Periodic Solutions to a Third-Order Conditional Difference Equation over the Integers

As mentioned in 1 , difference equations appear naturally as a discrete analogue and as a numerical solution of differential and delay differential equations having applications in various scientific branches, such as biology, ecology, physics, economy, technics, and sociology. The stability, asymptotic behavior, and periodic property of solutions to difference equations had been widely investigated, such as 2–14 . Recently, the study of max-type difference equation attracted a considerable attention, for example, 7, 11, 15–25 . Although max-type difference equations are relatively simple in form, it is unfortunately extremely difficult to understand thoroughly the behavior of their solutions. The max operator arises naturally in certain models in automatic control theory. On the other hand, there exists another kind of difference equations called conditional difference equations, which also have simple forms, but it is difficult to understand clearly the behavior of their solutions. From 2, 5 , we know that the following conditional difference equation


Introduction
As mentioned in 1 , difference equations appear naturally as a discrete analogue and as a numerical solution of differential and delay differential equations having applications in various scientific branches, such as biology, ecology, physics, economy, technics, and sociology. The stability, asymptotic behavior, and periodic property of solutions to difference equations had been widely investigated, such as 2-14 . Recently, the study of max-type difference equation attracted a considerable attention, for example, 7, 11, 15-25 . Although max-type difference equations are relatively simple in form, it is unfortunately extremely difficult to understand thoroughly the behavior of their solutions. The max operator arises naturally in certain models in automatic control theory. On the other hand, there exists another kind of difference equations called conditional difference equations, which also have simple forms, but it is difficult to understand clearly the behavior of their solutions.
From 2, 5 , we know that the following conditional difference equation a n ⎧ ⎨ ⎩ a n−1 a n−2 2 , if 2 | a n−1 a n−2 , a n−1 a n−2 , otherwise, Discrete Dynamics in Nature and Society with positive initial integers a 0 , a 1 , has the property that each positive integer solution a n is either stationary or unbounded. Later, Ladas 26 gave the conjecture also mentioned in 5 that all solutions to the conditional difference equation a n ⎧ ⎨ ⎩ a n−1 a n−2 3 , if 3 | a n−1 a n−2 , a n−1 a n−2 , otherwise,  Clark 3 studied periodic solutions to a n ca n−1 − a n−2 for various real c. Greene and Niedzielski 5 considered a generalization of 1.1 and 1.2 and studied the following conditional difference equation: a n ⎧ ⎨ ⎩ r a n−1 a n−2 , if r a n−1 a n−2 ∈ Z, a n−1 a n−2 , otherwise, where r is some fixed rational number. However, they pointed out that it was also very hard to characterize the periodic solutions to 1.3 . Hence, they addressed a different, easier question, and then did some research. Motivated by the above works, in this paper, we study the following conditional thirdorder difference equation, which is another generalization of 1.1 , 1.2 , and 1.3 : y n ⎧ ⎨ ⎩ r y n−1 y n−2 y n−3 , if r y n−1 y n−2 y n−3 ∈ Z, y n−1 y n−2 y n−3 , otherwise, where r is some appropriate rational number. We study this equation by transforming it into a first-order system. It is eventually proved that the equation has no period-two or three positive integer solutions. Besides, its all period-four and five positive integer solutions are derived under appropriate rational parameters. The main results are presented in Section 4.

Auxiliary Results
For the convenience of investigation, higher-order difference equations are usually converted into first-or lower-order difference equation systems. As we all know, second-order difference equations can be transformed into first-order difference equation system. First, we define a mapping F : Z 3 → Z 3 such that where x x, y, z T ∈ Z 3 , r is a rational parameter, and Applying the defined mapping F x , the third-order difference equation 1.4 can be converted into a corresponding first-order difference equation system as follows. Let x n x n , y n , z n T ∈ Z 3 , n ∈ N 0 and x n F n x 0 , n ∈ N, then we have the following first-order difference equation system: where x 0 y 0 , y 1 , y 2 T , y 0 , y 1 , y 2 are initial integers in 1.4 . Note that the condition r x n y n z n ∈ Z can be replaced by the equivalent matrix product condition r 1, 1, 1 x n ∈ Z. For instance, when r 1/5, then the periodic solution 1, 1, 1, 3, 1, 1, 1, 3, . . . to 1.4 corresponds to the following periodic solution to system 2.3 : Given an initial vector x 0 , each subsequent x n can be got from x 0 via a formula x n L n x 0 , where L n is an appropriate product of n matrices, each of which is C or D. Hence, for the above example, if x 0 1, 1, 1 T , then x 1 Cx 0 , x 2 Dx 1 DCx 0 , x 3 D 2 Cx 0 , x 4 D 3 Cx 0 , and so forth. Note that the matrices multiply x 0 from right to left and that x 4 x 0 , so D 3 Cx 0 x 0 . We have the following obvious result. Lemma 2.1. If system 2.3 has a periodic solution with period k and x is a vector in that periodic solution, then there exists a corresponding matrix L, which is a product of k matrices, each of which is C or D, such that x is an eigenvector of L with eigenvalue 1.
The proof of Lemma 2.1 is simple; here we point out that the converse of Lemma 2.1 is not true. The problem is that the sum of the entries in some x i may be divisible by the denominator of r at a time, when multiplication by C is called for.
Next, we present the following linear algebra facts, for example, 27, Chapter 7 , similar results presented in 5 , followed by the form most convenient to us.

Lemma 2.2. (i)
If v is an eigenvector with eigenvalue λ for a matrix L, then v is an eigenvector with eigenvalue λ k for L k .
(ii) The characteristic polynomial for a 3 × 3 matrix L has the form where tr L is the trace of L.

Discrete Dynamics in Nature and Society
(iii) The characteristic polynomials for LM and ML are identical for any n × n matrices L and M. In particular, tr LM tr ML). (iv) The trace is linear. That is, tr αL βM α tr L β tr M , for any n × n matrices L, M and scalars α, β.
(v) Every matrix satisfies its characteristic polynomial. In particular, for a 3 × 3 matrices L, one has Let the sequences H n n≥0 , J n n≥0 be two solutions to the difference equation y n y n−1 y n−2 y n−3 , n ≥ 3 2.7 with initial values H 0 H 1 0, H 2 1 and J 0 J 2 1, J 1 0, respectively. Then, we get the following result about the matrix C in 2.2 .
Proof. This result can be proved by induction. By 2.2 and the definitions of the sequences H n n≥0 , J n n≥0 , we have Therefore, 2.8 holds for n 1. Now, we assume that 2.8 holds for n ω, ω ∈ N. In the following, it suffices to prove that 2.8 holds for n ω 1. By 2.2 and the associative law of matrix multiplication, we have

2.10
Therefore, 2.8 holds for k ω 1. The proof is complete.
Discrete Dynamics in Nature and Society 5 Given a 3 × 3 matrix L N 1 N 2 · · · N k , k ∈ N, where each N j is either C or D, by Lemma 2.2 we define the following polynomial: where I is the identity matrix, and seek L for which P L r has rational zeros.

Some Other Properties
In this section, several properties of P L r defined in 2.11 are derived. They are used to derive restrictions on values of r that allow periodic solutions to 1.4 .

Lemma 3.1.
For the polynomial P L r in 2.11 , if P L r 0, then P L k r 0, for each k ∈ N.
and the result follows. , then tr C n 1 EC n 2 E · · · C n k E is a polynomial in x of degree k with nonnegative integer coefficients, with leading coefficient k j 1 H n j 3 , where n j ∈ N.

3.4
To obtain the leading coefficient, we may induct on the simple calculation

3.6
Thus the leading coefficient is The proof is complete.

Periodic Solutions
In this section, we prove that 1.4 has no periodic solution with prime period two or three and derive all periodic solutions to 1.4 with prime period four and five. Proof. Suppose that x ∈ Z 3 be a vector of a period-two solution to system 2.3 , then, for some matrix L 2 , we have x L 2 x; here the matrix L 2 has four possible cases C 2 , CD, DC and D 2 .
Take L 2 D 2 , for example, through some calculations, we get that

4.1
By solving the equation P D 2 r 3r 2 2r − 1 0, we get two real roots r 1 1/3 and r 2 −1. Through similar calculations, we can get Table 1.
The only value of r which may lead to period-two solutions to system 2.3 is r 1/3 since r / ∈ Z. Hence, by solving the following matrix equation with r 1/3 we get its all integer solutions x t, t, t T , t ∈ Z. Obviously, the solutions are equilibrium points which contradicts the assumption. The proof is complete.

Theorem 4.2. Suppose that r is a rational number and r /
∈ Z, then 1.4 has no periodic solutions with prime period three.
Proof. Assume that x ∈ Z 3 be a vector of a period-three solution to system 2.3 , then we get x L 3 x where L 3 is a matrix product of three matrices, each of which is C or D.
Obviously, L 3 has eight possible cases which can be divided into four categories such as three Cs, two Cs and one D, one C and two Ds, and three Ds. By 2.2 and through certain calculations, the matrices C 2 D, CDC, DC 2 are similar, and; thus, they generate the same polynomials P L 3 r , so do the matrices CD 2 , DCD, D 2 C.  In the following, we take L 3 D 3 , for example. Through some calculations, we have

4.3
By solving the equation P D 3 r 3r − 1 0, we get the only root r 1/3. Through similar calculations, we can obtain Table 2.
The only value of r which may lead to period-three solutions to system 2.3 is r 1/3 since r / ∈ Z. Hence, by solving the following matrix equation with r 1/3 we get its all integer solutions x t, t, t T , t ∈ Z. Obviously, the solutions are equilibrium points which contradicts the assumption. The proof is complete.
In the following, denote by S k, r the set of initial values y 0 , y 1 , y 2 which lead to period-k solutions to 1.4 with the parameter r.
Proof. Let x ∈ Z 3 be a vector of a period-four solution to system 2.3 , then we get x L 4 x, where L 4 is an appropriate product of four matrices, each of which is C or D. Through similar calculation to those in Theorems 4.1 and 4.2, Table 3 is derived.
The only values of r which possibly lead to period-four solutions to system 2.3 are r 1/3, −1/3, or 1/5 because of r / ∈ Z.
Discrete Dynamics in Nature and Society 9 Case 1 r 1/3 . By solving the following matrix equation we get its all integer solutions x t, t, t T , t ∈ Z. Obviously, the solutions are equilibrium points.
Case 2 r −1/3 . In this case, the matrix L 4 has four possible cases C 3 D, C 2 DC, CDC 2 , and DC 3 .
By solving the matrix equation we get its all integer solutions x t, t, t T , t ∈ Z. On the condition of that 3 t, then we can verify that the initial vector x 0 t, t, t T leads to a period-four solution to system 2.3 , such as the following: Similarly, for the following three matrix equations we derive its all integer solutions x −t, t, t T , x t, −t, t T , x t, t, −t T , t ∈ Z, respectively.
Note that, on the condition 3 t, the initial vectors x 0 −t, t, t T , t, −t, t T , or t, t, −t T also lead to period-four solutions to system 2.3 .
Case 3 r 1/5 . In this case, the matrix L 4 also has four possible cases CD 3 , DCD 2 , D 2 CD, and D 3 C. By solving the matrix equation we get its all integer solutions x t, t, 3t T , t ∈ Z. On the condition of that 5 t, then we can verify that the initial vector x 0 t, t, 3t T leads to a period-four solution to system 2.3 , such as the following:  Obviously, on the condition 5 t, the initial vectors x 0 t, 3t, t T , 3t, t, t T , or t, t, t T which are integer solutions to appropriate matrix equations corresponding to DCD 2 , D 2 CD, D 3 C, resp. also lead to period-four solutions to system 2.3 . The proof is complete.
To illustrate the results, we give two orbits see the following Figure 1 Proof. Let x ∈ Z 3 be a vector of a period-five solution to system 2.3 , then we get x L 5 x where L 5 is an appropriate product of five matrices, each of which is C or D. Through similar calculation to those in Theorems 4.1 and 4.2, Table 4 is obtained.
Apparently, the only values of r which possibly lead to period-three solutions to system 2.3 are r 1/3 or −7/15 because of r / ∈ Z. Case 2 r −7/15 . In this case, the matrix L 5 also has five possible cases C 4 D, C 3 DC, C 2 DC 2 , CDC 3 , and DC 4 . Solve the matrix equation we get its all integer solutions x 5t, t, 9t T , t ∈ Z. On the conditions that 3 t, 5 t, then we can verify that the initial vector x 0 5t, t, 9t T leads to a period-four solution to system 2.3 , such as the following:
To illustrate the results, we give two orbits see Figure 2 of period-five integer solutions to 1.4 of the particular cases t 2, 11 in Theorem 4.4.