Abstract
We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation: , where the parameters , and are nonnegative numbers with condition , , and the initial conditions , are arbitrary nonnegative numbers such that
1. Introduction and Preliminaries
In this paper we investigate the local and global dynamics of the following difference equation:where the parameters , and are nonnegative numbers with condition , , and the initial conditions , are arbitrary nonnegative numbers such that The change of variable transforms (1) to the following difference equation:where we assume that and that the nonnegative initial conditions , are such that for all . Thus the results of this paper extend to (2).
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 ,
Some special cases of (1) were investigated in [1], where special case , known as Riccati’s equation, was thoroughly investigated and its global dynamics was completely described. In fact, this is one of very few nonlinear difference equations whose solution can be explicitly found. As it was shown in [1] such equation can exhibit the whole range of different global behaviors such as global asymptotic stability of the equilibrium, global periodicity (i.e., all solutions are periodic with the same period), and chaos. To avoid this case we assume that . Furthermore, the special case is the first-order difference equation whose dynamics is well known and follows from Theorem , page 484 in [2]. Consequently, we will assume that . Finally, the special case , known as the Henon difference equation [3], was completely solved in [4], where the basins of attraction of all equilibrium points and the point at infinity were found and global dynamics was described in full detail. So, in this paper we assume that .
The first systematic study of global dynamics of a special quadratic fractional case of (3) where was performed in [5, 6]. Dynamics of some related quadratic fractional difference equations was considered in the papers [7–17]. In this paper we will perform the local stability analysis of all three equilibrium points of (1) and we will give the necessary and sufficient conditions for the equilibrium to be locally asymptotically stable, a saddle point, a repeller, or a nonhyperbolic equilibrium. The local stability analysis indicates that some possible dynamics scenarios for (1) include period doubling bifurcations, as in the case of equation considered in [18]. Another possible behavior is chaos as in the case of equation considered in [19]. Our local stability analysis will exclude the possibility of Neimark-Sacker bifurcation.
The paper is organized as follows. The rest of this section contains some global attractivity results which will be used in Section 4 to obtain global asymptotic stability results for some special cases of (1). Section 2 gives local stability analysis of all three equilibrium points when and Section 3 gives local stability analysis of all equilibrium points when . Finally, Section 4 gives some global attractivity results in some special cases. The global attractivity and global stability results that will be used in Section 4 are Theorems and from [1] and four results that follow.
The global attractivity results obtained specifically for complicated cases of (3), when many terms in numerator and denominator are present, are the following theorems [20].
Theorem 1. Assume that (3) has the unique equilibrium . If the condition holds, where and are lower and upper bounds of all solutions of (3), then is globally asymptotically stable.
Theorem 2. Assume that (3) has the unique equilibrium . If the condition holds, where and are lower and upper bounds of specific solution of (3), then is globally asymptotically stable on the interval .
In the case of (1) Theorems 1 and 2 give the following special results.
Corollary 3. If the condition holds, where and are lower and upper bounds of all solutions of (1), then is globally asymptotically stable.
Corollary 4. If the condition holds, where and are lower and upper bounds of specific solution of (1), then is globally asymptotically stable on the interval .
2. Case
This section gives complete local stability analysis for all equilibrium points (up to three) of (1) when .
2.1. Equilibrium Points
Equilibrium points of (1) are the positive solutions of the equationor equivalently
Theorem 5. (a) Equation (1) has a unique equilibrium point if one of the following conditions holds: (1) and or ;(2), , , and (3);(4). See Figures 1–3.(b) Equation (1) has two equilibrium points if , , , andorSee Figures 4(a) and 4(b).
(c) Equation (1) has three equilibrium points if , , , and See Figure 4(c).
(a)
(b)
(c)
(a)
(b)
(a)
(b)
(a)
(b)
(c)
Proof. Denote by It is easy to see that , , , and The critical points of satisfy and are given asNotice that , , and
(a)(1)Let (i)If , , then which implies that function intersects the positive part of -axis in one point since .(ii)If , , then , which implies that function intersects the positive part of -axis in one point since .(iii)If , , then Using the fact that we conclude that (1) has the unique equilibrium point.(iv)If , , then , since (1) has the unique positive equilibrium point.(2)Let If , , then , , and there exists the unique equilibrium point if or ; that is, which is equivalent to (3)(i)If , , which implies that is increasing function. Since there exists the unique positive equilibrium point.(ii)If then and which implies that is an inflection point of . Since , (1) has only one positive equilibrium point.(4)(i)If , , then and is increasing function. Since , there exists only one positive equilibrium point.(ii)If , , and which implies that there exists only one positive equilibrium point.(b) If and , , then both stationary points are positive. If orthen or and (1) has exactly two equilibrium points. In the first case . In the second case .
(c) If and , , then both stationary points are positive. If and which is equivalent to (1) has exactly three equilibrium points.
See Table 1 for existence of the positive equilibrium points of (1) when .
2.2. Local Stability Analysis
In this section we present the local stability of equilibrium points of (1) in the case
Set If denotes an equilibrium point of (1), then the linearized equation associated with (1) about the equilibrium point is where
Theorem 6. Assume that and let If any of the conditions (1),(2), , and ,(3), ,(4), , ,(5), , , and is satisfied, then the unique equilibrium point of (1) is(a)locally asymptotically stable if (b)a nonhyperbolic point if (c)a saddle point if
Proof. By using (26) and (27) we getIt is clear that If the hypotheses of theorem are satisfied, function is increasing as it passes through the unique equilibrium point Therefore which implies that
Nextare zeros of the quadratic function .
It is clear that , Notice that is equivalent to which is equivalent to
One can see that if and only if Similarly if which is equivalent to . Finally if which is equivalent to The conclusion follows from Theorem in [1].
Theorem 7. If , , and , then the unique positive equilibrium point is nonhyperbolic.
Proof. If we assume that , then . If then which means that is positive equilibrium point since . For the equilibrium point we have that which implies that is nonhyperbolic equilibrium point.
Lemma 8. Let be the partial derivative given by (27). Then
Proof. Inequality is equivalent to Using (10) we have that which is satisfied for all values of parameters.
Inequality is equivalent towhich is true.
The consequence of Lemma 8 is that cannot be a repeller.
Theorem 9. (a) If , , , and condition (13) is satisfied then (1) has two equilibrium points which is nonhyperbolic and which is locally asymptotically stable.
(b) If , , , and condition (14) is satisfied then the equilibrium point is locally asymptotically stable and the equilibrium point is nonhyperbolic.
Proof. (a) In this case the critical points of are given with (19) which yields where . Using condition (13) we obtain that which means that is an equilibrium point of (1). Since we have for the equilibrium point thatSo we conclude that the equilibrium point is nonhyperbolic.
Using the fact that function is increasing when it passes through the point , we conclude that for the equilibrium point Denote the zeros of the function as : It is clear that , Then is equivalent to which is equivalent to
By using (13), one can see that where Therefore is locally asymptotically stable.
(b) It is clear that Since we conclude that for the equilibrium point which implies that is locally asymptotically stable.
We have Using (14) we get that which means that is an equilibrium point of (1).
Since we have for the equilibrium point thatwhich implies that is a nonhyperbolic equilibrium point.
Theorem 10. Assume that and let
If , , , and then (1) has three equilibrium points: which is locally asymptotically stable, which is saddle point, and which is(i)locally asymptotically stable if (ii)nonhyperbolic if (iii)a saddle point if
Proof. Notice that , , where , , and are given by (19) and (33). Based on the properties of the cubic function we conclude that is increasing as it passes through the point which means that for the equilibrium point Furthermore is decreasing as it passes through the point which implies that for the equilibrium point The function is increasing as it passes through the point , which implies that for the equilibrium point Since we have that for the equilibrium points and So we prove that is locally asymptotically stable equilibrium point and is a saddle point.
Notice that for the equilibrium point Therefore the equilibrium point is locally asymptotically stable if nonhyperbolic if and a saddle point if
3. Case
This section gives complete local stability analysis for all equilibrium points (up to three) of (1) when . In this case (1) has always the zero equilibrium.
3.1. Equilibrium Points
Equilibrium points of (1) are the solutions of the equation or equivalently which means that the equilibrium points are and .
This implies the following:(a)If or and or and , there exists the unique equilibrium point (b)If and , there exist and the positive equilibrium point (c)If and , there exist and the positive equilibrium point (d)If , there exist and the positive equilibrium point (e)If and and , there exist and two positive equilibrium points
See Table 2 for all possible cases of existence of equilibrium points.
Remark 11. It should be noticed that conditions on existence of one, two, or three equilibrium points for (1) summarized in Tables 1 and 2 can be extended to the corresponding results for the general quadratic fractional difference equation (3). Indeed, the equilibrium points of (3) satisfy the equilibrium equationThus replacing the coefficients with , with , with , and with in Tables 1 and 2, we can obtain the explicit conditions for the existence of one, two, or three equilibrium points. The complement of the union of such conditions will give the condition for nonexistence of equilibrium points.
3.2. Local Stability Analysis
The linearized equation of (1) at the zero equilibrium is of the form where and , if we denote .
Now we have the following result on local stability of the zero equilibrium (see Theorem from [1]).
Proposition 12. The zero equilibrium of (1) is one of the following: (a)locally asymptotically stable if ,(b)nonhyperbolic of resonance type if ,(c)repeller if
The linearized equation of (1) at the positive equilibrium point is of the form
Indeed, since we have that
Now we have thatwhich shows that .
Theorem 13. (a) If and , then (1) has two equilibrium points: which is repeller and which is locally asymptotically stable.
(b) If and , then (1) has two equilibrium points: which is locally asymptotically stable and which is nonhyperbolic and locally stable.
(c) If and , then (1) has two equilibrium points: which is nonhyperbolic and locally stable and which is locally asymptotically stable.
Proof. (a) In view of Proposition 12 we have that is repeller. It follows from (67) and (70) that for the equilibrium point which shows that is locally asymptotically stable.
(b) By assumption we have that Because of that is locally asymptotically stable (see Proposition 12(a)).
For the equilibrium point we obtain from (67) and (70) that and .
The proof follows from Theorem [1].
(c) Since the zero equilibrium is nonhyperbolic and locally stable (see Proposition 12(b)). For the equilibrium point we have that which by (72) implies This means that the equilibrium point is locally asymptotically stable.
Theorem 14. Assume that . Then (1) has two equilibrium points which is repeller and which is one of the following: (i)If , then is locally asymptotically stable. (ii)If , then is nonhyperbolic. (iii)If , then is saddle point.
Proof. Proposition 12 implies that is a repeller. Note that for can take values of distinct signs. We haveSincethe proof follows from Theorem [1].
Theorem 15. Assume that , , and Then (1) has three equilibrium points: which is locally asymptotically stable, which is a saddle point, and which is locally asymptotically stable.
Proof. For the equilibrium point we have because which is satisfied under assumptions of theorem. It means that is locally asymptotically stable.
For the equilibrium point we have which implies that is saddle point.
The equilibrium point is locally asymptotically stable by Proposition 12.
4. Global Asymptotic Stability Results
In this section we give some global asymptotic stability results for some special cases of (1).
Theorem 16. Consider (1), where all coefficients are positive, subject to the conditionwhere and , and assume the hypotheses of Theorem 5(a). Then is globally asymptotically stable.
Proof. In view of Corollary 3 we need to find the lower and upper bounds for all solutions of (1) for . In this case the lower and upper bounds for all solutions of (1) for are derived as
Theorem 17. Consider (1), where and all other coefficients are positive, subject to the conditionwhere and and (i) and ; (ii), , and ; (iii)Then is globally asymptotically stable.
Proof. In this case, lower and upper bounds of all solutions of (1) are given with Now, an application of Theorem 5 and Corollary 3 completes the proof.
Theorem 18. Consider (1), where and all other coefficients are positive, subject to the conditionwhere and . Then is globally asymptotically stable.
Proof. In this case, in view of Theorem 5(a) part (), (1) has the unique equilibrium. Lower and upper bounds of all solutions of (1) are given with Now, an application of Theorem 5 part () and Corollary 3 completes the proof.
Theorem 19. Consider (1), where and all other coefficients are positive, subject to the conditionwhere and . Then is globally asymptotically stable.
Proof. In this case, in view of Theorem 5(a) part (), (1) has the unique equilibrium. Lower and upper bounds of all solutions of (1) are given withNow, an application of Corollary 3 completes the proof.
Theorem 20. Consider (1), where and all other coefficients are positive, subject to the conditionwhere , , and (i) and , , (ii) and , (iii), (iv), Then is globally asymptotically stable.
Proof. In these cases (1) has the unique equilibrium point and lower and upper bounds of all solutions of (1) are given with Now, an application of Corollary 3 completes the proof.
Theorem 21. Consider (1), where and all other coefficients are positive, subject to the conditionwhere and . Then is globally asymptotically stable.
Proof. In this case (1) has the unique equilibrium. Lower and upper bounds of all solutions of (1) are given with and an application of Corollary 3 completes the proof.
Simpler and more effective global asymptotic stability results (although less general) can be obtained by applying Theorems in [1]. For instance, consider (1), where and all other coefficients are positive; that is, consider the equationwhere we can assume that . Then the following result holds.
Theorem 22. Consider (92), subject to the conditionThen is globally asymptotically stable.
Proof. The function is decreasing in both arguments and it has an invariant interval with the property that . Furthermore is also an attracting interval; that is, , , for every solution of (92). In order to apply Theorem in [1] we need to show that the only solution of the systemis the equilibrium solution . System (95) implies and . If , then . If , then system (95) gives , which is impossible if . However, if , then and are solutions of the quadratic equation obtained by elimination of from system (95). Equation has no two real solutions if its discriminant is nonpositive.
Another special case of (1) which exhibits the global asymptotic stability of the unique equilibrium is the following equation:where we can assume that .
Theorem 23. Consider (98), subject to the conditionThen is globally asymptotically stable.
Proof. The function is increasing in and decreasing in and it has an invariant interval with the property that . Furthermore is also an attracting interval; that is, , , for every solution of (98). Equation (98) has only the zero equilibrium under the condition (99). Indeed the equilibrium solutions of (98) satisfy , and so there is no positive equilibrium if . In order to apply Theorem in [1] we need to show that the only solution of the systemis the equilibrium solution . System (95) implies and . If , then . If , then system (101) implies that and are solutions of the quadratic equation obtained by elimination of from system (101). Equation has no two real solutions if its discriminant is nonpositive, which is true if (99) holds.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.