Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control

Abstract

We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold . Treating the threshold as a bifurcation parameter that varies between 0 and , we work out a complete asymptotic and bifurcation analysis. Among other things, we show that all solutions either tend to a limit 1-cycle or to a limit 2-cycle and, we find the exact regions of attraction for these cycles depending on the size of the threshold. In particular, we show that when the threshold is either small or large, there is only one corresponding limit 1-cycle which is globally attractive. It is hoped that the results obtained here will be useful in understanding interacting network models involving piecewise constant control functions.

1. Introduction

Let . In [1], Ge et al. obtained a complete asymptotic and bifurcation analysis of the following difference equation:

(1.1)

where , , and is a nonlinear signal filtering control function of the form

(1.2)

in which the positive number can be regarded as a threshold bifurcation parameter.

By adding a positive constant to the right hand side of (1.1), we obtain the following equation:

(1.3)

Since can be an arbitrary small positive number, (1.1) may be regarded as a limiting case of (1.3). Therefore, it would appear that the qualitative behavior of (1.3) will "degenerate into" that of (1.1) when tends to 0. However, it is our intention to derive a complete asymptotic and bifurcation analysis for our new equation and show that, among other things, our expectation is not quite true and perhaps such discrepancy is due to the nonlinear nature of our model at hand.

Indeed, we are dealing with a dynamical system with piecewise constant nonlinearlities (see e.g., [2–6]), and the usual linear and continuity arguments cannot be applied to our (1.3). Fortunately, we are able to achieve our goal by means of completely elementary considerations.

To this end, we first recall a few concepts. Note that given , we may compute from (1.3) the numbers in a unique manner. The corresponding sequence is called the solution of (1.1) determined by or originated from the initial vector .

Recall also that a positive integer is a period of the sequence if for all and that is the least or prime period of if is the least among all periods of . The sequence is said to be -periodic if is its least period. The sequence is said to be asymptotically periodic if there exist real numbers , where is a positive integer, such that

(1.4)

In case is an -periodic sequence, we say that is an asymptotically -periodic sequence tending to the limit -cycle (This term is introduced since the underlying concept is similar to that of the limit cycle in the theory of ordinary differential equations.) . In particular, an asymptotically 1-periodic sequence is a convergent sequence and conversely.

Suppose that is the set of all solutions of (1.1) that tend to the limit cycle . Then, the set

(1.5)

is called the the region of attraction of the limit cycle . In other words, attracts all solutions originated from its region of attraction.

Equation (1.3) is related to several linear recurrence and functional inequality relations of the form

(1.6)

(1.7)

(1.8)

(1.9)

where and . Therefore, the following facts will be needed, which can easily be established by induction.

• (i) If is a sequence which satisfies (1.6), then

(1.10)

• (ii) If is a sequence which satisfies (1.7), then

(1.11)

• (iii) If is a sequence which satisfies (1.8), then

(1.12)

• (iv) If is a sequence which satisfies (1.9), then

(1.13)

We will discuss solutions of (1.3) originated from different and in . For this reason, we let and

(1.14)

Then, for ,

(1.15)

Since

(1.16)

we see that and

(1.17)

Similarly, let and

(1.18)

Then,

(1.19)

Since , we see further that

(1.20)

Note that (1.3) is equivalent to the following two dimensional dynamical system

(1.21)

by means of the identification for . Therefore, our subsequent results can be interpreted in terms of the dynamics of plane vector sequences defined by (1.21).

In particular, the following result states that a solution of (1.21) with will have one of its terms in .

Lemma 1.1.

Let be a solution of (1.3). If , then there is such that and .

Proof.

Suppose to the contrary that for all . Then, by (1.3),

(1.22)

This, in view of (1.10) and (1.11), leads us to

(1.23)

which is a contradiction. Thus, there is such that and . Furthermore,

(1.24)

The proof is complete.

In the following discussions, we will allow the bifurcation parameter to vary from to . Indeed, we will consider five cases: (i) , (ii) , (iii) , (iv) , and (v) and show that each solution of (1.1) tend to the limit cycles

(1.25)

Furthermore, in each case, we find the exact regions of attraction of the limit cycles. Then we describe our results in terms of our phase plane model (1.21) and compare them with what we have obtained for the phase plane model of (1.1).

We remark that since we need to find the exact regions of attraction, we need to consider initial vectors belonging to (up to 9) different parts of the plane. Therefore the following derivations will seem to be repetitive. Fortunately, the principles behind our derivations are quite similar, and therefore some of the repetitive arguments can be simplified.

For the sake of convenience, if no confusion is caused, the function is also denoted by in the sequel.

2. The Case Where  

In this section, we assume that .

Lemma 2.1.

Suppose that . Let be a solution of (1.3). Then, there is such that .

Proof.

If for all , then by (1.3), for . One sees from (1.10) and (1.11) that which is a contradiction. Hence, there must exist a such that . If , we are done. Otherwise, one sees that

(2.1)

Repeating the argument we either find such that , or one has that the subsequence lies in whereas . This would mean that the subsequence satisfies (1.6) or (1.7) for , and hence , a contradiction. The proof is complete.

Theorem A

Suppose . Then every solution of (1.3) converges to .

Proof.

In view of Lemma 2.1, we may suppose without loss of generality that . Since , we have

(2.2)

and by induction for all . Thus, by (1.3), for . In view of (1.10) and (1.11), . The proof is complete.

3. The Case Where  

In this section, we suppose that . Then, . Let and

(3.1)

Then,

(3.2)

For the sake of convenience, let us set

(3.3)

Lemma 3.1.

Suppose that . Let be a solution of (1.3). If , then there is such that .

Proof.

We break up into four different parts , , , and . We also let and .

Clearly, there is nothing to prove if .

Next, suppose that . Then for some . If , then by (1.3),

(3.4)

That is, . If for some , then

(3.5)

Hence, . By induction, we see that and hence, .

Suppose . Then by (1.3), . Hence,

Suppose that . Then by Lemma 1.1, there is such that .

Suppose that . Then for some . If , then in view of (1.3),

(3.6)

Hence, . If for some , then by (1.3),

(3.7)

Hence, , and by induction, . Thus .

Suppose . Then for some . As in the previous case, we may show by similar arguments that .

Therefore, in the last four cases, we may apply the first two cases to conclude our proof. The proof is complete.

Theorem B

Suppose that . Then, every solution of (1.3) with tends to .

Proof.

Indeed, in view of Lemma 3.1, we may assume without loss of generality that . Then, the same arguments in the proof of Theorem holds so that .

Lemma 3.2.

Suppose that . Let be a solution of (1.3). If , then there is such that and .

Proof.

We break up into five different parts , , , , and .

Clearly, there is nothing to prove if .

Next, suppose that . Then, by (1.3), . Hence, .

Next, suppose that . If for all , then, by (1.3), for . In view of (1.10) and (1.11),

(3.8)

which is a contradiction. Thus there is such that and . Hence,.

Next suppose that . Then, for some . If , then by (1.3),

(3.9)

Hence, . If for some , then

(3.10)

we see that . By induction, we may further see that . Hence, .

Next, suppose that . Then for some . By arguments similar to the previous case, we may then, show that . The proof is complete.

Theorem C

Suppose that . Then, any solution with tends to the limit 2-cycle .

Proof.

In view of Lemma 3.2, we may assume without loss of generality that and . Then, by (1.3),

(3.11)

and by induction and for all . Hence, by (1.3), and for . In view of (1.10) and (1.11), and . The proof is complete.

4. The Case Where  

In this section, we suppose . Then, .

Lemma 4.1.

Suppose that . Let be a solution of (1.3). Then, there is such that and .

Proof.

We break up the plane into seven different parts: , , , , , , and .

Clearly there is nothing to prove if .

Next, suppose that . Then, by (1.3), , and hence .

Next, suppose that . If for all , then by (1.3), for , which leads us to , which is a contradiction. Hence, there is such that and . Therefore, .

Next, suppose that . If for all , then, by (1.3), for , which leads us to the contradiction . Thus there is such that and . Then and hence, .

Next, suppose that . Then, by (1.3) and induction, it is easily seen that for all . If for all , then by (1.3),

(4.1)

In view of (1.12), , which is a contradiction. Hence, there is such that .

Next, suppose that . Then, . Hence, .

Finally, suppose that . Then, by Lemma 1.1, there is such that .

Therefore, in the last three cases, we may apply the conclusions in the first four cases to conclude our proof. The proof is complete.

Theorem D

Suppose that . Then any solution of (1.3) tends to the limit 2-cycle .

Proof.

Indeed, in view of Lemma 4.1, we may assume without loss of generality that and . Then the same arguments in the proof of Theorem then shows that and .

5. The Case Where  

In this section, we assume that . Then .

Lemma 5.1.

Suppose that . Let be a solution of (1.3). If , Then, there is such that and .

Proof.

We break up the set into eight different parts: , , , , , , , and .

Clearly, there is nothing to prove if .

Next, suppose that . If for all , then by (1.3), for . In view of (1.10) and (1.11), we obtain the contradiction.

(5.1)

Hence, there is such that and . Thus .

Next, suppose that . Then, by (1.3), . Hence, .

Next, suppose that . Then, for some . If , then

(5.2)

Hence, . If for some , then

(5.3)

we see that . By induction, we see that . Hence, .

Next, suppose that . Then for some . If , then

(5.4)

Hence, . If for some , then

(5.5)

we see that . By induction, . Hence .

Next, suppose that . Then, by (1.3), . Hence, .

Next, suppose that . Then, , and hence, .

Finally suppose . Then, by Lemma 1.1, there is such that .

Therefore, in the fourth, sixth, seventh, and the eigth cases, we may use the conclusions in the other cases to conclude our proof. The proof is complete.

Theorem E

Suppose that . Then, any solution with tends to the limit 2-cycle .

Proof.

Indeed, in view of Lemma 5.1, we may assume without loss of generality that . Then the same arguments in the proof of Theorem shows that and .

Theorem F

Suppose that . Then, any solution of (1.3) with tends to .

Proof.

By (1.3),

(5.6)

and by induction, for all . Hence, for , which leads us to . The proof is complete.

6. The Case Where  

In this section, we suppose that . Then .

Lemma 6.1.

Suppose that . Let be a solution of (1.3). Then there is such that .

Proof.

If for all , then by (1.3), for all . In view of (1.12) and (1.13), this is impossible, and hence, there must exist a such that . Then . By induction, we see that the subsequence lies in , and hence for all . In view of (1.11), . The proof is complete.

Theorem G

Suppose that . Then, every solution of (1.3) converges to .

Proof.

Indeed, in view of Lemma 6.1, we may assume without loss of generality that . Then the same arguments in the proof of Theorem shows that .

7. Phase Plane Interpretation and Comparison Remarks

We first recall that (1.1) and (1.3) are equivalent to

(7.1)

(7.2)

respectively, by means of the identification for

Then, Theorem states that when , all solutions of (7.2) tends to the point , or equivalently, all solutions of (7.2) are "attracted" to the limit 1-cycle , or equivalently, the limit 1-cycle is a global attractor. For the sake of convenience, let us set

(7.3)

Then the above statements can be restated as follows.

• (i) If , then the limit 1-cycle attracts all solutions of (7.2).

• Similarly, we may restate the other Theorems obtained previously as follows.

• (ii) If , then the limit 1-cycle attracts all solutions of (7.2) originated from , and the limit 2-cycle attracts all other solutions of (7.2).

• (iii) If , then the limit 2-cycle attracts all solutions of (7.2).

• (iv) If , then the limit 1-cycle attracts all solutions of (7.2) originated from (see (3.3)), and the limit 2-cycle attracts all other solutions.

• (v) If , then the limit 1-cycle attracts all solutions of (7.2).

  For comparison purposes, let us now recall the asymptotic results in [1]. Let us set

  

  (7.4)

• (vi) If , then the limit 1-cycle attracts all solutions of (7.1) originated from , and the limit 2-cycle attracts all other solutions.

• (vii) If , then the limit 1-cycle attracts all solutions of (7.1) originated from ; the limit 2-cycle attracts all solutions of (7.1) originated from , and the limit 1-cycle attracts all other solutions.

• (viii) If , then the limit 1-cycle attracts all solutions of (7.1) originated from ; and the limit 1-cycle attracts all other solutions.

In view of these statements, we see that for a small positive , all solutions of (7.2) tend to a unique "lower" state vector, and for large , to another unique "higher" state vector. On the other hand, for a small positive , there are always solutions of (7.1) which tend to a limit 2-cycle, and solutions which tend to the limit 1-cycle , and for a large , there are solutions of (7.1) which tend to the limit 1-cycle and solutions to the limit 1-cycle . These observations show that it is probably not appropriate to call (1.1) the limiting case of (1.3)!

Finally, we mention that network models such as the following

(7.5)

can be used to describe competing dynamics and it is hoped that our techniques, and results here will be useful in these studies.