Stability, Neimark–Sacker Bifurcation, and Approximation of the Invariant Curve of Certain Homogeneous Second-Order Fractional Difference Equation

Equation (3) is a general homogeneous rational difference equation with quadratic terms and it is very complicated for investigation in many special cases. +e function associated with the right side of equation (3) is of the form f(u, v) � (Au2 + Buv + Cv2)/(au2 + buv + cv2), with the following monotonicity property: it is either monotonically decreasing in the first variable and monotonically increasing in the second one or monotonically increasing in the first variable and monotonically decreasing in the second variable. Using the theory of monotone maps, it was possible to investigate the global dynamics of some special cases of equation (3) [1–4] in the situations when corresponding function f is monotonically decreasing in the first variable and monotonically increasing in the second variable. In [4], the authors investigated the local and global character of the unique equilibrium point and the existence of Neimark–Sacker and period-doubling bifurcations of equation (3). +ey have also studied the local and global stability of the minimal period-two solutions for some special cases of the parameters. +e special case when A � B � 0 and C � 1, i.e.,

Equation (3) is a general homogeneous rational difference equation with quadratic terms and it is very complicated for investigation in many special cases. e function associated with the right side of equation (3) is of the form f(u, v) � (Au 2 + Buv + Cv 2 )/(au 2 + buv + cv 2 ), with the following monotonicity property: it is either monotonically decreasing in the first variable and monotonically increasing in the second one or monotonically increasing in the first variable and monotonically decreasing in the second variable. Using the theory of monotone maps, it was possible to investigate the global dynamics of some special cases of equation (3) [1][2][3][4] in the situations when corresponding function f is monotonically decreasing in the first variable and monotonically increasing in the second variable.
In [4], the authors investigated the local and global character of the unique equilibrium point and the existence of Neimark-Sacker and period-doubling bifurcations of equation (3). ey have also studied the local and global stability of the minimal period-two solutions for some special cases of the parameters. e special case when A � B � 0 and C � 1, i.e., x n+1 � x 2 n− 1 ax 2 n + bx n x n− 1 + cx 2 n− 1 , n � 0, 1, 2, . . . , (4) was considered in [2]. It was shown that equation (4) is characterized by three types of global behavior with respect to the existence of a unique positive equilibrium and existence of one or two minimal period-two solutions, one of which is locally asymptotically stable and the other is a saddle point. An important feature of this equation is the coexistence of an equilibrium and the minimal period-two solution which are both locally asymptotically stable. Also, the basins of attraction of these solutions are described in detail.
In [3], a special case of equation (3) was considered when B � c � 0, i.e., It was shown that (5) exhibits period-two bifurcation and that stable manifolds of the minimal period-two solutions represent the boundaries of the basin of attraction of locally stable equilibrium point and the basins of attraction of points (0, ∞) and (∞, 0). Further, in the situations when equilibrium is a saddle point, corresponding stable manifold separates the basins of attraction of the points (0, ∞) and (∞, 0). e investigation of the special cases of equation (3) when corresponding function f is monotonically increasing in the first variable and monotonically decreasing in the second variable is significantly harder, since by now, there is no any general result for this type of monotonicity. Some initial steps, regarding this, were taken in [5], where the considered case was In [5], the authors have successfully used the embedding method to demonstrate the boundedness of the solutions, and then they determined the invariant interval of equation (6). at was a crucial idea for proving that local asymptotic stability (which holds when p > 1) implies global asymptotic stability of the unique positive equilibrium point when . Also, the existence of Neimark-Sacker bifurcation is shown and asymptotic approximation of the invariant curve is computed. e investigation of the dynamics of equation (4) and its special cases has been the subject of many research studies for the last ten years. Some of these cases, as we have seen, have been successfully realized. However, only the ideas from [5] finally made the problem of investigating the behavior of equation (1) or equation (2) solvable. Namely, note that equation (2) has the form where (a/A) � p, (b/A) � 1, and (c/A) � q. erefore, we will consider equation (7) instead of equation (1). However, the application of the embedding method on equation (7) is significantly harder compared with application on equation (6) (because the form is more complicated). We will investigate local and global stability of a unique equilibrium point and boundedness of the solutions of equation (7) and examine the existence of Neimark-Sacker bifurcation. Also, in the situation when Neimark-Sacker bifurcation appears, we will give the asymptotic approximation of the invariant curve. e special case of equation (3), when C � c � 1 and B � b � 0, was considered in [1]. In the region of parameters where a < A, corresponding function f is monotonically increasing by first variable and monotonically decreasing by second variable, and through very complicated calculations, using the so-called "M-m" theorems, it is shown that in some areas of parametric space of parameters A and a, unique positive equilibrium is globally stable. e existence of period-doubling bifurcation is proved in the case A < a, when corresponding function f is monotonically decreasing by first variable and monotonically increasing by second variable. Using the theory of monotone maps, the global stability of minimal period-two solution is shown for some special values of parameters.
Notice that equation (3) is a special case (and probably the most complicated equation of the form (3, 3)) of the following general second-order rational difference equation with quadratic terms that has caught the attention of mathematical researchers over the last ten years ( [1][2][3][4][5][6][7][8][9][10][11][12]). e following lemma gives us the type of local stability of a unique positive equilibrium point of equation (7) depending on different values of parameters p and q.

Lemma 1. Equation (7) has a unique equilibrium point
e linearized equation associated with equation (7) about the equilibrium point has the form Since the conclusion follows.

Global Asymptotic Stability
In this section, we will show that all solutions of equation (7) are bounded, and using the so-called "M-m" theorem, we will obtain sufficient conditions for unique positive equilibrium to be globally asymptotically stable. Similar to [5,13], we apply the method of embedding. First, we substitute in equation (7) and obtain en, by substituting in equation (13), we have Notice that the solutions of equation (15) are bounded: that is, Since every solution x n ′ ∞ n�− 1 of equation (7) is also a solution x n ∞ n�− 2 of equation (15) with initial values 2 , we see that the solutions of equation (7) are also bounded.
From (7) and (15), we get which implies that is, By replacing (20), we obtain the following equation: that is, Remark 1. Note that equation (22) has the unique equilibrium point x � p + q + 1, which is the same equilibrium as in equation (7). It follows from Discrete Dynamics in Nature and Society Furthermore, notice that every solution x n ′ ∞ n�− 1 of equation (7) is also a solution x n ∞ n�− 2 of equation (22) with initial values is an invariant interval for the function f.
then we obtain f(u, v, w) ≤ U. It further implies that for every p > q + 1, there exists such U, which means that I is invariant for the function f, where U ≥ p(q + pq + p 2 + q 2 )/(p − q − 1)(p + q). Since we can assume that p > q + 1 and U � p(q + pq + p 2 + q 2 )/ (p − q − 1)(p + q), I is the invariant for f. □ Lemma 3. Interval I � [p, U] is an attracting interval for equation (22).
Proof. It is clear that we need to show that every solution of equation (22) must enter interval I. Note that for arbitrary initial conditions If x 1 , x 2 , x 3 ∈ I, p > q + 1, then x n ∈ I for n > 3, by Lemma 2. Otherwise, if x 3 > (p(q + pq + p 2 + q 2 )/ (p − q − 1)(p + q)), let us prove that there must be some k > 3 such that x n ∈ I for all n ≥ k. Namely, suppose that So, from equation (22), we obtain which implies that is, By induction, we conclude Since (p 2 /(p + q + q 2 )) > 1 for p > q + 1, the right side in (33) is decreasing sequence converging to en, from (33), we get lim n⟶∞ x n ≤ U.
Since the case lim n⟶∞ x n � U is not possible (otherwise there would exist another positive equilibrium different from x), it implies that there is some k > 3 such that for all n ≥ k, i.e., every solution of equation (22) must enter the interval I. □ Theorem 1. If p ≥ ������������ � (q + 1)(2q + 1), then the equilibrium point x of equation (7) is globally asymptotically stable.
Proof. It is enough to prove that x is an attractor of equation (22). Since there exists the invariant and attracting interval I � [p, p(q + pq + p 2 + q 2 )/(p − q − 1)(p + q)] when p > q + 1, we need to check the conditions of eorem A.0.5 in [14]: By subtracting the second equation in (38) from the first, we get from which m � M or for m ≠ M, If m � p, then Since By substituting (42) into (38), we obtain the following quadratic equation: whose discriminant has the form D(p, q) � q 3q − p 2 + 2q 2 + 1 q + 3p 2 q + 4p 3 + 3q 2 + 2q 3 .

Neimark-Sacker Bifurcations
e following results are obtained by applying the algorithm from eorem 1 and Corollary 1 in [5] (see also [22]). If we make a change of variable y n � x n − x, we will shift the equilibrium point to the origin. en, the transformed equation is given by for n � 0, 1, . . . , and write equation (1) in the equivalent form: Let F denote the corresponding map defined by en, the Jacobian matrix of F is given by Discrete Dynamics in Nature and Society e eigenvalues of Jac F (0, 0) are μ(p) and μ(p) where (50) Lemma 4. If p � p 0 � q, then F has equilibrium point at (0, 0) and eigenvalues of Jacobian matrix of F at (0, 0) are μ and μ where Moreover, μ satisfies the following: such that pA � μp, Aq � μq, and pq � 1, where A � Jac F (0, 0).

□
Let p � p 0 + η, where η is a sufficiently small parameter. From Lemma 4, we can transform system (47) into the normal form  and ω(λ) so that in polar coordinates, the function F(λ, x) is given by (57) Now, we compute a(p 0 ) following the procedure in [22]. Notice that p � p 0 if and only if η � 0. First, we compute K 20 , K 11 and K 02 defined in [22]. For p � p 0 � q, we have    Discrete Dynamics in Nature and Society  Finally, we get If x is fixed point of F, then invariant curve Γ(λ) can be approximated by (69) us, we prove the following result.
Theorem 2. Let x � p + q + 1; then, there is a neighborhood U of the equilibrium point x and δ > 0 such that for |p − q| < δ and x 0 , x − 1 ∈ U, ω-limit set of solution of equation (7) is the equilibrium point if p > q and belongs to a closed invariant C 1 curve Γ(p) encircling the equilibrium point if p < q. Furthermore, Γ(q) � 0 and invariant curve Γ(p) can be approximated by Since a(p 0 ) ≠ 0, a nondegenerate Neimark-Sacker bifurcation occurs at the critical value p 0 (� q). We proved a(p 0 ) < 0 and d(p 0 ) < 0, so it implies if p < p 0 , then an attracting closed curve exists, surrounding the unstable fixed point, when the parameter p crosses the bifurcation value p 0 (supercritical Neimark-Sacker bifurcation). As p increases, the attracting closed curve decreases in size and merges with the fixed point at p � p 0 , leaving a stable fixed point (subcritical Neimark-Sacker bifurcation). All orbits starting outside or inside the closed invariant curve, except at the origin, tend to the attracting closed curve. e asymptotic approximation of the invariant curve is shown in Figure 2, and some orbits and trajectories in the case where the unique equilibrium point is stable or nonhyperbolic are given in Figures 3 and 4. Using parameter value q � 3 and decreasing the dynamical parameter p, the unique positive equilibrium point loses its stability via Neimark-Sacker bifurcation leading to chaos as depicted in Figure 5(a). Also, the Lyapunov exponents corresponding to Figure 5(a) are shown in Figure 5(b), which verifies the existence of chaos after the occurrence of Neimark-Sacker bifurcation.

Conclusion
e investigation of the dynamic stability of homogeneous difference equation (3) and all its special cases is very complicated. e corresponding function f(u, v) associated with the right side of equation (3) is monotonically increasing in the first variable and monotonically decreasing in the second variable or monotonically decreasing in the first variable and monotonically increasing in the second variable or it switches its type of monotonicity between the first and second case or vice versa, depending on the parameters which appear in the function. e theory of monotone maps (or more precisely, the theory of competitive maps) was used for determining the dynamics of equation (3) or some special case of equation (3), in the scenario when corresponding function f(u, v) is monotonically decreasing in the first variable and monotonically increasing in the second variable (see [2,3]). e investigation of the special cases of equation (3) when corresponding function f is monotonically increasing in the first variable and monotonically decreasing in the second variable is significantly harder because there is no general result for this type of monotonicity.
In every situation, when the corresponding equation does not possess minimal period-two solutions, the global stability of a unique equilibrium usually can be determined by applying the so-called "M-m" theorems and finding an invariant interval of the map f before that, of course. However, it is very often impossible or extremely complicated to conduct, as we saw in [1]. at is the case with equation (1) which does not possess minimal period-two solution (since the corresponding function is monotonically increasing by first and monotonically decreasing by second variable). So, for that reason, we studied equation (7) instead of equation (1). By using the method of embedding, we were able to connect equation (7) with equation (15). Namely, we have shown that every solution x n ′ ∞ n�− 1 of equation (7) is also a solution x n ∞ n�− 2 of equation (15) with initial con- 2 , and x 0 � p + (x 1 ′ /x 0 ′ ) + q(x 1 ′ /x 0 ′ ) 2 . Furthermore, we have shown that every solution of equation (15) is bounded, which implies that every solution of equation (7) is also bounded. After that, we linked equation (15) with equation (22) and showed that every solution x n ′ ∞ n�− 1 of equation (7) is also a solution of equation (22) with initial conditions Additionally, we determined the invariant and attracting interval for the function that is associated with the right side of equation (22) and successfully applied "M-m" theorem to get conditions for parameters p and q under which the equilibrium of equation (22) and therefore of equation (7) is globally asymptotically stable. e area of the regions in (p, q) plane where unique equilibrium is locally asymptotically stable and is not globally asymptotically stable is small (see Figure 1). We expect to prove Conjecture 1 in some of our future studies.
Finally, using Neimark-Sacker theorem [15,[18][19][20][21], we gave reduction to the normal form and performed computation of the coefficients of bifurcation, based on the computational algorithm developed in [22]. Furthermore, we determined the asymptotic approximation of the invariant curve and provided visual evidence (see . Also, based on the computational algorithm in [23], we calculated the Lyapunov exponents corresponding to Figure 5(a) to confirm the existence of chaos after the occurrence of Neimark-Sacker bifurcation as in [24,25] (see Figure 5(b)).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.