Stability and Bifurcations Analysis of Discrete Dynamical SystemsView this Special Issue
Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation
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 .
The special case of Equation (3) when , i.e.,was studied in . 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 . 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 .
Two interesting special cases of Equation (2) are the following difference equations:studied in , andstudied in . 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  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.
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 .
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 .
2. Linearized Stability
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
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.
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)).
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.
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 obtainwhere