Analytical Solution to a Third-Order Rational Difference Equation

Inspired by some open conjectures in a rational dynamical system by G. Ladas and Palladino, in this paper, we consider the problem of solving a third-order difference equation. We comment the conjecture by Ladas. A third-order rational difference equation is solved analytically. The solution is compared with the solution to the linearized equation. We show that the solution to the linearized equation is not good, in general. The methods employed here may be used to solve other rational difference equations. The period of the solution is calculated. We illustrate the accuracy of the obtained solutions in concrete examples.


Introduction
Te use of recurrences to solve mathematical problems dates back to Babylon in 2000 B.C.E. in the context of the approximate resolution of algebraic equations and the approximate calculation of square roots. In Greek times, the Pythagoreans (ffth century B.C.E.) implicitly used nonautonomous diference equations to study the numbers associated with fgures or pentagonal numbers.
Te Fibonacci sequence, continued fractions, binomial coefcients, the calculus of fnite diferences, the Newton-Raphson method, and the numerical methods to approximate the solutions of a diferential equation are just some of them (see [1,2] for more details). In the frst half of the twentieth century, great interest arose in the development of numerical methods, which was greatly enhanced by the appearance of powerful computer calculation tools.
In the 50s of the last century, moreover, nonlinear diference equations began to be used as applied models, especially in ecology. Later, the discovery that even the simplest models exhibit enormous complexity led to the introduction of mathematical chaos and renewed interest in the theory of diference equations.
Nonlinear diference equations and their systems are hot topics that have attracted the attention of several researchers. A signifcant number of papers are devoted to this feld of research. One can consult, for example, the papers [3][4][5][6][7][8][9][10], where one can fnd concrete models of such equations and systems, as well as understand the techniques used to solve them and investigate the behavior of their solutions.
Recently, an increased interest has been witnessed in studying the theory of discrete dynamical systems, specifcally of their associated diference equations. A sizable number of works on the behavior and properties of pertaining solutions (boundedness and unboundedness) have been published in various areas of applied mathematics and physics. Te theory of diference equations fnds many applications in almost all areas of natural science [11]. Te diference equations with discrete and continuous arguments play important role for understanding nonlinear dynamics and phenomena [8]. Te increased interest in diference equations is partly due to their ease of handling.
Although diference equations have very simple forms, it is extremely difcult to completely understand the global behavior of their solutions. One can refer to [4][5][6] and the references therein. Diference equations have always played an important role in the construction and analysis of mathematical models of biology, ecology, physics, and economic processes. Te study of nonlinear rational difference equations of higher order is of paramount importance, since we still know little about such equations.
Let m ≥ 1 be a natural number. Given f: R m ⟶ R which we will call iteration function, a diference equation (DE) of order m in explicit form is any expression like the following: x n+1 � f x n−m+1 , . . . , x n . (1) Te above formula allows us to build a family of sequences called the set of solutions of the DE, whose defnition is as follows: fxed a vector X � (x −m+1 , . . . , x 0 ), the solution of (1) from initial conditions X or generated by the initial conditions X is the sequence (x j ) ∞ j�−m+1 whose frst m terms are the components of X and the rest are obtained inductively by formula (1). When for some r ≥ 0, the vector (x r−m+1 , . . . , x r ) does not belong to the domain of defnition of f, the construction of (x j ) ∞ j�−m+1 cannot be realized. In such a case, we say that X is an element of the forbidden set of (1), denoting it by P.
Te expression solution of the diference equation is reserved for the sequences generated from the elements of B � R n /P, called the good set of the DE. Occasionally, the term fnite solution is used. When X ∈ P and r is the largest integer such that x m is well defned, refer to (x j ) r j�−m+1 . But, unless otherwise indicated, the word "solution" is associated with sequences of infnite terms. To emphasize this diference, we will sometimes say that such solutions are well defned. Solutions of a DE are also called trajectories or orbits. Such nominations are inspired by the terminology of dynamic systems.
In this paper, we will consider the following third-order rational diference equation: Te most important solutions to equation (2) are the periodic solutions, those formed by a quantity fnite number of terms which repeat itself indefnitely. Teir relevance lies in the fact that, on many occasions, the equation can be described qualitatively by identifying its periods and the behavior of the rest of the solutions with respect to them. For example, a common situation is that some periods behave as attractors of the rest of the solutions, which implies that the model associated with DE will consist, in the long run, of a certain cycle.
Even when the dynamics of the ED are not so clear, the determination of the periodic solutions is still relevant information to give us an idea of what is happening.
In [7],Abo-Zeid has discussed the global behavior of all solutions of the diference equation: where a and b are real numbers and the initial conditions x 1 and x 0 are real numbers. A class of third-order rational diference equations of form (2) with nonnegative coefcients is considered in [12].

(5)
We defne the residual as Ten, where z j � r n j (j � 1, 2, 3) and 2 Te Scientifc World Journal Te number μ 0 is an equilibrium point, and it satisfes the quadratic equation Te numbers r 1 , r 2 , and r 3 bare the roots to the cubic equation Te constants μ i,j,k are obtained from the system κ i,j,k � 0(i, j, k � 0, 1, 2). Tey read Finally, the constants μ 1 , μ 2 , and μ 3 are obtained from the initial conditions

Some Particular Cases
Ladas-Palladino conjecture claims that the solutions to third-order DE are bounded if β � c. In this case, Let us fnd an approximate solution for this DE.
Tus, we have one real root and two complex roots.
First case (β ≠ c). In this case, at least one of the roots of the cubic in (16) will have magnitude greater than the unity. Te approximate solution will be unbounded. We will not consider this case.
Second case (β � c). In this case, one of the roots of the cubic in (16) equals −1 and the other two are complex and they lie on the unit circle |z| � 1. Tat is, all roots have magnitude 1. Te approximate solution will be bounded. In order to simplify the matters, let x n � z n /β , and x 0 � (1/2) + � 2 √ .

Conclusions and Future Work
We have shown in our paper that approximate analytical solutions of a rational dynamical system, namely, thirdorder diference equation, are periodic and bounded but this may not happen to the exact solution of such a rational dynamical system. We may use the same methods of linearization to predict orbits and boundedness solutions for other rational dynamical systems such as diference equations of a fourth degree or more, rather than that we may prove or disprove other open conjectures in rational dynamical systems which are proposed by G. Ladas.

Data Availability
No data were used for supporting this paper.

Conflicts of Interest
Te authors declare that they have no conficts of interest.