductively as follows:

and, for any index $m = 1, 2, \cdots, k-1$ as

where $R_{0}: = R_k.$

It is clear that in order, these items to be well defined, the condition

must be satisfied^†.

^†The case $c_m^{(j)} = 1$, for some indices $j, m$, will be discussed later at the end of this work.

To continue, we are firstly going to express the term $y_{n+1}^{(j)}$ of the $j_{th}$- coordinate as a function of the term $y_{n+1-k}^{(j)}$ of the same coordinate. To do that, we observe that the transformed system (2.2), for all $n = 0, 1, 2, \cdots$ and $j\in\overline{1, k}$, becomes

Then,

To predict the rule, we do one more substitution and obtain that, for $n = 2, 3, \cdots,$

Inductively, for $n = k-1, k, \cdots$, we obtain

The final step we arrive at is

because $j-k = j(\text{mod}^*(k))$. Before we continue to our discussion, we must notice that the relation (2.6) gives a well defined value whenever $y_{n-k+1}^{(j)}\neq c_{k-1}^{(j)},$ for all $j\in\overline{1, k}$ and $n = k-2, k-1, \cdots.$ Indeed, if this statement is not true, then due to (2.4) the value $y_{0}^{(j-1)}$ is equal to $c_0^{(j)} = 0$, which contradicts the fact that $y_{0}^{(j-1)}$ belongs to the active set.

The advantage of relation (2.6) is that it is a relation involving the $j_{th}$-coordinate only.

Next, we set

Since the system has $k$ coordinates, it is convenient to define the two-parameter sequence

which, obviously, satisfies the relation

Finally, we make one more substitution by setting $s^{(l, j)}_n: = \prod_{j = 0}^nr^{(l, j)}_j.$ Then, we have

The new sequence satisfies the linear difference equation

for all $l = 0, 1, 2, \cdots, k-1$ and $j = 1, 2, \cdots, k.$ To solve this equation, we consider its characteristic equation

and obtain its (complex) characteristic roots $\lambda_1^{(j)}$, $\lambda_2^{(j)}$. We distinguish the following cases.

Case 1: Assume that the characteristic values are not equal. Then, the general solution of Eq (2.7) can be written in the form

where the coefficients $\sigma^{(l, j)}_1, \sigma^{(l, j)}_2$ are complex numbers being not both equal to zero, and they depend on the initial values of the solution. So, in order to express them in terms of the initial value $Y_0: = (y_0^{(1)}, y_0^{(2)}, \cdots, y_0^{(k)})^T$ we observe that

and

Since $\sigma^{(l, j)}_1, \sigma^{(l, j)}_2$ are not both equal to zero, it follows that one of the two fractions $\sigma^{(l, j)}_1/\sigma^{(l, j)}_2,$ and $\sigma^{(l, j)}_2/ \sigma^{(l, j)}_2$ is a finite number. If we assume that the quantity $\xi^{(l, j)} = {\sigma^{(l, j)}_1}/{\sigma^{(l, j)}_2}$ is a finite number, we can write the previous relation as

If the fraction $\zeta^{(l, j)}: = \sigma_2^{(l, j)}/\sigma_1^{(l, j}$ is a finite number, we can write it as

and we can work analogously.

To proceed, we keep in mind (2.10) and discuss the two cases $n = 0$ and $n\geq 1,$ separately. First, we observe that

and

Setting $n = 1$ and $l = 0$ in relation (2.12) and taking into account (2.6), we obtain

From this equation, we obtain

Substitute this value in (2.12) and get the expression of the values of the sequence $(y^{(j)}_{kn}).$

Next, fix any $l\in\{1, 2, \dots, k-1\}$. Setting $l = m$ in the place of $n+1$ in (2.5), we get

due to (2.11). Therefore,

On the other hand, from (2.12), for $n = 1,$ (2.6) and (2.11) we have

namely

Solving the system of these last two equations in terms of the initial value $y_0^{(j-i)},$ we obtain

and

from which the value of the item $\xi^{(i, j)} = {\sigma_i^{(1, j)}}/{\sigma_2^{(i, j)}}$ is obtained. Taking into account the values $\xi^{(0, j)}$ from (2.13) and relations (2.16) and (2.17), from (2.12), the expression of the solution $y_{n}^{(j)}$ follows.

Case 2: Next, assume that $\lambda^{(j)}_1 = \lambda^{(j)}_2 = :\lambda^{(j)}.$ Then the solution can be written in the form

where, again, the coefficients $\sigma^{(l, j)}_1, \sigma^{(l, j)}_2$ are complex numbers and not both equal to zero.

In this case we have $r^{(l, j)}_0 = s_0^{(l, j)} = \sigma^{(l, j)}_1$ and

If we assume, as previously, that the quantity

is a finite complex number, we can write the previous relation as

The case where ${\sigma^{(l, j)}_2}/{\sigma^{(l, j)}_1}$ is finite, is analogous.

Now, we observe that

and

Setting $n = 1$ and $l = 0$ in relations (2.6) and (2.19) we obtain

Therefore

Next, fix any $l\in\{1, 2, \dots, k-1\}$. Setting in (2.5), $l = m$ in the place of $n+1$ we get

due to (2.18). Therefore,

On the other hand, from (2.19), for $n = 1,$ and from (2.6) and (2.18) we have

From here, due to (2.21), we obtain

Taking into account the values $\xi^{(0, j)}$ from (2.20) and $\xi^{(l, j)}$ from relations (2.21), (2.22), we can go to (2.19) and obtain the expression of the $j_{th}$-coordinate of the solution.

The results we derived so far are summarised in the following theorem:

Theorem 2.2. Assume that (2.3) is satisfied and consider the algebraic equation (2.8).

1) If Eq (2.8) has unequal roots, the solution of the system is given by (2.11) and (2.12) where $\xi^{(l, j)} = \sigma_1^{(l, j)}/\sigma_2^{(l, j)}$ and $\sigma_1^{(l, j)}, \sigma_2^{(l, j)}$ are given in (2.16), (2.17) and $d_{l}^{(j)}$ is defined in (2.15).

2) If Eq (2.8) has equal roots, then the solution of system (1.6) is given by (2.18) and (2.19), where $\sigma_1^{(l, j)}, \sigma_2^{(l, j)}$ are given by (2.21) and (2.22).

Remark. Here and in all the sequel the reader can see that analogous results hold if we use the item $\zeta^{(l, j)}: = \sigma_2^{(l, j)}/\sigma_1^{(l, j)}$.

Before we close the subject of the solvability of the solutions, we must notice that one could ask how many initial values we need to express the solutions. Indeed, as we have seen the number of the quantities $d_l^{(j)}$ is equal to $k^2$, though we need only $k$ of them. But we can observe that each $d_l^{(j)}$ depends on the initial value $y_0^{(j-l)}.$ Therefore this initial value determines all parameters of the form $d_{l+m}^{(j+m)},$ for any $m = 0, 1, 2, \cdots, k-1.$

2.3.   Asymptotic behaviour of the solutions

In this subsection we assume, again, that the initial values of the solutions do not belong to the forbidden sets. Then, we can use the expression of the solutions as it is obtained above. We have the following result.

Theorem 2.3. Let $\lambda^{(j)}_-,$ $\lambda^{(j)}_+$ be the roots of the algebraic equation (2.8) and let $(y_n^{(j)})$ be the $j$-th coordinate of the solution of system (2.2) with initial value in the active set.

1) a) If $|\lambda^{(j)}_-| < |\lambda^{(j)}_+|$ and $\xi^{(l, j)}\neq 0$, then it holds

while if $\xi^{(l, j)} = 0$, then

b) If $|\lambda^{(j)}_+| < |\lambda^{(j)}_-|$, and $\xi^{(l, j)}\neq 0$, then

while if $\xi^{(l, j)} = 0$, then

2) If the roots are equal, say, to $\lambda^{(j)}$, then $\lim_{n\to+\infty}y_{kn+l}^{(j)} = \lambda^{(j)}+c_{k-1}^{(j)}.$

3) If the roots are of the form $\lambda_1^{(j)} = q_je^{i\omega_j}$ and $\lambda_2^{(j)} = q_je^{i\phi_j}$, then, the $j_{th}$- coordinate of the solution has the form

Hence, if $\xi^{(l, j)} = 0,$ then

for all $l, n$, namely, it is constant.

Assume that $\xi^{(l, j)}\neq 0.$

i) If $(\phi_j-\omega_j)/\pi$ is a rational number, the sequence $(y_n^{(j)})|_{n\geq k}$ is periodic.

ii) Let $\theta_j/\pi: = (\phi_j-\omega_j)/\pi$ be irrational. Then the terms of the sequence form a subset of the complex plane, which is dense in the straight line

in case $|\xi^{(l, j)}| = 1,$ dense in the circle $q_je^{i\omega_j}C+c_{k-1}^{(j)},$ where $C$ is the circle

in case $|\xi^{(l, j)}|\neq1.$

Proof. 1) a) Assume that $|\lambda^{(j)}_-| < |\lambda^{(j)}_+|$. From (2.12), we observe that for each $n$ and $l, j$, it holds

b) If it holds $|\lambda^{(j)}_+| < |\lambda^{(j)}_-|,$ then, from the previous relation, we obtain

2) If the roots are equal to $\lambda^{(j)},$ then, from (2.19), we obtain

3) In the third case, the expression of the solutions follows from (2.11) and (2.12).

Let $\xi^{(l, j)}\neq0$.

ⅰ) Assume that $\theta_j/\pi$ is a rational real number ${p_j}/{r_j},$ say. Define the positive integer $N_j: = 2kr_j$ and consider an index $m\geq k.$ Then, we can write it in the form $m = kn+l$, for some indices $n, l.$ From the expression

we obtain

due to the fact that $e^{i(n+2r_j)\theta_j} = e^{in\theta_j}e^{i2\pi p_j} = e^{in\theta_j}.$ This proves that the sequence $y^{(j)}$ is periodic.

ⅱ) Next, assume that $\theta_j/\pi$ is irrational. Then, it is well known (see, for instance, Prop. 4.1, p. 10 in ^[7]) that the set $\{e^{in\theta_j}:\; n = 1, 2, \cdots\}$ is dense in the unit circle $C$. Let $z\in C$. Then $z$ is the limit of a sequence of the form $e^{in_m\theta_j}, \; m = 1, 2, \cdots$. First we are going to find the set of all $\zeta_j\in \mathbb{C}$ such that

From here we obtain that

and so

This is equivalent to the relation

Inversely, if $\zeta$ is a complex number satisfying Eq (2.25) then, the number $\frac{(\zeta_j-1)\xi^{(l, j)}}{1-\zeta_j e^{-i\theta_j}}$ is a point in $C.$ Now, if $|\xi^{(l, j)}| = 1,$ the first factor in (2.26) vanishes, and we get a straight line given by (2.23). Otherwise, we have the circle given by (2.24). Now, the proof follows by observing that dilations of straight lines and circles produce straight lines and circles, respectively. □

3.   The real case

As we proved in the previous section, in case condition (2.3) holds, there exists a limit of all solutions of system (1.5). This limit turns out to be a fixed point of the system. In this section, we restrict ourselves to the case where all Ricatti numbers are negative real numbers and we shall try to locate the existence of a fixed point of system (1.5) in the positive orthant of the space $({\mathbb{R}^+})^k$. Moreover, this fixed point is unique. The method we use relies on the classical fixed point of Brouwer.

By setting $p_j: = -R_j,$ $j = 1, 2, \cdots, k,$ we have $p_j > 0$ and system (2.2) is written as

We shall work on the set

The main result of this section refers to the existence and uniqueness of a fixed point of the system lying in $D$ and it is the following.

Theorem 3.1. Assume that (2.1) holds and the Riccati numbers $R_j, \; j = 1, 2, \cdots, k$ are real negative. Then system (2.2) admits a unique fixed point in the set $D,$ which attracts all solutions starting from positive states. It turns out that system (1.5) admits a fixed point in the set

which attracts all solutions with initial states in

Proof. We write the system of Eq (3.1) in the form of an equation in the $k$-dimensional space. To do that consider the function

and the sequence of vectors $Y_n: = (y_n^{(1)}, y_n^{(2)}, \cdots, y_n^{(k)})^T,$ where the superscript $T$ denotes transposition. Then, the system (3.1) takes the form

We observe that for all $z\in[1, 1+p_{j}]$ it holds

which means that the function $\phi$ maps $D$ into $D$. Since $\phi$ is a continuous function and $D$ is a closed, bounded and convex set, applying Brouwer's fixed-point theorem, we conclude that there is a fixed point $V\in D$ of equation

Vector $V: = (v_1, v_2, \cdots, v_k)^T$ is such that

where $v_{0}: = v_k$ and $p_0: = p_k.$

Next let $Y_n: = (y_n^{(1)}, y_n^{(2)}, \cdots, y_n^{(k)})^T$ be a solution. From (3.1), we have

for all $j = 1, 2, \cdots, k$ and $n = 0, 1, 2, \cdots.$ Here we have put $y_n^{(0)} = y_n^{(k)}.$ If $V$ is a fixed point, as above, we observe that it holds

and so $|v_j-y_{n+1}^{(j)}|\leq\mu|v_j-y_n^{(j)}|, \; j = 1, 2, \cdots, \; n = 0, 1, 2, \cdots$ where

The latter implies that

from which we conclude that $\lim_{n\to+\infty}y_{n}^{(j)} = v_j, \; j = 1, 2, \cdots, k.$ After this fact the proof of the theorem is complete. □

Remark 1. It is easy to see that in case $k = 2$, the fixed point has coordinates given by

In case $k = 3$,

Remark 2. Since the Riccati numbers $R_j$ are reals and have negative signs, it is not hard to see that, the roots $\lambda^{(j)}_-, \lambda^{(j)}_+$ of the characteristic equation (2.8) are reals and they satisfy the case 1) a) of Theorem 2.3. This means that the eigenvalues $\lambda^{(j)}_+$ (2.8) are the coordinates of the fixed point $V$.

A special case

The question what can we say when, for some (or for all) indices $j$ and $m,$ it holds $c_m^{(j)} = 1?$ As we see, if $1-c_{m-1}^{(j)} = 0$, the quantity $c_m^{(j)}$ is not defined. If $R_{j-m-1} = 1,$ then $c_m^{(j)} = 1$, if $c_{m-1}^{(j)} = 0.$ Since the Riccati numbers cann't be zero, we have $m = 1.$ So, if $R_{j-2} = 1,$ then $c_1^{(j)} = 1,$ and this is the only value of the sequence of parameters $c$ which exists, after $c_0^{(j)}.$ As we shall see in this case all solutions are periodic.

To prove it, consider the circular system

and we shall discuss the $j_{th}$- coordinate. Since the system is circular, without loss of generality we can assume that $j = 1.$ After some manipulations and applying induction^‡, we obtain that

^‡We can find analogous expressions for all coordinates and not only for $j = k.$

for all $n\geq \max\{4k-2, 3k\}.$ This means that the sequence $(y_n)$ is periodic with period $3k$.

4.   Discussion

We have discussed a circular system of Riccati type complex difference equations. First, we presented the forbidden set of the system. Then we discussed the solvability of it, by expressing the solutions in terms of the initial values. The expression of the solutions helps to obtain their asymptotic behavior. In the case of negative real Riccati numbers, it is shown that there is a unique fixed point of the system, which attracts all solutions starting from positive values.

5.   Conclusions

The forbidden set of the circular system (1.5) is given in terms of the elements of an auxiliary system obtained from the Riccati numbers. Also, the expression of the solutions is given by using an auxiliary finite system of sequences, which follows from these numbers. In the case, of negative real Riccati numbers, we show the convergence of the solutions (with positive initial values) to a fixed point, which is unique. In case $R_j = 1$, for all $j = 1, 2\cdots,$ it is shown that the solutions are periodic.

Aknowledgments

The author would like to thank the anonymous referees for their valuable comments and for her/his helpful suggestions which led to the improvement of this paper.

Conflict of interest

The author declares that he does not have competing interests.

The Prof. George L. Karakostas is the Guest Editor of special issue "Recent contributions to difference equations" for AIMS Mathematics. The Prof. George L. Karakostas was not involved in the editorial review and the decision to publish this article.