Global asymptotic stability and naimark-sacker bifurcation of Global asymptotic stability and naimark-sacker bifurcation of certain mix monotone difference equation certain mix monotone difference equation

We investigate the global asymptotic stability of the following second order rational difference equation of the form 𝑥 𝑛+1 = (𝐵𝑥 𝑛 𝑥 𝑛−1 + 𝐹)/(𝑏𝑥 𝑛 𝑥 𝑛−1 + 𝑐𝑥 2𝑛−1 ), 𝑛 = 0, 1, .. ., where the parameters 𝐵 , 𝐹 , 𝑏 , and 𝑐 and initial conditions 𝑥 −1 and 𝑥 0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

was studied in [8].The authors performed the local stability analysis of the unique equilibrium point and gave the necessary and sufficient conditions for the equilibrium to be locally asymptotically stable, a repeller or nonhyperbolic equilibrium.Also, it was shown that Equation (4) exhibits the Naimark-Sacker bifurcation.
The special case of Equation (3) (when  =  = 0 and  = 1) is the following equation: where the parameters , , and  are nonnegative numbers with condition  +  > 0,  ̸ = 0 and the initial conditions  −1 ,  0 arbitrary nonnegative numbers such that  −1 + 0 > 0. Equation ( 5) is a perturbed Sigmoid Beverton-Holt difference equation and it was studied in [9].The special case of Equation (5) for  = 0 is the well-known Thomson equation where the parameters  and  are positive numbers and the initial conditions  −1 ,  0 are arbitrary nonnegative numbers, is used in the modelling of fish population [13].
The dynamics of ( 6) is very interesting and follows from the dynamics of related equation ,  = 0, 1, . . . .
Indeed ( 6) is delayed version of (7) and so it exhibits the existence of period-two solutions.Two interesting special cases of Equation ( 2) are the following difference equations: ,  = 0, 1, . . ., studied in [14], and studied in [5].In both equations, ( 8) and ( 9), the associated map changes its monotonicity with respect to its variable.
In this paper, in some cases when the associated map changes its monotonicity with respect to the first variable in an invariant interval, we will use Theorems 1 and 2 below in order to obtain the convergence results.However, if  =   = (/) 3 , we would not be able to use this method, so we will use the semicycle analysis; see [15] to show that each of the following four sequences { 4 } ∞ =1 , { 4+1 } ∞ =0 , { 4+2 } ∞ =0 , { 4+3 } ∞ =0 converges to the unique equilibrium point.Also, we will show that Equation (1) exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.
Note that the problem of determining invariant intervals in the case when the associated map changes its monotonicity with respect to its variable has been considered in [17,18].
In this paper, we will use the following well-known results, Theorem 2.22, in [16], and Theorem 1.4.7 in [19].
en, ( ) has a unique equilibrium  ∈ [, ] and every solution of Equation ( ) converges to .

Remark .
As is shown in [20] the unique equilibrium in Theorems 1 and 2 is globally asymptotically stable.
The rest of this paper is organized as follows.The second section presents the local stability of the unique positive equilibrium solution and the nonexistence of the minimal period-two solution.The third section gives global dynamics in certain regions of the parametric space.The results and techniques depend on monotonic character of the transition function (, ) which is either decreasing in both arguments or increasing in first and decreasing in second argument.In simpler situations Theorems 1 and 2 are sufficient to prove global stability of the unique equilibrium.In more complicated situations we use the semicycle analysis, which is extensively used in [15,19] for many linear fractional equations, to prove that every solution has four convergent subsequences, which leads to the conclusion that every solution converges to period-four solution.In some parts of parametric space we prove that there is no minimal periodfour solution and so every solution converges to the equilibrium, while in other parts of parametric space we prove that the period-four solution exists.The semicycle analysis presented here uses innovative techniques based on analysis of systems of polynomial equations which coefficients depend on four parameters.Finally in the region of parameters complementary to the one where the period-four solution exists we prove that the Naimark-Sacker bifurcation takes place which produces locally stable periodic solution.All numerical simulations indicate that the equilibrium solution is globally asymptotically stable whenever it is locally asymptotically stable and that the dynamics is chaotic whenever the equilibrium is repeller.An interesting feature of Equation ( 1) is that it gives an example of second order difference equation with period-four solution for which period-two solution does not exist.The global dynamics of Equation (11) when the transition function (, ) is either increasing in both arguments or decreasing in the first and increasing in the second argument is fairly simple as every solution {  } breaks into two eventually monotonic subsequences { 2 } and { 2+1 }; see [21][22][23].The global dynamics of Equation ( 11) when the transition function (, ) is either decreasing in both arguments or increasing in the first and decreasing in the second argument could be quite complicated ranging from global asymptotic stability of the equilibrium, see [19,21,22,[24][25][26] to conservative and nonconservative chaos, see [3,19,26].Interesting applications can be found in [27].

Linearized Stability
In this section, we present the local stability of the unique positive equilibrium of Equation ( 1) and the nonexistence of the minimal period-two solution of Equation (1).
In view of the above restriction on the initial conditions of Equation ( 1), the equilibrium points of Equation ( 1) are the positive solutions of the equation or equivalently Equation ( 1) has the unique positive solution  given as where Now, we investigate the stability of the positive equilibrium of Equation (1).Set and observe that The linearized equation associated with Equation (1) about the equilibrium point  is where Theorem 4. Let  0 = (/) 3 .e unique equilibrium point  of Equation ( ) given by ( ) is Proof.In view of we have that and and so || < |1 − |.Also, we have Since || < |1 − |, the equilibrium point  will be nonhyperbolic if  = −1 and || < 2. From  = −1 we obtain and by using (14), we have Now, and the characteristic equation of ( 19) is of the form from which that is,  is nonhyperbolic equilibrium point.Let us denote Then, and The condition || < |1 − | = 1 −  is always satisfied.Hence, it holds: the equilibrium solution  is locally asymptotically stable if i.e.,  <  0 and a repeller if  > −1, which is equivalent with  > 3 √/ =  ℎ , i.e.,  >  0 .See Figure 1.
which implies  = .So, there is no a minimal period-two solution.

Global Results
In this section, we prove several global attractivity results in the parts of parametric space.We notice that the function (, V) is always decreasing with respect to the second variable and can be either decreasing or increasing with respect to the first variable, depending on the sign of the nominator of    .Therefore, and the function (, V) is nonincreasing in both variables if V ≤ √/, and nondecreasing with respect to the first variable and nonincreasing with respect to the second variable if V ≥ √/.Since if we denote   = (/) 3 , we can have three possible cases: As we have been seen, the nature of the local stability of the equilibrium point depends on the parameter  0 , so we distinguish the following scenarios: (1)   ≤  0 , (2)   >  0 .
(a)  <   ≤  0 .If  <   ≤  0 , the function (, V) is nondecreasing with respect to the first variable and nonincreasing with respect to the second variable on the invariant interval of Equation ( 1) which is given by and which is true for  ≤  and  <   .
Also, since  < ( 3 / 3 ) = , we obtain This means that the equilibrium point  belongs to the invariant interval [, ].
Theorem 6.If  <   ≤  0 , then the equilibrium point  is globally asymptotically stable.
Proof.The system of algebraic equations is reduced to the system which yields Since  ̸ = , then it implies that  =  = .Now, by using Theorems 1 and 4, the conclusion follows.
For some numerical values of parameters we give a visual evidence for Theorem 6 which indicates that in the case when  <   <  0 , the corresponding orbit converges very quickly (see Figure 2 Proof.From (48) we have that from which  =  =  or ( +  − ( + )) = 0.If ( + − ( + )) = 0, then  = /(− + (+ )).Since  = ( 2 + )/ 2 ( + ) > /( + ) (see the first equation of system (49)), then  > 0. After substituting  in the second equation of system (49), we get from which we have that Straightforward calculation show that  1 =  2 and  2 =  1 .Notice that the solution ( 2 ,  2 ) is exactly the same as the solution ( 1 ,  1 ), and that system (48) has a unique solution which means that the interval [, ] = [/, √/] is an invariant interval.Indeed, since the function  is nonincreasing in both variables on the invariant interval, then and we obtain that and Hence, The following calculation will show that   <  3 /.Indeed, which is true.Also, since (√/)(/) < 0, it means that the equilibrium point  belongs to the invariant interval [, ].Now, by using Lemma 7, Theorems 2 and 4, we get the conclusion that the equilibrium  is globally asymptotically stable.
For some numerical values of parameters we give a visual evidence for Theorem 8. See Figure 3. Proof.Since the map associated with the right-hand side of Equation ( 1) is always decreasing in the second variable, we have that Note that under assumption of Lemma 9, the following inequality holds: (c)  =   <  0 .By substituting parameter  =   = (/) 3  =  3 , where  =  = /, in Equation ( 1), we obtain where  = .By eliminating  and  we obtain where the functions (, ) and (, ) can be written in the polynomial form as where Since  ̸ = 0 and  ̸ = 0, from system (64), we obtain the following four cases: (1) The system implies  =  = , and by using (63), we get  =  = .
(2) The system implies  =  and if  < .If  = , then (, ) = 0 is satisfied for every  > 0, and by using system (63), it follows that the periodic solution of the minimal period four is of the form (62). (3) The system implies  =  and so the conclusion is the same as in the previous case.(4) The system demands more detailed analysis.
(a) Assume that  > .Then we can write  =  + ,  > 0. Consider the polynomials (, ) and (, ) as polynomials in one variable : where If  ∈ [ 2 /( + ) 2 , ], then   ⩾ 0, for  = 0, 1, 4 and  2 > 0,  3 > 0, so we have that Since (, ) and (, ) are polynomials of the fifth and fourth degrees, respectively, the resultant of these polynomials is the determinant of the ninth degree: from which we obtain where If the equation   (, ) = 0 has solutions for variable , then they are the common roots of both equations in system (73) for a fixed value of .One of these positive roots is  1 = , but for  =  and  > 0 system (73) has no solutions since (, ) > 0, see (76).Therefore, in this case, Equation (61) has no minimal period-four solution.
The positive solution of the equation We will show later that  2 can not be a component of any positive solutions of system (73).
Also, the positive solution of the equation Λ 2 () = 0 is Note that  3 =  2 .Now, we prove that ( 2 ,  2 ) can not be solution of system (73).Indeed, suppose the opposite, i.e., which is a contradiction with the assumption that  > 0 and  > 0.
Consequently system (73) does not have positive solutions when  > .
(b) Assume that  = .Then, system (73) (, , , ) = 0,  (, , , ) = 0, (99) is of the form and combining those equations, we have the following four cases: (i) and the solution in this case is  =  = , (ii) and substituting  by  we obtain from which we get that the solution is  =  = , (iii) and the solution is  =  = , (iv) (106) By subtracting we get i.e.,
Theorem 11.Assume that  =   = (/) 3  <  0 .en, the unique equilibrium point  = / of Equation ( ) is globally asymptotically stable.Also, every solution of Equation ( ) oscillates about the equilibrium point  with semicycles of length two.
Proof.Notice that i.e.,  +1 and  −1 are from the different sides of the equilibrium point (see also Lemma 9, when √/ = /).Also, that means  +1 and  +5 are always from the same side of the equilibrium point  = /.Since where  = ( 3  +3  +2 + 2 )− 3 ( +3  +2 +( +2 ) 2 )  is a linear function in variable   , it can be seen that  = 0 ⇐⇒  +4 =   = / because Equation (61) has no period-two solutions nor period-four solutions (and it holds that   = / ⇒  +2 = / ⇒  +4 = /, see Lemma 9).Also, which means that every sequence is monotone and bounded.That implies that each of the sequences is convergent.Since, by Lemmas 5 and 10, Equation (61) has neither minimal period-two nor period-four solutions, it holds lim which implies that equilibrium  is an attractor and by using Theorem 4, which completes the proof of the theorem.
For some numerical values of parameters we give a visual evidence for Theorem 11.See Figure 4.
In this case, by using Lemma 10, we see that Equation (61) has minimal period-four solutions of the form (62). Based on our many numerical simulations and the proof of Theorem 11, we believe that the following conjectures are true.
Conjecture 13.If  =   =  0 (that is  = ), then every solution of Equation ( ) converges to some period-four solution of the form ( ) or to the equilibrium point .
For some numerical values of parameters we give a visual evidence for this case.See Figures 6 and 7. Case ( <  0 <   ).We give a visual evidence for some numerical values of parameters which indicates very interesting behaviour and verifies our suspicion that the equilibrium point  is globally asymptotically stable in this case also.See Figures 8 and 9.

Naimark-Sacker Bifurcation for 𝑏 ̸ = 𝑐
In this section, we consider bifurcation of a fixed point of map associated with Equation (1) in the case where the eigenvalues are complex conjugates and of unit module.We use the following standard version of the Naimark-Sacker result, see [28,29] Theorem 15 (Naimark-Sacker or Poincare-Andronov-Hopf Bifurcation for maps).Let  : R × R 2 → R 2 ; (, ) →  (, ) be a  4 map depending on real parameter  satisfying the following conditions: If ( 0 ) > 0, then there is a neighborhood  of the origin and a  > 0 such that for | −  0 | <  and  0 ∈ , then -limit set of  0 is the origin if  <  0 and belongs to a closed invariant  1 curve Γ() encircling the origin if  >  0 .Furthermore, Γ( 0 ) = 0.
If ( 0 ) < 0, then there is a neighborhood  of the origin and a  > 0 such that for | −  0 | <  and  0 ∈ , then -limit set of  0 is the origin if  >  0 and belongs to a closed invariant Then the coefficient ( 0 ) of the cubic term in Equation (116) in polar coordinate is equal to where and )) . ( Theorem 16.Assume that , ,  > 0,  0 = (/)   then Let us define the function Then (, V) has the unique fixed point (0, 0).The Jacobian matrix of (, V) is given by and its value at the zero equilibrium is i.e., The eigenvalues (), (), using (128), are Discrete Dynamics in Nature and Society because Then where Denote  0 = (/) 3 .For  =  0 we have  = 3 √ 0 / = /.The eigenvalues of  0 are ( 0 ) and ( 0 ) where The eigenvectors corresponding to ( 0 ) and ( 0 ) are V( 0 ) and V( 0 ) where Further, and   ( 0 ) ̸ = 1 for  = 1, 2, 3, 4 for  > 0,  > 0, and  ̸ = .For  =  0 and  = /  ( and Hence, for  =  0 system (125) is equivalent to ) . (140) Then system (125) is equivalent to its normal form where Let By the straightforward calculation we obtain that     and that completes the proof of the theorem.