Complete Asymptotic Analysis of a Nonlinear Recurrence Relation with Threshold Control

Abstract

We consider a three-term nonlinear recurrence relation involving a nonlinear filtering function with a positive threshold . We work out a complete asymptotic analysis for all solutions of this equation when the threshold varies from to . It is found that all solutions either tends to 0, a limit 1-cycle, or a limit 2-cycle, depending on whether the parameter is smaller than, equal to, or greater than a critical value. It is hoped that techniques in this paper may be useful in explaining natural bifurcation phenomena and in the investigation of neural networks in which each neural unit is inherently governed by our nonlinear relation.

1. Introduction

Let In [1], Zhu and Huang discussed the periodic solutions of the following difference equation:

(1.1)

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

(1.2)

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

In this paper, we consider the following delay difference equation:

(1.3)

where and . Besides the obvious and complementary differences between (1.1) and our equation, a good reason for studying (1.3) is that the study of its behavior is preparatory to better understanding of more general (neural) network models. Another one is that there are only limited materials on basic asymptotic behavior of discrete time dynamical systems with piecewise smooth nonlinearities! (Besides [1], see [2–6]. In particular, in [2], Chen considers the equation

(1.4)

where is a nonnegative integer and is a McCulloch-Pitts type function

(1.5)

in which is a constant which acts as a threshold. In [3], convergence and periodicity of solutions of a discrete time network model of two neurons with Heaviside type nonlinearity are considered, while "polymodal" discrete systems in [4] are discussed in general settings.) Therefore, a complete asymptotic analysis of our equation is essential to further development of polymodal discrete time dynamical systems.

We need to be more precise about the statements to be made later. To this end, we first note that given we may compute from (1.3) the numbers in a unique manner. The corresponding sequence is called the solution of (1.3) determined by the initial vector For better description of latter results, we consider initial vectors in different regions in the plane. In particular, we set

(1.6)

which is the complement of nonpositive orthant and contains the positive orthant Note that is the union of the disjoint sets

(1.7)

(1.8)

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 the least period of The sequence is said to be asymptotically periodic if there exist real numbers where is a positive integer, such that

(1.9)

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 -periodic sequence is a convergent sequence and conversely.

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

(1.10)

by means of the identification for Therefore our subsequent results can be interpreted in terms of the dynamics of plane vector sequences defined by (1.10). For the sake of simplicity, such interpretations will be left in the concluding section of this paper.

To obtain complete asymptotic behavior of (1.3), we need to derive results for solutions of (1.3) determined by vectors in the entire plane. The following easy result can help us to concentrate on solutions determined by vectors in

Theorem 1.1.

A solution of (1.3) with in the nonpositive orthant is nonpositive and tends to

Proof.

Let Then by (1.3),

(1.11)

and by induction, for any we have

(1.12)

Since we have

(1.13)

The proof is complete

Note that if we try to solve for an equilibrium solution of (1.3), then

(1.14)

which has exactly two solutions , when and has the unique solution when However, since is a discontinuous function, the standard theories that employ continuous arguments cannot be applied to our equilibrium solutions or to yield a set of complete asymptotic criteria. Fortunately, we may resort to elementary arguments as to be seen below.

To this end, we first note that our equation is autonomous (time invariant), and hence if is a solution of (1.3), then for any the sequence defined by for is also a solution. For the sake of convenience, we need to let

(1.15)

Then

(1.16)

(1.17)

(1.18)

We also let

(1.19)

Then

(1.20)

(1.21)

(1.22)

(1.23)

2. The Case  

Suppose Then

(2.1)

We first show the following.

Lemma 2.1.

Let If is a solution of (1.3) with then there exists an integer such that

Proof.

From our assumption, we have Let be a solution of (1.3) with Then there are eight cases.

Case 1.

If our assertion is true by taking

Case 2.

Suppose Then Furthermore, in view of (1.17) and (2.1),

(2.2)

If then by (1.3),

(2.3)

This means that our assertion is true by taking Next, if then by (1.3) and (1.18),

(2.4)

Thus our assertion holds by taking If where is an arbitrary positive integer, then by (1.3),

(2.5)

By induction,

(2.6)

Thus our assertion holds by taking

Case 3.

Suppose We assert that there is a nonnegative integer such that for and Otherwise we have for It follows that

(2.7)

By induction, for any we have

(2.8)

which implies

(2.9)

This is contrary to the fact that for

Now that there exists an integer such that and it then follows

(2.10)

If then our assertion holds by taking If then Thus

(2.11)

If then our assertion holds by taking If we have Hence

(2.12)

Repeating the procedure, we have

(2.13)

If then our assertion holds by taking Otherwise,

(2.14)

for all But this is contrary to (2.1). Thus we conclude that for some Our assertion then holds by taking

Case 4.

Suppose As in Case 2,

(2.15)

If then by (1.3),

(2.16)

Thus our assertion holds taking If then by (1.3),

(2.17)

Thus our assertion holds by taking If where is an arbitrary positive integer, then by (1.3),

(2.18)

Thus our assertion holds by taking

Case 5.

Suppose Then by (1.21) and (1.23),

(2.19)

If then by (1.3),

(2.20)

Thus our assertion holds by If then by (1.3),

(2.21)

Thus our assertion holds by taking If where is an arbitrary positive integer, then by (1.3), we have

(2.22)

That is, Therefore we may conclude our assertion by induction.

Case 6.

Suppose Since

(2.23)

we see that

(2.24)

If then by (1.3),

(2.25)

That is, We may thus apply the conclusion of Case 5 and the time invariance property of (1.3) to deduce our assertion. If where is an arbitrary nonnegative integer, then by (1.3), we have

(2.26)

That is, We may thus use induction to conclude our assertion.

Case 7.

Suppose As in Case 5,

(2.27)

If then by (1.3),

(2.28)

Thus our assertion holds by taking If then by (1.3),

(2.29)

That is, Thus our assertion holds by taking

If where is an arbitrary positive integer, then by (1.3), we have

(2.30)

That is, Thus our assertion follows from induction.

Case 8.

Suppose Then

(2.31)

If then by (1.3),

(2.32)

That is, We may now apply the assertion in Case 5 to conclude our proof. If where is an arbitrary nonnegative integer, then by (1.3), we have

(2.33)

That is, We may thus complete our proof by induction.

Theorem 2.2.

Suppose then a solution of (1.3) with will tend to .

Proof.

In view of Lemma 2.1, we may assume without loss of generality that From our assumption, we have Furthermore, by (1.3),

(2.34)

By induction, for any we have

(2.35)

and similarly

(2.36)

Thus for any and

(2.37)

The proof is complete.

3. The Case  

We first show that following result.

Lemma 3.1.

Let If is a solution of (1.3) with there exists an integer such that and (or and ).

Proof.

From our assumption, we have Let be the solution of (1.3) determined by Then there are eight cases to show that there exists an integer such that and .

Case 1.

Suppose Then our assertion is true by taking

Case 2.

Suppose By (1.3)

(3.1)

This means that our assertion is true by taking

Case 3.

Suppose If for any then by (1.3),

(3.2)

By induction, for any we have

(3.3)

Hence

(3.4)

But this is contrary to our assumption that Hence there exists an integer such that and Thus our assertion holds by taking

Case 4.

Suppose As in Case 3 of Lemma 2.1, we may show that if for all then it follows that

(3.5)

But this is contrary to the fact that for Hence there exists an integer such that and it then follows

(3.6)

This means that our assertion is true by taking

Case 5.

Suppose . Then by (1.21) and (1.23),

(3.7)

If then by (1.3),

(3.8)

When we have

(3.9)

That is, We may thus apply the conclusion of Case 3 to deduce our assertion.

Suppose If then we have

(3.10)

We may apply the conclusion of Case 3 to deduce our assertion. If we have

(3.11)

Thus our assertion holds by taking If then by (1.3),

(3.12)

That is, In view of the above discussions, our assertion is true. If where is an arbitrary positive integer, then by (1.3), we have

(3.13)

That is, Therefore we may conclude our assertion by induction.

Case 6.

Suppose

(3.14)

As in Case 6 of Lemma 2.1, if then by (1.3), we have We may thus apply the conclusion of Case 5 to deduce our assertion. If where is an arbitrary nonnegative integer, then by (1.3), we have We may thus use induction to conclude our assertion.

Case 7.

Suppose By (1.3), we have

(3.15)

That is, We may thus apply the conclusion of Case 5 to deduce our assertion.

Case 8.

Suppose Then

(3.16)

As in Case 8 of Lemma 2.1, if then by (1.3), we have We may now apply the assertion in Case 5 to conclude our proof. If where is an arbitrary nonnegative integer, then by (1.3), we have We may thus complete our proof by induction.

Theorem 3.2.

Let Then any solution of (1.3) with is asymptotically -periodic with limit -cycle

Proof.

In view of Lemma 3.1, we may assume without loss of generality that and Then by (1.3),

(3.17)

By induction, for any we have

(3.18)

Thus and for any Then

(3.19)

4. The Case  

Suppose Then We need to consider solutions with initial vectors in or defined by (1.7) and (1.8), respectively.

Lemma 4.1.

Let If is a solution of (1.3) with then there exists an integer such that

The proof is the same as the discussions in Cases 5 through Case 8 in the proof of Lemma 2.1, and hence is skipped.

Theorem 4.2.

Suppose then a solution of (1.3) with will tend to

Proof.

In view of Lemma 4.1, we may assume without loss of generality that By (1.3),

(4.1)

By induction, for any we have

(4.2)

and similarly

(4.3)

Thus for any Thus (2.37) hold so that

(4.4)

The proof is complete.

Theorem 4.3.

Suppose then any solution of (1.3) with is asymptotically -periodic with limit -cycle

Proof.

We first discuss the case, where By (1.3),

(4.5)

By induction, for any we have

(4.6)

Thus and for any Then

(4.7)

If then by (1.3),

(4.8)

That is, We may thus apply the previous conclusion to deduce our assertion.

If then similar to the discussions of Case 3 of Lemma 2.1, there exists an integer such that and That is, In view of the previous case, our assertion holds. The proof is complete.

5. Concluding Remarks

The results in the previous sections can be stated in terms of the two-dimensional dynamical system (1.10). Indeed, a solution of (1.10) is a vector sequence of the form that renders (1.10) into an identity for each It is uniquely determined by

Let us say that a solution of (1.10) eventually falls into a plane region if for all large that it is eventually falls into two disjoint plane regions and alternately if there is some such that and for all and that it approaches a limit -cycle if there is some such that and as Then we may restate the previous theorems as follows.

• (i) The vectors and form the corners of a square in the plane.

• (ii) A solution of (1.10) with in the nonpositive orthant (is nonpositive and) tends to

• (iii) Suppose , then a solution of (1.10) with in will (eventually falls into and) tend to

• (iv) Suppose , then a solution of (1.10) with in will (eventually falls into and alternately and) approach the limit -cycle

• (v) Suppose , then a solution of (1.10) with in will (eventually falls into ) tend to

• (vi) Suppose , Then a solution of (1.10) with in will (eventually falls into and alternately) approach the limit -cycle

Since we have obtained a complete set of asymptotic criteria, we may deduce (bifurcation) results such as the following.

If then all solutions originated from the positive orthant approach the limit -cycle if then all solutions originated from the positive orthant tend to if then all solutions originated from the positive orthant tend to if and approach the limit cycle otherwise.

Roughly the above statements show that when the threshold parameter is a relatively small positive parameter, all solutions from the positive orthant tend to a limit -cycle; when it reaches the critical value some of these solutions (those from ) switch away and tend to a limit -cycle, and when drifts beyond the critical value, all solutions tend to the limit -cycle. Such an observation seems to appear in many natural processes and hence our model may be used to explain such phenomena. It is also expected that when a group of neural units interact with each other in a network where each unit is governed by evolutionary laws of the form (1.3), complex but manageable analytical results can be obtained. These will be left to other studies in the future.