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Complete Asymptotic Analysis of a Nonlinear Recurrence Relation with Threshold Control
Advances in Difference Equations volume 2010, Article number: 143849 (2010)
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:

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

in which the positive number can be regarded as a threshold parameter.
In this paper, we consider the following delay difference equation:

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

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

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

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


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

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:

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),

and by induction, for any we have

Since we have

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

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

Then



We also let

Then




2. The Case
Â
Suppose Then

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),

If then by (1.3),

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

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

By induction,

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

By induction, for any we have

which implies

This is contrary to the fact that for
Now that there exists an integer such that
and
it then follows

If then our assertion holds by taking
If
then
Thus

If then our assertion holds by taking
If
we have
Hence

Repeating the procedure, we have

If then our assertion holds by taking
Otherwise,

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,

If then by (1.3),

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

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

Thus our assertion holds by taking
Case 5.
Suppose Then by (1.21) and (1.23),

If then by (1.3),

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

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

That is, Therefore we may conclude our assertion by induction.
Case 6.
Suppose Since

we see that

If then by (1.3),

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

That is, We may thus use induction to conclude our assertion.
Case 7.
Suppose As in Case 5,

If then by (1.3),

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

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

That is, Thus our assertion follows from induction.
Case 8.
Suppose Then

If then by (1.3),

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

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),

By induction, for any we have

and similarly

Thus for any
and

The proof is complete.
3. The Case
Â
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)

This means that our assertion is true by taking
Case 3.
Suppose If
for any
then by (1.3),

By induction, for any we have

Hence

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

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

This means that our assertion is true by taking
Case 5.
Suppose . Then by (1.21) and (1.23),

If then by (1.3),

When we have

That is, We may thus apply the conclusion of Case 3 to deduce our assertion.
Suppose If
then we have

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

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

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

That is, Therefore we may conclude our assertion by induction.
Case 6.
Suppose

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

That is, We may thus apply the conclusion of Case 5 to deduce our assertion.
Case 8.
Suppose Then

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),

By induction, for any we have

Thus and
for any
Then

4. The Case
Â
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),

By induction, for any we have

and similarly

Thus for any
Thus (2.37) hold so that

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),

By induction, for any we have

Thus and
for any
Then

If then by (1.3),

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.
References
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Acknowledgment
This project was supported by the National Natural Science Foundation of China (10661011).
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Ge, Q., Hou, C. & Cheng, S. Complete Asymptotic Analysis of a Nonlinear Recurrence Relation with Threshold Control. Adv Differ Equ 2010, 143849 (2010). https://doi.org/10.1155/2010/143849
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DOI: https://doi.org/10.1155/2010/143849