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Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control
Advances in Difference Equations volume 2010, Article number: 542073 (2010)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ1_HTML.gif)
where ,
, and
is a nonlinear signal filtering control function of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ2_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ10_HTML.gif)
-
(ii)
If
is a sequence which satisfies (1.7), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ11_HTML.gif)
-
(iii)
If
is a sequence which satisfies (1.8), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ12_HTML.gif)
-
(iv)
If
is a sequence which satisfies (1.9), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ13_HTML.gif)
We will discuss solutions of (1.3) originated from different
and
in
. For this reason, we let
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ14_HTML.gif)
Then, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ15_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ16_HTML.gif)
we see that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ17_HTML.gif)
Similarly, let and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ18_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ19_HTML.gif)
Since , we see further that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ20_HTML.gif)
Note that (1.3) is equivalent to the following two dimensional dynamical system
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ21_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ22_HTML.gif)
This, in view of (1.10) and (1.11), leads us to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ23_HTML.gif)
which is a contradiction. Thus, there is such that
and
. Furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ25_HTML.gif)
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
Â
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ27_HTML.gif)
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
Â
In this section, we suppose that . Then,
. Let
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ28_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ29_HTML.gif)
For the sake of convenience, let us set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ30_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ31_HTML.gif)
That is, . If
for some
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ32_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ33_HTML.gif)
Hence, . If
for some
, then by (1.3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ34_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ35_HTML.gif)
which is a contradiction. Thus there is such that
and
. Hence,
.
Next suppose that . Then,
for some
. If
, then by (1.3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ36_HTML.gif)
Hence, . If
for some
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ37_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ38_HTML.gif)
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
Â
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ39_HTML.gif)
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
Â
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.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ40_HTML.gif)
Hence, there is such that
and
. Thus
.
Next, suppose that . Then, by (1.3),
. Hence,
.
Next, suppose that . Then,
for some
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ41_HTML.gif)
Hence, . If
for some
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ42_HTML.gif)
we see that . By induction, we see that
. Hence,
.
Next, suppose that . Then
for some
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ43_HTML.gif)
Hence, . If
for some
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ44_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ45_HTML.gif)
and by induction, for all
. Hence,
for
, which leads us to
. The proof is complete.
6. The Case Where
Â
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ46_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F542073/MediaObjects/13662_2010_Article_1307_Equ50_HTML.gif)
can be used to describe competing dynamics and it is hoped that our techniques, and results here will be useful in these studies.
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Hou, C., Han, L. & Cheng, S. Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control. Adv Differ Equ 2010, 542073 (2010). https://doi.org/10.1155/2010/542073
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DOI: https://doi.org/10.1155/2010/542073