Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation
We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form where the parameters ,, and are positive numbers and the initial conditions and are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.
We investigate global behavior of the equation where the parameters,, andare positive numbers and the initial conditionsandare arbitrary nonnegative numbers such that. Equation (1) is a special case of equations Some special cases of (3) have been considered in the series of papers [1–5]. Some special second-order quadratic fractional difference equations have appeared in analysis of competitive and anticompetitive systems of linear fractional difference equations in the plane; see [6–10]. Describing the global dynamics of (3) is a formidable task as this equation contains as a special case many equations with complicated dynamics, such as the linear fractional difference equation which dynamics was investigated in great detail in  and in many papers which solved some conjectures and open problems posed in . Equation (2) can be brought to the form and one can take the advantage of this auxiliary equation to describe the dynamics of (2). This approach was used in [1–4, 12]. In this paper, we take a different approach based on the theory of monotone maps developed in [13, 14] and use it to describe precisely the basins of attraction of all attractors of this equation. The special case of (1) whenis the linear fractional difference equation whose global dynamics is described in . We show that (1) exhibits three types of global behavior characterized by the existence of a unique positive equilibrium solution and one or two minimal period-two solutions, one of which is locally stable and the other is a saddle point. The unique feature of (1) is the coexistence of an equilibrium solution and the minimal period-two solution both being locally asymptotically stable. This new phenomenon is caused by the presence of quadratic terms and did not exist in the case of (4).
Our results will be based on the following theorem for a general second-order difference equation see .
Theorem 1. Letbe a set of real numbers and letbe a function which is nonincreasing in the first variable and nondecreasing in the second variable. Then, for every solutionof the equation the subsequencesandof even and odd terms of the solution do exactly one of the following:(i)eventually, they are both monotonically increasing;(ii)eventually, they are both monotonically decreasing;(iii)one of them is monotonically increasing and the other is monotonically decreasing.
The consequence of Theorem 1 is that every bounded solution of (7) converges to either equilibrium or period-two solution or to the point on the boundary, and the most important question becomes determining the basins of attraction of these solutions as well as the unbounded solutions. The answer to this question follows from an application of theory of monotone maps in the plane which will be presented in Section 2.
We now give some basic notions about monotone maps in the plane.
Consider a partial orderingon. Two pointsare said to be related ifor. Also, a strict inequality between points may be defined asifand. A stronger inequality may be defined asifwithand.
A map on a nonempty set is a continuous function . The map is monotone if implies that for all , and it is strongly monotone on if implies that for all . The map is strictly monotone on if implies that for all . Clearly, being related is invariant under iteration of a strongly monotone map.
Throughout this paper, we will use the North-East ordering (NE) for which the positive cone is the first quadrant; that is, this partial ordering is defined byifandand the South-East (SE) ordering defined asifand.
A map on a nonempty set which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive.
If is a differentiable map on a nonempty set , a sufficient condition for to be strongly monotone with respect to the SE ordering is that the Jacobian matrix at all points has the following sign configuration: provided that is open and convex.
For , define for to be the usual four quadrants based at and numbered in a counterclockwise direction; for example, . Basin of attraction of a fixed point of a map , denoted as , is defined as the set of all initial points for which the sequence of iterates converges to . Similarly, we define a basin of attraction of a periodic point of period . The next five results, from [13, 14], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by Smith in [5, 16].
Theorem 2. Letbe a competitive map on a rectangular region . Let be a fixed point ofsuch thatis nonempty (i.e.,is not the NW or SE vertex of ), andis strongly competitive on. Suppose that the following statements are true.(a)The maphas aextension to a neighborhood of .(b)The Jacobian of at has real eigenvalues , such that , where , and the eigenspace associated with is not a coordinate axis.Then, there exists a curve through that is invariant and a subset of the basin of attraction of , such that is tangential to the eigenspace at , and is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of in the interior of are either fixed points or minimal period-two points. In the latter case, the set of endpoints of is a minimal period-two orbit of .
We will see in Theorem 4 that the situation where the endpoints of are boundary points of is of interest. The following result gives a sufficient condition for this case.
Theorem 3. For the curve of Theorem 2 to have endpoints in, it is sufficient that at least one of the following conditions is satisfied.(i)The maphas no fixed points nor periodic points of minimal period-two in.(ii)The maphas no fixed points in,, andhas no solutions.(iii)The maphas no points of minimal period-two in,, andhas no solutions.
For maps that are strongly competitive near the fixed point, hypothesis () of Theorem 2 reduces just to. This follows from a change of variables  that allows the Perron-Frobenius theorem to be applied. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis.
The next result is useful for determining basins of attraction of fixed points of competitive maps.
Theorem 4. (A) Assume the hypotheses of Theorem 2, and let be the curve whose existence is guaranteed by Theorem 2. If the endpoints of belong to , then separates into two connected components, namely, such that the following statements are true.(i)is invariant, andasfor every.(ii)is invariant, andasfor every.(B) If, in addition to the hypotheses of part (A),is an interior point of andisand strongly competitive in a neighborhood of, thenhas no periodic points in the boundary ofexcept for, and the following statements are true.(iii)For every, there existssuch thatfor.(iv)For every, there existssuch thatfor.
Ifis a map on a set and ifis a fixed point of, the stable setofis the setand unstable setofis the set
Whenis noninvertible, the setmay not be connected and made up of infinitely many curves ormay not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on , the setsandare the stable and unstable manifolds of.
Theorem 5. In addition to the hypotheses of part (B) of Theorem 4, suppose that and that the eigenspace associated with is not a coordinate axis. If the curve of Theorem 2 has endpoints in , then is the stable set of , and the unstable set of is a curve in that is tangential to at and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of in are fixed points of .
Remark 6. We say thatis strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivativenegative and first partial derivativepositive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of (7) follows from the fact that ifis strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to (7) is a strictly competitive map on; see .
Setandin (7) to obtain the equivalent system Let. The second iterateis given by and it is strictly competitive on(see ).
Remark 7. The characteristic equation of (7) at an equilibrium point, has two real rootswhich satisfyand, wheneveris strictly decreasing in first and increasing in second variable. Thus, the applicability of Theorems 2–5 depends on the nonexistence of minimal period-two solution.
There are several global attractivity results for (7). Some of these results give the sufficient conditions for all solutions to approach a unique equilibrium and they were used efficiently in . The next result is from .
Theorem 8 (see ). Consider (7) whereis a continuous function andis decreasing in the first argument and increasing in the second argument. Assume thatis a unique equilibrium point which is locally asymptotically stable and assume thatandare minimal period-two solutions which are saddle points such that
Then, the basin of attraction of is the region between the global stable sets and . More precisely,
The basins of attraction and are exactly the global stable sets of and .
Ifor, thenconverges to the other equilibrium point or to the other minimal period-two solutions or to the boundary of the region.
3. Local Stability Analysis
Lemma 9. Equation (1) has a unique positive equilibrium solution.(i)If, then equilibrium solutionis locally asymptotically stable.(ii)If, then equilibrium solutionis a saddle point.(iii)If, then equilibrium solutionis nonhyperbolic (with eigenvaluesand).
Proof. By (17), a linearization of (1) is of the form Its characteristic equation is with eigenvalues, where. It is clear thatand. Now, we prove thatand Namely,(i), which is always satisfied;(ii)also, (iii)and.Also, if, then, and we have.
4. Periodic Solutions
In this section, we present results for the existence of minimal period-two solutions of (7).
Proof. Suppose that there is a minimal period-two solutionof (1), whereandare distinct nonnegative real numbers such that. Then,satisfy
from which we obtain three cases:
Conclusion () follows from (27) and (28).
Subtracting (30) from (29), we have that is, for. Substituting (32) in (29), we obtain from which Equation (32) implies that It is clear that If, then which is a contradiction.
By using substitution equation (1) becomes the system of equations
Notice that periodic solutions,,, andof (1) are equilibrium points of the map.
Now, we have
Theorem 11. (i) The minimal period-two points
are locally asymptotically stable.
(ii) If and , then the minimal period-two points whereandsatisfy (25), are saddle points.
Proof. (i) Since, for periodic point, we have
with eigenvalues, which implies thatis locally asymptotically stable.
Similarly, since, for periodic solution, we have with eigenvalues, which implies thatis locally asymptotically stable.
(ii) By (26), we have that By (50) and (51), we obtain that is, Similarly, we have so that
Now, we obtain that Jacobian matrix of the mapat the pointis of the form
The corresponding characteristic equation is where that is, where
Notice that so that
We need to show that (i) Consider that which is satisfied because(ii)Notice that
This implies that which is satisfied.
5. Global Results and Basins of Attraction
In this section, we present global dynamics results for (1).
Notice that, , and, .
Theorem 12. If , then (7) has a unique equilibrium point , which is locally asymptotically stable, and has the minimal period-two solution , , which is a saddle point and has the minimal period-two solution , which is locally asymptotically stable. The basin of attraction of is the region between the global stable sets and . The basins of attraction and are exactly the global stable sets of and . Furthermore, the basin of attraction of the minimal period-two solution , is the union of the regions above and below in SE ordering; that is,(i)if, thenand;(ii)if, thenand.
Proof. Using assumption(and its consequences:) and (25), it is easy to check that. It is easy to check that the equilibrium pointis locally asymptotically stable for the strictly competitive mapas well. Equation (1) is equivalent to the system of difference equations (40) which can be decomposed into the system of the even-indexed and odd-indexed terms as follows:
The conclusion follows from Lemma 9 and from Theorems 10, 11, and 8 and using the facts that(i)if, then (ii)if, then
It means that(i)if, thenand, that is, (ii)if, thenand, that is, (see Figure 1).
Theorem 13. If , then (1) has a unique equilibrium point which is saddle point and has the minimal period-two solution , which is locally asymptotically stable. There exists a set which is an invariant subset of the basin of attraction of . The set is a graph of a strictly increasing continuous function of the first variable on an interval and separates , where , into two connected and invariant components and which satisfy that(i)if, thenand;(ii)if, thenand.
Proof. It is easy to check thatis a saddle point for the strictly competitive mapas well. The existence of the set with stated properties follows from Lemma 9 and Theorems 2, 4, and 10. Therefore, using (68), we obtain that(i)ifthen (ii)if, then Consequently,(i)if, thenand; that is, (ii)if, thenand; that is, (see Figure 2).
Theorem 14. If , then (1) has a unique equilibrium point which is nonhyperbolic and has two minimal period-two points , which are locally asymptotically stable points. There exists a set which is an invariant subset of the basin of attraction of . The set is a graph of a strictly increasing continuous function of the first variable on an interval and separates , where , into two connected and invariant components and which satisfy that(i)if, thenand;(ii)if,and.
Proof. In view of Lemma 9, the eigenvalues of the mapat the equilibrium pointare and , which means thatandare the eigenvalues of the mapUsing (51), (52), (54), and (56), we obtain
whereA straightforward calculation yields that the eigenvector corresponding to the eigenvalueis of the form
We see that eigenvectoris not parallel to coordinate axes. Therefore, all conditions of Theorem 2 are satisfied for the mapwith . As a consequence of this and using (68), we have that(i)ifthenand;(ii)if, thenand.
It means that(i)if, thenand; that is, (ii)if, thenand; that is, (see Figure 3).
Remark 15. As one may notice from the figures all stable manifolds of either saddle point equilibrium points or saddle period-two solutions are asymptotic to the origin, which is the point where (1) is not defined. These manifolds cannot end in any other point on the axes since the union of axes without the origin is an invariant set. Thus, the limiting points of the global stable manifolds of either saddle point equilibrium points or saddle period-two solutions have endpoints atand.
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