Abstract
We investigate the global asymptotic stability of the following second order rational difference equation of the form where the parameters , , , and and initial conditions and 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.
1. Introduction and Preliminaries
In this paper, we investigate the local and global dynamics of the following difference equation:where the parameters are positive real numbers and initial conditions and are arbitrary positive real numbers. Equation (1) is the special case of a general second order quadratic fractional equation of the formwith nonnegative parameters and initial conditions such that , and . Several global asymptotic results for some special cases of Equation (2) were obtained in [1–11]. Also, Equation (1) is a special case of the equationwith positive parameters and nonnegative initial conditions , . Local and global dynamics of Equation (3) was investigated in [12].
The special case of Equation (3) when , i.e.,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 and ) is the following equation:where the parameters , , and are nonnegative numbers with condition , and the initial conditions , arbitrary nonnegative numbers such that . Equation (5) is a perturbed Sigmoid Beverton-Holt difference equation and it was studied in [9]. The special case of Equation (5) for is the well-known Thomson equationwhere the parameters and are positive numbers and the initial conditions , 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 equationIndeed (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:studied in [14], andstudied 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 , 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 , , , 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].
Theorem 1. Let be a compact interval of real numbers and assume that is a continuous function satisfying the following properties:(a) is nondecreasing in for each , and is nonincreasing in for each ;(b)If is a solution of the system then .Thenhas a unique equilibrium and every solution of Equation (11) converges to .
Theorem 2. Let be an interval of real numbers and assume that is a continuous function satisfying the following properties:(a) is nonincreasing in both variables(b)If is a solution of the system then .Then, (11) has a unique equilibrium and every solution of Equation (11) converges to .
Remark 3. 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 period-four 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 and ; see [21–23]. 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, 24–26] to conservative and nonconservative chaos, see [3, 19, 26]. Interesting applications can be found in [27].
2. 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 equationor equivalentlyEquation (1) has the unique positive solution given aswhereNow, we investigate the stability of the positive equilibrium of Equation (1). Setand observe that The linearized equation associated with Equation (1) about the equilibrium point iswhere
Theorem 4. Let The unique equilibrium point of Equation (1) given by (15) is(i)locally asymptotically stable if ,(ii)a repeller if ,(iii)a nonhyperbolic point of elliptic type if .
Proof. In view ofwe have thatandand so
Also, we haveSince , the equilibrium point will be nonhyperbolic if and . From we obtainand by using (14), we haveNow,and the characteristic equation of (19) is of the form from which that is, is nonhyperbolic equilibrium point. Let us denote Then,The condition is always satisfied. Hence, it holds: the equilibrium solution is locally asymptotically stable ifi.e., and a repeller if , which is equivalent with , i.e., . See Figure 1.
Lemma 5. Equation (1) has no minimal period-two solution.
Proof. Otherwise Equation (1) has a minimal period-two solution which satisfies Then,which yieldswhich implies . So, there is no a minimal period-two solution.
3. Global Results
In this section, we prove several global attractivity results in the parts of parametric space.
We notice that the function 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 is nonincreasing in both variables if , and nondecreasing with respect to the first variable and nonincreasing with respect to the second variable if . Sinceif we denote , 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 , so we distinguish the following scenarios:(1),(2)
Case 1 (). Notice first that implies and that implies . Now, we observe three subcases.
(a). If , the function 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 i.e., it holdsIndeed, sincewe have thatand which is true for and .
Also, since , we obtain This means that the equilibrium point belongs to the invariant interval .
Theorem 6. If , then the equilibrium point is globally asymptotically stable.
Proof. The system of algebraic equationsis reduced to the systemwhich yieldsSince , 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 , the corresponding orbit converges very quickly (see Figure 2(a)), and in the case when , the corresponding orbit converges significantly slower (see Figure 2(b)).
(a)
(b)
(b)
Lemma 7. If , then the system of algebraic equationshas the unique solution
Proof. From (48) we have thatthat is, from which or .
If , then . Since (see the first equation of system (49)), then . After substituting in the second equation of system (49), we get from which we have that Straightforward calculation show that and Notice that the solution is exactly the same as the solution , and that system (48) has a unique solution if .
Theorem 8. If , where , then the equilibrium is globally asymptotically stable.
Proof. If , thenwhich means that the interval is an invariant interval.
Indeed, since the function is nonincreasing in both variables on the invariant interval, thenand we obtain thatandHence,The following calculation will show that . Indeed, which is true.
Also, since , 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.
Lemma 9. Assume that .(i)If , then .(ii)If , then .(iii)If , then .
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) . By substituting parameter , where , in Equation (1), we obtain
Lemma 10. (i) Assume that , i.e., . Then Equation (61) does not possess a minimal period-four solution.
(ii) Assume that , i.e., . Then Equation (61) has the minimal period-four solutions of the formwhere and is an arbitrary number depending on initial conditions and .
Proof. Suppose that Equation (61) has a minimal period-four solution ; then it holdswhere . By eliminating and we obtainwhere the functions and can be written in the polynomial form aswhereSince and , 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 is satisfied for every , 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 , . Consider the polynomials and as polynomials in one variable :whereIf , then , for and , so we have thatSince 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 obtainwhereIf the equation 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 , but for and system (73) has no solutions since , see (76). Therefore, in this case, Equation (61) has no minimal period-four solution.
The positive solution of the equation isWe will show later that can not be a component of any positive solutions of system (73).
The positive solution of the equation isand . Namely,which is true, andwhich is also true. So, system (73) has no solutions since , see (76).
Now, we prove that the eventually positive roots of the equation for variable can belong only to the interval and that every can not be a component of any positive solution of system (73). First, we prove that for . Let , ; then we obtaini.e., . It means that the function eventually has the positive roots in the interval . Since we already considered the case when , now we investigate the existence of the positive roots of the equation for . As we have seen, for and , so system (73) has no solution in the case when the equation has the positive roots in the interval . This implies that Equation (61) has no minimal period-four solution whenever any root of equation lies in the interval .
Now, we prove that the equation has no root for variable if and . It is easy to see the following: For we haveandNow, it is sufficient to prove in . Sincewherethen for , , we obtainwithSimilarly, now we will consider and as polynomials in the variable (with the coefficients , , , ). The resultant of these polynomials is whereIf the equation 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 , and for and system (73) has no solutions since for . It means that Equation (61) has no minimal period-four solution.
Similarly, for the second root , we obtain ,where . It means that Equation (61) has no minimal period-four solution in this case. As we have already seen, the equation has eventually positive roots only in the interval , . Then,This implies for and , so system (73) has no solution in the case when the equation has the positive roots in the interval , which further means that Equation (61) has no minimal period-four solution.
Also, the positive solution of the equation isNote that . Now, we prove that can not be solution of system (73). Indeed, suppose the opposite, i.e.,which is a contradiction with the assumption that and .
Consequently system (73) does not have positive solutions when .
(b) Assume that . Then, system (73)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) i.e., and the solution is ,(iv) By subtracting we get i.e., Hence, the solution is This means that is a solution of system (73) and that Equation (61) does not possess a minimal period-four solution.Thus, if , then Equation (61) does not possess a minimal period-four solution. Consequently if , then Equation (61) has the minimal period-four solutions of the form (62).
Theorem 11. Assume that . Then, the unique equilibrium point of Equation (61) is globally asymptotically stable. Also, every solution of Equation (61) oscillates about the equilibrium point with semicycles of length two.
Proof. Notice that i.e., and are from the different sides of the equilibrium point (see also Lemma 9, when ). Also, that means and are always from the same side of the equilibrium point . Sincewhere is a linear function in variable , it can be seen that because Equation (61) has no period-two solutions nor period-four solutions (and it holds that , 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 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.
Remark 12. One can see from Theorems 6, 8, and 11 that the equilibrium point is globally asymptotically stable for all values of parameter such that , where , i.e., (see Figure 5(a)) and for all values of parameter such that , i.e., (see Figure 5(b)).
(d) . Since implies , Equation (61) is of the formIn 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.
(a)
(b)
Conjecture 13. If (that is ), then every solution of Equation (61) converges to some period-four solution of the form (62) or to the equilibrium point .
For some numerical values of parameters we give a visual evidence for this case. See Figures 6 and 7.
(a) = (1.5, 1)
(b) = (5.9, 5)
Case 2 (). 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.
Conjecture 14. If (that is, ), then the equilibrium point of Equation (1) is globally asymptotically stable.
4. Naimark-Sacker Bifurcation for
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). Letbe a map depending on real parameter satisfying the following conditions:(i) for near some fixed ;(ii) has two nonreal eigenvalues and for near with ;(iii) at ;(iv) for Then there is a smooth -dependent change of coordinate bringing into the formand there are smooth function , and so that in polar coordinates the function is given byIf , then there is a neighborhood of the origin and a such that for and , then -limit set of is the origin if and belongs to a closed invariant curve encircling the origin if . Furthermore, .
If , then there is a neighborhood of the origin and a such that for and , then -limit set of is the origin if and belongs to a closed invariant curve encircling the origin if . Furthermore, .
Consider a general map that has a fixed point at the origin with complex eigenvalues and satisfying and By putting the linear part of such a map into Jordan Canonical form, we may assume to have the following form near the origin:Then the coefficient of the cubic term in Equation (116) in polar coordinate is equal towhereand
Theorem 16. Assume that , , and .(i)If or , where and are positive solutions of the equation , then there is a neighborhood of the equilibrium point and such that for and then -limit set of solution of Equation (1), with initial condition is the equilibrium point if and belongs to a closed invariant curve encircling the equilibrium point if Furthermore, (ii)If or , then there is a neighborhood of the equilibrium point and a such that for and then -limit set of is the equilibrium point if and belongs to a closed invariant curve encircling the equilibrium point if Furthermore, .
Proof. See Figures 10 and 11 for visual illustration. In order to apply Theorem 15 we make a change of variableSet thenLet us define the function Then has the unique fixed point . The Jacobian matrix of is given byand its value at the zero equilibrium isi.e.,The eigenvalues , , using (128), arebecause Thenwhere Denote . For we have . The eigenvalues of are and where The eigenvectors corresponding to and are and where Further, and for for , , and . For and andHence, for system (125) is equivalent to Let , whereThen system (125) is equivalent to its normal form whereLetBy the straightforward calculation we obtain that where and Further,Now, by using (118), (119), (120), (121), and (122) we obtainand finally,If we substitute with we obtainSo,Solutions, determined numerically, are , , and . Since and it must be . Now,Further,and , i.e., . By differentiating the equilibrium equation with respect to and solving for we obtain , i.e., . Now, By substituting in the above expression and considering the fact that , we obtaini.e.,and that completes the proof of the theorem.
(a)
(b)
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The article is a joint work of all four authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
Acknowledgments
This work is supported by the Fundamental Research Funds of Bosnia and Herzegovina .