Asymptotic Dynamics of a Class of Third Order Rational Difference Equations

The asymptotic dynamics of the classes of rational difference equations (RDEs) of third order defined over the positive real-line as xn+1 = xn axn + bxn−1 + cxn−2 , xn+1 = xn−1 axn + bxn−1 + cxn−2 , xn+1 = xn−2 axn + bxn−1 + cxn−2 and xn+1 = axn + bxn−1 + cxn−2 xn , xn+1 = axn + bxn−1 + cxn−2 xn−1 , xn+1 = axn + bxn−1 + cxn−2 xn−2 is investigated computationally with theoretical discussions and examples. It is noted that all the parameters a, b, c and the initial values x−2, x−1 and x0 are all positive real numbers such that the denominator is always positive. Several periodic solutions with high periods of the RDEs as well as their inter-intra dynamical behaviours are studied.


Introduction
Discrete dynamical systems is becoming very popular as it comes naturally in various biological systems including many other applied disciplines [1], [2]. In the past three decades, the theory of discrete dynamical systems and difference equations developed significantly. The theory of difference equations occupies a central position in applied sciences [3]. Undoubtedly, the theory of difference equations will continue to play an important role in mathematics as a whole. The nonlinear difference equations of higher orders come quite naturally in modelling various systems we come across such as ecology, physiology, physics, engineering, economics, probability theory, genetics, psychology and resource management [4], [5], [6]. It is very interesting to investigate the behaviour of solutions of a higher-order rational difference equation and to discuss the local asymptotic stability of its equilibrium points [7], [8], [9]. Studying non-linear rational difference equations of higher orders is not though easy but very interesting with enriched phenomena [10].
There are several textbooks available where the basic theory of rational difference equations of second and third order over the real lines are discussed [11], [12], [13]. We are primarily concerned with the stability of the equilibrium points, the periodic character of the equation, and with convergence to periodic solutions and other dynamical behaviors if exists any [14].
There are 225 third order non-linear rational difference equations for which several thought-provoking open problems and conjectures on the boundedness character, the global stability, and the periodic behavior of their solutions are yet to close [15], [16], [17], [18], [19]. In this present study, the rational difference equations (RDEs) we are interested to investigate are the following: x n+1 = x n ax n + bx n−1 + cx n−2 (1) x n+1 = x n−1 ax n + bx n−1 + cx n−2 (2) x n+1 = x n−2 ax n + bx n−1 + cx n−2 These RDEs let us call Type-I and following RDEs, let us named as Type-II x n+1 = ax n + bx n−1 + cx n−2 x n (4) x n+1 = ax n + bx n−1 + cx n−2 x n−1 (5) x n+1 = ax n + bx n−1 + cx n−2 x n−2 (6) It is mentioned that all the parameters a, b, c and the initial values x −2 , x −1 and x 0 are all positive real numbers such that the denominator is always positive. It is noted that the rational difference equations 1, 2 and 3 of Type-II are same as 4, 5 and 6 respectively.
2 Local Asymptotic Stability of the Equilibrium points of RDEs of Type-I and Type-II The only positive equilibrium point of the equations 1, 2, 3 of Type-I and equations 4, 5, 6 of Type-II are 1 a+b+c and a + b + c respectively. In the following two sub-sections we would see the local asymptotic stability of the equilibriums of the two types of RDEs.
Before we proceed into the subsections, we wish to present two important theorems which would be used [2]. Theorem 2.1. Assume that a 2 , a 1 , and a 0 are real numbers. Then a necessary and sufficient condition for all roots of the equation λ 3 + a 2 λ 2 + a 1 λ + a 0 = 0 to lie inside the unit disk is |a 2 + a 0 | < 1 + a 1 , |a 2 − 3a 0 | < 3 − a 1 and a 2 0 + a 1 − a 0 a 2 < 1 Theorem 2.2. Assume that a 0 , a 1 , a 2 are real numbers such that then all roots of equation λ 3 + a 2 λ 2 + a 1 λ + a 0 = 0 lie inside the unit disk.

Local Asymptotic Stability of the Type-I RDEs
The non-zero positive equilibrium point (x) of 1, 2 and 3 is the solutions of the equation It turns out that 1 a+b+c is the only equilibrium point of the Type-I RDEs. The linearized equation of the rational difference equation 1 with respect to the equilibrium point 1 a+b+c is with associated characteristic equation Proof. By setting the conditions as stated in the Theorem 2.1 into the characteristic equation 18 and by simplification once can easily derive the necessary and sufficient condition for the local asymptotic stability of the equilibrium point 1 a+b+c of the 1 of Type-I is The linearized equation of the rational difference equation 2 with respect to the equilibrium point 1 with associated characteristic equation Theorem 2.4. The necessary and sufficient for local asymptotic stability of the equilibrium point 1 a+b+c of the 2 of Type-I is a + c < b The linearized equation of the rational difference equation 2 with respect to the equilibrium point 1 with associated characteristic equation Theorem 2.5. The necessary and sufficient for local asymptotic stability of the equilibrium point 1 a+b+c of the 3 of Type-I is We have found computationally 10000 points (a, b, c) of the RDEs of the Type-I such that each of the Theorems 2.3, 2.4 and 2.5 hold good. The points (a, b, c) are plotted in three dimension as shown in Fig.1.

Local Asymptotic Stability of the Type-II RDEs
The non-zero positive equilibrium point (x) of 4, 5 and 6 is the solutions of the equation It turns out that a + b + c is the only equilibrium point of the Type-II RDEs. The linearized equation of the rational difference equation 4 with respect to the equilibrium point a + b + c is with associated characteristic equation Theorem 2.6. The necessary and sufficient for local asymptotic stability of the equilibrium point a+b+c of the 4 of Type-II is The linearized equation of the rational difference equation 5 with respect to the equilibrium point with associated characteristic equation Theorem 2.7. The necessary and sufficient for local asymptotic stability of the equilibrium point a+b+c of the 5 of Type-II is a + 2b > √ 5 The linearized equation of the rational difference equation 6 with respect to the equilibrium point with associated characteristic equation Theorem 2.8. The necessary and sufficient for local asymptotic stability of the equilibrium point a+b+c of the 6 of Type-II is We have found computationally 10000 points (a, b, c) for all the RDEs of Type-II such that each of the Theorems 2.
Here we list a set of parameters a, b and c for which the Theorem 2.9 holds good.
Theorem 2.10. The equilibrium point a + b + c of all the three RDEs 4, 5 and 6 of Type-II is locally asymptotically stable simultaneously if and only if

Sufficient Condition for Local Asymptotic Stability of RDEs
Here we shall derive the sufficient conditions for local asymptotic stability of the equilibria 1 a+b+c for RDEs of Type-I and a + b + c for RDEs of Type-II.
Theorem 2.14. The equilibrium 1 a+b+c is locally asymptotically stable for the RDE 1 if b + c < a.
Theorem 2.15. The equilibrium 1 a+b+c is locally asymptotically stable for the RDE 2 if a + c < b.
Theorem 2.16. The equilibrium 1 a+b+c is locally asymptotically stable for the RDE 3 if a + b < c. Theorem 2.17. The equilibrium a + b + c is locally asymptotically stable for the RDE 4 if b + c < a.
Theorem 2.18. The equilibrium a + b + c is locally asymptotically stable for the RDE 5 if a + c < b.
Theorem 2.19. The equilibrium a + b + c is locally asymptotically stable for the RDE 6 if a + b < c.

Periodic Solutions of the RDEs
The necessary condition for the existence of prime period-two solutions of the general third order rational equation with nonnegative parameters α, β, γ, A, B, C, D and with arbitrary non-negative initial conditions x −2 , x −1 , x 0 such that the denominator is always positive is where φ and ψ are prime period two solutions of the general third order RDE [11], [13], [20], [21].
Let us discover the existence of the prime period two solutions of the RDEs of Type-I and Type-II before we actually proceed to discover the prime period two solutions of the RDEs. Now by comparing the RDEs of Type-I and Type-II with the general third order RDE, we have the parameters as the following shown in the Table-1:

Periodic Solutions of the RDEs of Type-I
The RDE 2 has prime period two solutions as confirmed theoretically and computationally. Let φ, ψ, φ, ψ, . . . , . . . is a prime period two solution of the RDE 2. Then it must satisfy the identity b(φ + ψ) = 1. if φ = 0 then by the identity we get ψ = 1 b . Hence 0, 1 b , 0, 1 b , . . . . . . turns out to be a prime period two solution of the RDE 2.      Table 3 and the corresponding prime period two solutions plot are shown in the Fig. 14. φ, ψ = 1 − φ Remarks a=0.4301, b=1, c=0.5722 φ = 0.4921, ψ = 0.5067 as shown in Fig. 14 (left) a=0.3367, b=1, c=0.6624 φ = 0.6927, ψ = 0.3077 as shown in Fig. 14 (middle) a=0.9345, b=1, c=0.1079 φ = 0.5272, ψ = 0.4517 as shown in Fig. 14 (right)      In the Example 3.3 and 3.4, the trajectories in the Fig.15 and Fig.16 are turning out to be a self-similar fractal [23]. The fractal dimensions of the trajectories of the figures are enumerated through the Benoit software (for quick reference, the snapshot of the calculation is given in the Fig.17) and found to be 1.862 and 1.871 respectively [24].  It is noted that the initial values x −2 , x −1 and x 0 are taken from the -neighbourhood of 0, 0, 1 c respectively.  As usual the initial values x −2 , x −1 and x 0 are taken from the -neighbourhood of 0, 0, 1 c respectively. Remark 3.1. As observed in the Examples 3.5 and 3.6, it is observed that the period three solutions 0, 0, 1 c , 0, 0, 1 c , . . . , . . . is locally asymptotically stable. We also encounter some quasi-periodic (near-periodic) with high quasi-period (near-period) solutions of the RDE 3 of Type-I as shown in the following two examples.

Periodic Solutions of the RDEs of Type-II
We have seen the prime period two solutions of the RDEs 4 and 6 of Type-II exist whenever b = a + c. This is illustrated here through the following examples as shown in Table 4.    Here We have omitted examples in this case. Rather we give a set of parameters such that the prime period solutions happen. Also such parameters as points (a, b, c) in three dimension are plotted with in two dimension the prime period two solutions are plotted in the Fig. 22.

A Comparative View of Dynamics of RDEs
In this section, we shall describe a comparative view through examples of the dynamical behavior of the rational difference equations of both the Type-I and Type-II [22].

Conclusions & Future Endeavours
This present article describes the dynamical behavior of the class of rational difference equations of two types. They exhibit various dynamics with variations. This is possibly mathematical unity in diversity. For certain parameters the RDEs agree with local asymptotic stability and there are parameters under which each difference equations are self-sensitive. It is noted that no chaotic trajectory is seen in these rational difference equations for positive real parameters and initial conditions. We expect various exotic dynamical behaviors such as (fractal, chaotic, high periods etc) if parameters and initial values can be made over entire real line. There are certainly many other aspect of the dynamical behavior which are due to be seen in near future. There several other class of rational difference equations could be studied computationally in order to apprehend their comparative dynamics.