Periodic and Chaotic Orbits of a Discrete Rational System

Lazaryan, N.; Sedaghat, H.

doi:https://doi.org/10.1155/2015/519598

Discrete Dynamics in Nature and Society

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Volume 2015 | Article ID 519598 | https://doi.org/10.1155/2015/519598

Periodic and Chaotic Orbits of a Discrete Rational System

N. Lazaryan¹and H. Sedaghat¹

Academic Editor: Cengiz Çinar

Received29 Aug 2014

Revised09 Jan 2015

Accepted15 Jan 2015

Published23 Feb 2015

Abstract

We study a rational planar system consisting of one linear-affine and one linear-fractional difference equation. If all of the system’s parameters are positive (so that the positive quadrant is invariant and the system is continuous), then we show that the unique fixed point of the system in the positive quadrant cannot be repelling and the system does not have a snap-back repeller. By folding the system into a second-order equation, we find special cases of the system with some negative parameter values that do exhibit chaos in the sense of Li and Yorke within the positive quadrant of the plane.

1. Introduction

In this paper we examine the possible occurrence of coexisting cycles and Li-Yorke type chaos (see [1–5]) for the discrete planar system:where all 9 parameters are real numbers.

Rational equations and systems appear with increasing frequency in applications; see [6–10] for some instances of rational systems and equations in biological modeling. Specifically, a system of types (1a) and (1b) models certain stage-structured populations in biology; see [11–14]. If and denote the populations of adults and juveniles in period , respectively, then the following special case of (1a) and (1b): is known as a stage-structured, Beverton-Holt type population model with survival rates and fertility and competition coefficients , . Many more discrete planar and higher dimensional models in biology, economics, and other areas involve other types of rational and nonlinear difference equations; see [15–17].

The occurrence of chaotic orbits for (1a) and (1b) is far from obvious. It is well known that a system of linear difference equations with constant coefficients does not have chaotic orbits. On the other hand, if one of the equations of the system is a polynomial of degree greater than 1, then the system may possess chaotic orbits within a bounded invariant set, as in the case of the familiar logistic map on the real line or the Hénon map in the plane; see, for example, [15, 18].

We study the occurrence of complex behavior in (1a) and (1b). Prior studies of linear-fractional equations and systems (see [19] and references therein) have not been focused on demonstrating the occurrence of chaos or coexisting cycles and recent works [20, 21] that investigate homogeneous rational systems did not consider chaotic behavior. Studies of chaos in rational or planar systems generally do exist in the literature as indicated in the references below; see, for example, [22–26]. In particular, in [26] the occurrence of chaos in homogeneous rational systems in the plane is established.

Since (1b) is discontinuous on the plane (unlike polynomial equations), the existence of solutions is guaranteed for (1a) and (1b) only if division by zero is avoided at every step of the iteration. In typical studies of rational systems, it is generally assumed that all nine parameters and the initial values are nonnegative (we refer to this as the positive case) to avoid possible occurrence of singularities in the positive quadrant of the plane. This quadrant is also the part of the plane that is naturally of greatest interest in modeling applications such as the aforementioned adult-juvenile model. But the type of nonlinearity exhibited by linear-fractional equations is of a particular kind that tends to be mild in nature away from singularities. This may be one reason for the relatively well-behaved orbits in the positive case rather than complex orbits that tend to be associated with rapid rates of change.

To be more precise, we show that in the positive case any fixed point of (1a) and (1b) in the positive quadrant () must be nonrepelling; that is, it is not true that both of the eigenvalues of the system’s linearization at have modulus greater than 1. This implies that is not a snap-back repeller in the positive case.

We consider cases where some of the 9 system parameters are negative and allow singularities to occur in the positive quadrant . For instance, if , then the straight line , which is part of the singularity or forbidden set of the system in this case, crosses the positive quadrant so if any point of an orbit of (1a) and (1b) falls on this line, then division by zero occurs. With negative parameters, it is necessary to either determine the forbidden sets or find a way of avoiding them. Determination of forbidden sets has been done for some higher order equations; see, for example, [27–29]. But this is a difficult task for systems like (1a) and (1b). To identify special cases of (1a) and (1b) where orbits avoid such singularities we fold the system, that is, transform it into a second-order quadratic-fractional equation and then find special cases in which the occurrence of Li-Yorke type chaos can be established in the positive quadrant. As a bonus, we find special cases of (1a) and (1b) that have periodic solutions of all possible periods in the positive quadrant. Obtaining these results would be quite difficult without folding.

Folding is applicable generically to systems, whether continuous or discrete, and used, though not by this name, in diverse areas from control theory to the study of chaos in differential systems; see [26] for an introduction in the planar case.

2. Folding and Fixed Points

In this section, we discuss the folding of (1a) and (1b) and its fixed points in the positive quadrant . To avoid reductions to linear or to triangular systems, we assume throughout this paper that the parameters of (1a) and (1b) satisfy

2.1. Folding the System

To fold (1a) and (1b), we solve (1a) for to obtain

Now

Using (1b) and (4) to eliminate yields

Through combining terms and simplifying, we obtain the rational, second-order equation: where

We refer to the pair of (4) and (7) as a folding of (1a) and (1b). Note that (4) is a passive equation in the sense that it yields without further iterations once a solution of (7) is known. In this sense, we may think of (7) as a reduction of (1a) and (1b) to a scalar difference equation.

Equation (7) is a quadratic-fractional equation. This class of difference equations extend the linear-fractional equation and have been subjects of increasing study; see, for example, [27, 29–31].

Remark 1. A routine calculation shows that the orbits of (1a) and (1b) correspond to the solutions of (7) in the sense that if is a solution of (7) with given initial values and and is given by (4) for , then is an orbit of (1a) and (1b). Conversely, if is an orbit of (1a) and (1b) from an initial point and , then is a solution of (7).

2.2. Fixed Points in the Positive Quadrant

The fixed points of (1a) and (1b) satisfy the following equations:

From (9a),

Before calculating the values of the - and -components, we note the following facts about the solutions of the system (1a) and (1b).

Lemma 2. Assume that all system parameters are nonnegative and satisfy (3); that is, (a)If there is a fixed point of the system in the positive quadrant (i.e., ), then and .(b)If , then every orbit of (1a) and (1b) in the positive quadrant is unbounded.

Proof. (a) Let be a fixed point of the system in the positive quadrant. Then, by (9a) since by hypothesis and (11). Since , it follows that or .
(b) From (1a), it follows that for all
By induction, for all and it follows that the orbit is unbounded if .

Now, to calculate the fixed points, from (10) and (9b), we obtain

Multiplying and rearranging the terms yield a quadratic equation in given by where

Depending on whether some of the last 3 parameters are zeros or not, a number of possibilities for fixed points occur. Since we are only interested in the fixed points in the positive quadrant, it is relevant to point out that so by Lemma 2 . Assuming that to ensure the existence of real solutions for (15), we calculate the roots:

These roots can be expressed more succinctly using the parameters of the folding. We use the notation for the root with the positive sign:with given by (10) and use to denote the root with the negative sign:with again given by (10). It is an interesting fact that of the two fixed points above only one of them can be in the positive quadrant.

Lemma 3. Let all system parameters in (1a) and (1b) be nonnegative and satisfy (11). If (1a) and (1b) have a fixed point in , then that fixed point is and it is unique with given by (20) and given by (10).

Proof. Lemma 2 and the above discussion indicate that a necessary condition for the existence of fixed points in the positive quadrant is that holds. We found two possible fixed points given by (20) and (21) plus (10). Both of these are well defined if and only if (18) holds. Now, again by Lemma 2, the fixed point is in the positive quadrant if ; that is,
Similarly for , it is required that
The preceding covers all possible fixed points in the first quadrant under the hypotheses of the lemma. We now show that (23) cannot hold, thus leaving as the only possible fixed point in the first quadrant. Note that
Therefore,
Since the last quantity is nonnegative under the hypotheses, it follows that (23) does not hold and the proof is complete.

3. Nonexistence of Repellers

We see in the proof of Lemma 3 that exists in the positive quadrant if (18) and (22) both hold. Of particular interest to us is whether can be repelling under the hypotheses of Lemma 3. We recall that a fixed point is repelling if all eigenvalues of the linearization of the system at that point have modulus greater than 1.

Theorem 4. Let all system parameters in (1a) and (1b) be nonnegative and satisfy (11). If (1a) and (1b) have a fixed point in , then it is uniquely and this is not a repelling fixed point.

Proof. The first assertion follows from Lemma 3. To show that is not repelling, we examine the eigenvalues of the linearization of (1a) and (1b) at . The Jacobian matrix of (1a) and (1b) evaluated at the fixed point is where
Since by (9b) the above expressions for and reduce to
The characteristic equation of the above Jacobian is where
Let and write (30) as
The roots of (33) are the eigenvalues; that is,
When (or ), the eigenvalues are complex and their common modulus is . So both eigenvalues have modulus greater than 1 if and only if
If , then both eigenvalues are real with . By considering the 3 possible cases, routine calculations show that both eigenvalues have modulus greater than 1 if and only if
With regard to (35), note that by (10) so
By (28), so
It follows that (35) does not hold and further so that the first of the inequalities in (37) also does not hold. To check the remaining inequality , it is more convenient if we rewrite the expressions for in terms of the folding parameters, using (10) to eliminate
Note that so if and only if which reduces to
However, (20) implies that
So (44) is false, and thus or equivalently . Hence, is not repelling in the positive quadrant.

The above theorem shows that any fixed point of the system in the positive quadrant is nonrepelling if all system parameters are nonnegative; in particular, there are no snap-back repellers in the positive case (though unstable saddle fixed points exist for some parameter values).

4. Cycles and Chaos in the Positive Quadrant

If , then (7) reduces to the linear-fractional equation:

This type of linear-fractional equation has been studied extensively under the assumption of nonnegative parameters; see, for example, [19]. Although many questions remain to be answered about (46), chaotic solutions for it have not been found. To assure the occurrence of limit cycles and chaos and to avoid reductions to linear systems or to triangular systems where one of the equations is single variable, we assume that

If some of the parameters in (7) are negative, then even the existence and boundedness of solutions are nontrivial issues. Our aim here is to show that special cases of (7) with some negative coefficients exhibit Li-Yorke chaos in the positive quadrant. We note that (7) reduces to a first-order difference equation if

In this case, we define and to obtain

The theory of one-dimensional maps may be applied to (49). To simplify calculations, we assume in addition to (48) that which reduce (49) to where and .

Note that if , then (51) is affine and as such it does not have chaotic solutions.

A comprehensive study of (51) appears in [30]. The following is a consequence of the results in [30]. We point out that if is the minimal period of a solution of (51) with , then the sequence also has minimal period and by (4) has period . It follows that the orbit has minimal period .

Theorem 5. Assume that conditions (47), (48), and (50) hold with the (normalized) values and and define .(a)If , then all orbits of (1a) and (1b) with are well defined and bounded. If also , then these orbits are contained in .(b)If , then all orbits of (1a) and (1b) with converge to the unique fixed point of (1a) and (1b).(c)If , then (1a) and (1b) have an asymptotically stable 2-cycle where is given by (4) and (d)If and , then the points , , constitute a stable orbit of period 3 for (1a) and (1b) where is given by (4) and (e)If , then orbits of (1a) and (1b) with include cycles of all possible periods.(f)For , orbits of (1a) and (1b) with are bounded and exhibit chaotic behavior.

Proof. Statements (a)–(f) follow largely from Theorems 4–6 in [30]. It only remains to show that orbits whose initial points satisfy are contained in and to determine the unique fixed point. Since for all , (4) and (51) imply under the assumptions in (a) that
If , then it follows that for all and the proof of (a) is complete. Finally, in (b) we see that the fixed point of (1a) and (1b) when is determined from (9a) and (9b), (48), and (50) as which is in the positive quadrant if .

Example 6. To illustrate Theorem 5, consider the following special case of (1a) and (1b): which satisfies Part (c) of Theorem 5 () and there exists an asymptotically stable 2-cycle (a limit cycle) for this system. Different parameter values yield the following system which satisfies Parts (e) and (f) of Theorem 5 with :
This special case of (1a) and (1b) has periodic orbits of all periods (depending on initial points) and exhibits Li-Yorke type chaos. This fact is far from obvious and even the existence of cycles in the first quadrant for these equations is quite difficult to prove without folding.
We also mention that in both of the above systems so every orbit whose initial point satisfies is contained in the positive quadrant .

The hypotheses of Theorem 5 are sufficient but not necessary for the occurrence of complex behavior in the positive quadrant. In fact, due to the continuity of rational expressions in terms of their parameters, the conclusions of Theorem 5 hold if the quantities , and are sufficiently small but not necessarily zero. Numerical simulations indicate that Li-Yorke chaos persists in the positive quadrant if the parameters in the last system above are slightly perturbed. Caution is needed though because if we deviate too much from the conditions of Theorem 5, then the nontrivial nature of the singularity set must be taken into account before a claim of the occurrence of chaos can be verified.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors appreciate beneficial comments by an anonymous referee.

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Copyright

Copyright © 2015 N. Lazaryan and H. Sedaghat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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