Stability of the Non-Hyperbolic Zero Equilibrium of Two Close-to-Symmetric Systems of Difference Equations with Exponential Terms

There has been interest in difference equations of biological models for a long period of time. Some of the research can be found in [1–8]. The most interesting special cases of such equations are usually those whose characteristic polynomials of their linearizations have characteristic zeros belonging to the unit circle. Some classical results in this direction can be found in [9,10]. The case in which general difference equations of biological models have unity as a characteristic zero has been thoroughly investigated in [6]. Some interesting concrete systems, which naturally extend one of the basic biological models of this type and which appear also in [7], can be found in [11,12]. In [2], the authors obtained results concerning the global behavior of the positive solutions for the difference equation: xn+1 = axn + bxn−1en , n = 0, 1, ...

In [22,23], the authors studied analogous results for the following close-to-symmetric systems of difference equations: respectively, where a, b, c, and d are positive constants and the initial values x −1 , x 0 , y −1 , and y 0 are also positive numbers.Now in this paper, using the center manifold theorem (see [9,10,29]), we study the stability of the zero equilibrium of the following close-to-symmetric systems of difference equations: and where a, b, c, and d are real constants.In [24], the authors studied the instability of Equations ( 1) and (2).It is known that if the zero equilibrium of Equation (1) (resp.Equation ( 2)) is hyperbolic, that is, the coefficient matrix of the linearized system of Equation (1) (resp.Equation ( 2)) has all the eigenvalues inside the unit circle, then it is easy to determine the stability of the zero equilibrium of Equation (1) (resp.Equation ( 2)) (see Theorem 4.11 of [29]).In the case for which the zero equilibrium of the above systems in non-hyperbolic, the dynamics of the center manifold plays an important role in the determination of the stability of the zero equilibrium of the systems.More precisely, using the center manifold theorem (see [9,10,29]), the dynamics of the systems can be obtained by studying a one-dimensional equation that contains an approximation of the center manifold.
Regarding the asymptotic behavior of the positive solutions of some scalar equations related to the above systems, we note that the most interesting case is when the sum of the coefficients is equal to one (see [6]).For this goal, asymptotic methods and their applications were employed.For some results on the methods and on the existence of specific types of solutions, see, for example, [30][31][32] and the references therein.Some of the equations related to that in [2] have appeared in mathematical biology (see, e.g., [6][7][8]).We also note that results concerning symmetric and cyclic systems of difference equations, for study that seemed to have been initiated in [33], are included in the papers [5,[13][14][15][16][20][21][22][23][24][34][35][36][37][38][39][40][41][42] and the related references therein.Finally, we note that, because difference equations have several applications in applied sciences, there exists a rich bibliography concerning theory and applications (see ).

Stability of Zero Equilibrium of Equation (1)
In the following, we find conditions for the stability of the zero equilibrium of Equation (1) using center manifold theory.Proposition 1.Consider Equation ( 1), in which a, b, c, and d are real constants and x 0 and y 0 are also real numbers.Suppose that the following relations hold: Then the matrix has one eigenvalue λ 1 = −1 and the other eigenvalue λ 2 < 1. Suppose also that one of the following holds: where or Equation ( 3) and the relations where or Equation ( 3) and the relations Then the zero equilibrium of Equation ( 1) is stable.
Because the characteristic equation of J is p(λ) = λ 2 − λ(a + c) + ac − bd = 0 and the conditions for Equation (3) hold, we have that λ 1 = −1 is an eigenvalue of J.Moreover, we obtain p(λ) = (λ + 1)(λ − a − c − 1), and therefore λ 2 = a + c + 1 is also an eigenvalue of J.Because Equation ( 3) is satisfied, it is clear that |λ 2 | < 1.We now let where T is the matrix that diagonalizes J defined by Then Equation (8) can be written as where We now let v = h(u) with h(u) = ψ(u) + O(u 4 ) and ψ(u) = Au 2 + Bu 3 , where A and B are real numbers, the center manifold.The use of ψ(u) as an approximation of h(u) is justified by Theorem 7 of [9].Consequently, according to Theorem 8 of [9] (see also Theorem 5.2 of [29]), and using Equations ( 9) and ( 10), the study of the stability of the zero equilibrium of Equation ( 1) reduces to the study of the stability of the scalar equation The map G can also be written in the following form: where c i , i = 1, 2, ..., 12 are real constants depending on a, c, b, d, and A. In what follows in this section, we use the following constants: We need to compute the constant A of the center manifold.From Equation ( 9), we take which implies that Then we take From Equation (12), we see that G (0) = −1 and Then from Equations ( 13)-( 15) and from Equation (3), d = (a+1)(c+1) b , we obtain We suppose first that Equations ( 3) and ( 4) are satisfied, considering that ρ 2 is a root of the quadratic polynomial h(b).We now show that We consider We can easily prove that Because a > −2+ √ 4−3c 2  3   , we obtain 3a 2 + 4a + c 2 > 0.Moreover, because c < 0, we have Then, if a < 0, from the inequality of Equation ( 19), we obtain Equations ( 3), (18), and (20) imply that ∆ > 0. This means that D, as defined in Equation ( 5), is a real number.Finally, from the third inequality of Equation ( 4), we have that Equation ( 17) is true.Now, if a > 0, from Equation (3), we have and thus from Equations ( 3) and ( 18), we obtain ∆ > 0. Thus, D is a real number.Hence from the third inequality of Equation ( 4), we have that Equation ( 17) is satisfied.Therefore, from Equations ( 3), (4), and ( 16), we obtain G (0) > 0. This implies that SG(0) = −G (0) − 3/2(G(0) ) 2 < 0, where SG(0) is the Schwarzian derivative.Hence the zero equilibrium is stable for the scalar Equation ( 11).Thus, from Theorem 8 of [9] (see also Theorem 5.2 of [29]), the zero equilibrium of the original system (Equation ( 1)) is stable.

Stability of Zero Equilibrium of Equation (2)
In the following, we study the stability of the zero equilibrium of Equation (2) using center manifold theory.Proposition 2. Consider Equation ( 2) where a, b, c, and d are arbitrary real constants and the initial values x 0 and y 0 are also real numbers.Suppose that the following relations hold: Then the matrix J = b a c d has one eigenvalue λ 1 = −1 and the other eigenvalue λ 2 < 1. Suppose also that either Equation ( 22) and b > 0, −1 hold, or that Equation ( 22) and the relations hold.Then the zero equilibrium of Equation ( 2) is stable.
Because the characteristic equation of J is p(λ) = λ 2 − λ(b + d) + bd − ac = 0 and the Equation ( 22) conditions hold, we have that λ 1 = −1 is an eigenvalue of J.Moreover, we obtain p(λ) = (λ + 1)(λ − b − d − 1), and therefore λ 2 = b + d + 1 is also an eigenvalue of J.Because Equation ( 22) is satisfied, it is clear that |λ 2 | < 1.We let now where T is the matrix that diagonalizes J, defined by Therefore, Equation ( 25) can be written as where We now let v = h(u) with h(u) = ψ(u) + O(u 4 ) and ψ(u) = Au 2 + Bu 3 , where A and B are real numbers, the center manifold.The use of ψ(u) as an approximation of h(u) is justified by Theorem 7 of [9].Consequently, according to Theorem 8 of [9] (see also Theorem 5.2 of [29]) and using Equations ( 26) and ( 27), the study of the stability of the zero equilibrium of Equation ( 2) reduces to the study of the stability of the following scalar equation: The map G can also be written in the following form: where c i , i = 1, 2, ..., 12 are real constants that depend on a, b, c, d, and A. In what follows, we use the following constants: We need to compute the constant A. From Equation ( 26), we take which implies that Then we take From Equation ( 29), we see that G (0) = −1, and Equation ( 15) is satisfied.Then from Equations ( 15), (30), and (31), we obtain Suppose firstly that Equations ( 22) and ( 23) hold.We now show that G (0) > 0. From Equation ( 23 Therefore, from Equations ( 32), (33), and (34), we have G (0) > 0.
Finally suppose that Equations ( 22) and ( 24) are satisfied.From Equation ( 24 Hence, we have SG(0) < 0, where SG(0) is the Schwarzian derivative.This implies that the zero equilibrium is stable for the scalar Equation (28).Thus, from Theorem 8 of [9] (see also Theorem 5.2 of [29]), the zero equilibrium of the original system (Equation ( 2)) is also stable.
Author Contributions: All authors contributed in preparing the final manuscript and the associated theoretical research.More precisely, G.P. conceived the idea, gave useful insights during analysis, and submitted the paper.C.M. performed the theoretical analysis and contributed in the writing of the manuscript.C.S. and N.P. proved their expertise in revising the final article.