Liping Dou 1 and Chengmin Hou 1 and Sui Sun Cheng 2
Academic Editor:Garyfalos Papashinopoulos
1, Department of Mathematics, Yanbian University, Yanji 133002, China
2, Department of Mathematics, Tsing Hua University, Hsinchu 30043, Taiwan
Received 18 January 2015; Revised 10 March 2015; Accepted 11 March 2015; 22 April 2015
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.
1. Introduction
It is of great interest to see how an artificial neural network uses components which are closer to "biological" components. One particular important component can be described by means of the step (activation) function defined by [figure omitted; refer to PDF] with a nonnegative (threshold) real parameter [figure omitted; refer to PDF] . Roughly, the activation function mimics a so called McCulloch neuron that may receive an excitatory value (indicated by 1) if the input signal has strength within limits 0 and [figure omitted; refer to PDF] , and otherwise it remains intact with an inhibitory value (indicated by 0).
If we let [figure omitted; refer to PDF] be the state value of a neural unit during the time period [figure omitted; refer to PDF] , then the recurrence relation [figure omitted; refer to PDF] may be used to describe a one neuron McCulloch-Pitts system where the state value is updated from the two most recent state values.
Let [figure omitted; refer to PDF] . Zhu and Huang [1] discussed the "limit cycles" of the recurrence relation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is defined by (1), in which the positive threshold [figure omitted; refer to PDF] can be regarded as a bifurcation parameter. Then, Chen [2] considered the following recurrence relation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are 2-periodic sequences with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Asymptotic behaviors of these equations reflect their differences (see [1, 2]). A good reason for studying (4) is that the constants [figure omitted; refer to PDF] and [figure omitted; refer to PDF] used in the physical model described by (3) may not be truly constant but exhibit fluctuating behaviors between two limits. Since in (4) we have chosen to consider the case where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are replaced by 2-periodic sequences, the question then arises as to what will happen if we choose general periodic sequences.
In this paper, we offer partial answers by considering the difference equation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -periodic sequences with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . By studying this equation, we hope that the subsequent results will lead to much more general ones for complex systems involving similar periodic parameters and discontinuous controls.
In order to study the asymptotic behavior of (5), let us first note that it is a three-term recurrence relation so that, given [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we may calculate [figure omitted; refer to PDF] , and so forth in a sequential manner. The resulting sequence [figure omitted; refer to PDF] is naturally called a solution of (5). For example, when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -periodic sequences, we may write [figure omitted; refer to PDF] This motivates us to define a vector equation. Given a sequence [figure omitted; refer to PDF] , its Casoratian vector sequence is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then (5) is equivalent to the asynchronous vector equation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Note that, given [figure omitted; refer to PDF] , we may use (7) to generate [figure omitted; refer to PDF] which, when "lined up," yields the same [figure omitted; refer to PDF] as described above. For this reason, the sequence [figure omitted; refer to PDF] will be called the solution of (7) determined by [figure omitted; refer to PDF] .
Therefore, to obtain complete asymptotic behaviors of (5), we need to derive the behaviors of solutions of (7) determined by vectors [figure omitted; refer to PDF] in the entire plane.
The following result, however, can help us concentrate on solutions determined by vectors [figure omitted; refer to PDF] in [figure omitted; refer to PDF] .
Theorem 1.
A solution [figure omitted; refer to PDF] of (7) with [figure omitted; refer to PDF] in the nonpositive orthant [figure omitted; refer to PDF] is nonpositive and tends towards [figure omitted; refer to PDF] .
Proof.
Let [figure omitted; refer to PDF] . Then, by (7) and by induction, for any [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we may get [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] as [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] , we see that [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] . The proof is complete.
Next, note that our system is autonomous (time invariant), and hence, if [figure omitted; refer to PDF] is a solution of (7), then, for any [figure omitted; refer to PDF] , the sequence, [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , is also a solution of (7).
Suppose [figure omitted; refer to PDF] is a solution of (7). Then, we say that
[figure omitted; refer to PDF] : approaches a limit 1-cycle [figure omitted; refer to PDF] if [figure omitted; refer to PDF] ;
: [figure omitted; refer to PDF] approaches a limit 2-cycle [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .
For the sake of convenience, we also need to introduce some notations. The numbers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are defined as [figure omitted; refer to PDF] while the numbers [figure omitted; refer to PDF] and their properties are listed in the Appendix. These numbers are introduced in order to break the plane into different parts such that the behavior of each solution of (7) which originates from each part may be traced.
In the sequel, we distinguish the cases (i) [figure omitted; refer to PDF] , (ii) [figure omitted; refer to PDF] , and (iii) [figure omitted; refer to PDF] , and then, for different [figure omitted; refer to PDF] , we find the asymptotic behaviors of the corresponding solutions.
2. The Case [figure omitted; refer to PDF]
We begin with the following.
Lemma 2.
Let [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] , then there exist integers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .
Proof.
From our assumption, we have [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be a solution of (7) with [figure omitted; refer to PDF] . Then, there are eight cases.
Case 1 . If [figure omitted; refer to PDF] , then by (7) we may easily obtain [figure omitted; refer to PDF] , our assertion is true by taking [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .
Case 2 . Suppose [figure omitted; refer to PDF] . Then by (7) and induction we may see that [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; and hence, [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] . Then, there exists enough large [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .
Case 3 . Suppose [figure omitted; refer to PDF] . As in Case 2, we may show that there exists enough large [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Our assertion is thus true.
Case 4 . Suppose [figure omitted; refer to PDF] . We may first show that there is a [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Otherwise, for any [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] . Then by (7) we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] as [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] , we see that [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] , which is a contradiction. The rest of the proof comes from the assertions in Cases 2 and 3.
Case 5 . Suppose [figure omitted; refer to PDF] . Then by (7) and induction we may see that [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; and hence, [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] . Then, there exists enough large [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .
Case 6 . Suppose [figure omitted; refer to PDF] . As in Case 5, we may show that there exists enough large [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Our assertion is thus true.
Case 7 . Suppose [figure omitted; refer to PDF] . In view of (A.3), there exist [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Next, we consider two cases.
(I) Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then by (7) we have [figure omitted; refer to PDF]
: that is, [figure omitted; refer to PDF] . Our assertion is true by Case 6.
(II) Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then by (7) we have [figure omitted; refer to PDF]
: and, by induction, we have [figure omitted; refer to PDF]
: that is, [figure omitted; refer to PDF] . Our assertion is true by means of Case 6.
Case 8 . Suppose [figure omitted; refer to PDF] . Then, there exist [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . As in Case 7, we may show that [figure omitted; refer to PDF] . Our assertion is true by means of Case 5.
Theorem 3.
Let [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] , then it approaches the limit 1-cycle [figure omitted; refer to PDF] .
Proof.
By Lemma 2, we may assume without loss of generality that [figure omitted; refer to PDF] . Then, by (7) and induction, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] as [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] , we see that [figure omitted; refer to PDF] tends towards the limit 1-cycle [figure omitted; refer to PDF] . The proof is complete.
3. The Case [figure omitted; refer to PDF]
In this section, we assume [figure omitted; refer to PDF] . Set [figure omitted; refer to PDF]
Note that a simple plot of the regions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in the plane shows that they are disjoint and form the complement of the region [figure omitted; refer to PDF] .
Lemma 4.
Let [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] , then there exist integer [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .
The proof is similar to those discussed in Cases 5 through 8 in the proof of Lemma 2 and, hence, is skipped.
Theorem 5.
Let [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] , then it approaches the limit 1-cycle [figure omitted; refer to PDF] .
Proof.
By Lemma 4, we may assume without loss of generality that [figure omitted; refer to PDF] . Then, similar to the proof of Theorem 3, we have [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] as [figure omitted; refer to PDF] tends towards [figure omitted; refer to PDF] , we see that [figure omitted; refer to PDF] tends towards the limit 1-cycle [figure omitted; refer to PDF] . The proof is complete.
Theorem 6.
A solution [figure omitted; refer to PDF] of (7) with [figure omitted; refer to PDF] tends towards the limit 2-cycle [figure omitted; refer to PDF] .
Proof.
By (7), we have [figure omitted; refer to PDF] and by induction, [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] tends towards the limit 2-cycle [figure omitted; refer to PDF] .
Theorem 7.
A solution [figure omitted; refer to PDF] of (7) with [figure omitted; refer to PDF] tends towards the limit 2-cycle [figure omitted; refer to PDF] .
The proof is similar to Theorem 6 and is skipped.
Theorem 8.
Suppose that [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then,
(i) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ;
(ii) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ;
(iii): [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ;
(iv) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .
Proof.
Suppose that [figure omitted; refer to PDF] .
(i) We distinguish two different cases.
Case 1 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF]
Case 2 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] and by induction, we have [figure omitted; refer to PDF]
(ii) Similar to (i), by distinguishing two different cases and by induction, we have the following.
Case 1 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .
Case 2 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .
(iii) We distinguish four different cases.
Case 1 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . By (7), [figure omitted; refer to PDF]
Case 2 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, by (7), [figure omitted; refer to PDF]
Case 3 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] and, by induction, we have [figure omitted; refer to PDF]
Case 4 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] and, by induction, we have [figure omitted; refer to PDF]
(iv) Similar to (iii), by distinguishing four different cases and by induction, we have the following.
Case 1 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .
Case 2 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .
Case 3 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .
Case 4 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .
Note that, in view of (A.3) and (A.4), the above result handles all solutions of (7) originated from [figure omitted; refer to PDF] . Together with Theorems 6 and 7, we have taken care of all the cases where [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is the complement of the region [figure omitted; refer to PDF] , Theorems 1 and 5-7 can be used to take care of all possible solutions of (7).
4. The Case [figure omitted; refer to PDF]
In this section, we assume that [figure omitted; refer to PDF] and we set [figure omitted; refer to PDF] if [figure omitted; refer to PDF] . We continue to use those notations in the case where [figure omitted; refer to PDF] .
Lemma 9.
Let [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] , then there exist integers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .
Proof.
There are seven cases.
(i) If [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , then, by (7), we may easily see that [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .
(ii) Suppose [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] , which is a contradiction. Therefore, there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]
(iii) Suppose [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] , which is a contradiction. Therefore, there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]
(iv) If [figure omitted; refer to PDF] , then, in view of (A.15), there exist [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Furthermore, if [figure omitted; refer to PDF] , we distinguish two cases.
Case 1 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF]
Case 2 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF]
Next, suppose [figure omitted; refer to PDF] .
Case 1 . Consider [figure omitted; refer to PDF] . From [figure omitted; refer to PDF] , we may get [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF]
Case 2 . Consider [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . From [figure omitted; refer to PDF] , we may get [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF]
Theorem 10.
Let [figure omitted; refer to PDF] . Then, any solution [figure omitted; refer to PDF] of (7) tends asymptotically to the 2-cycle [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .
Proof.
In view of Lemma 9, we may assume without loss of generality that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then, by (7), [figure omitted; refer to PDF] and by induction, we have [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Therefore, [figure omitted; refer to PDF] tends towards the limit 2-cycle [figure omitted; refer to PDF] . The proof is complete.
By reviewing Theorems 1, 3, 5-8 and 10 carefully, we may see that all solutions of (7) tend towards the limit 1-cycles [figure omitted; refer to PDF] or [figure omitted; refer to PDF] or towards the limit 2-cycles [figure omitted; refer to PDF] or [figure omitted; refer to PDF] . In Theorems 1, 3, 5-8, we are precise about the limit cycle for each solution from [figure omitted; refer to PDF] , but not in Theorem 10. We can also be precise in the last case. There are more technical details, however. Hence, we refer the readers who are interested in the precise "initial vector-limit vector" relations to the Appendix.
5. Discussions
The results in the previous sections are stated in terms of the [figure omitted; refer to PDF] -dimensional asynchronous dynamical system (7). Note that (7) is asynchronous in the sense that, given [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the subsequent [figure omitted; refer to PDF] are calculated one by one from the first, second, third, [figure omitted; refer to PDF] equation of (7), respectively.
We may also regard the original equation (5) as the following 2-dimensional asynchronous system: [figure omitted; refer to PDF]
Let us say that a solution [figure omitted; refer to PDF] of (39) approaches a limit 1-cycle [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and a limit 2-cycle [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Let us further say that a solution [figure omitted; refer to PDF] of (39) eventually falls into a region [figure omitted; refer to PDF] if [figure omitted; refer to PDF] for all large [figure omitted; refer to PDF] and eventually alternates between two regions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all large [figure omitted; refer to PDF] .
Then, we may restate the previous theorems as follows.
(i) The vectors [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] form the corners of a square in the plane.
(ii) A solution [figure omitted; refer to PDF] of (39) with [figure omitted; refer to PDF] in the nonpositive orthant [figure omitted; refer to PDF] is nonpositive and tends towards the limit 1-cycle [figure omitted; refer to PDF] .
(iii): Suppose [figure omitted; refer to PDF] . Then, a solution [figure omitted; refer to PDF] of (39) with [figure omitted; refer to PDF] will tend towards the limit 1-cycle [figure omitted; refer to PDF] .
(iv) Suppose [figure omitted; refer to PDF] . Then, a solution [figure omitted; refer to PDF] of (39) with [figure omitted; refer to PDF] will tend towards the limit 2-cycle [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .
(v) Suppose [figure omitted; refer to PDF] . Then, a solution [figure omitted; refer to PDF] of (39) with [figure omitted; refer to PDF] will eventually fall into [figure omitted; refer to PDF] and approach the limit 1-cycle [figure omitted; refer to PDF] .
(vi) Suppose [figure omitted; refer to PDF] . Then, a solution [figure omitted; refer to PDF] of (39) with [figure omitted; refer to PDF] will eventually alternate between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and approach the limit 2-cycle [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .
Since we have obtained a complete set of asymptotic criteria, we may deduce (bifurcation) results such as the following. If it [figure omitted; refer to PDF] , then all solutions [figure omitted; refer to PDF] which originated from the positive orthant approach a limit 2-cycle; if [figure omitted; refer to PDF] , then all solutions that originated from the positive orthant tend towards the limit 1-cycles [figure omitted; refer to PDF] ; if [figure omitted; refer to PDF] , then all solutions originated from the positive orthant tend towards the limit 1-cycle [figure omitted; refer to PDF] or 2-cycles [figure omitted; refer to PDF] or [figure omitted; refer to PDF] . Roughly, the above statements show that when the threshold parameter [figure omitted; refer to PDF] is a relatively small positive parameter, all solutions from the positive orthant tend towards a limit 2-cycle; when the threshold parameter reaches the critical value [figure omitted; refer to PDF] , some of these solutions drift away and tend towards a limit 1-cycle, and when [figure omitted; refer to PDF] drifts beyond the critical value, all solutions tend towards the limit 1-cycle [figure omitted; refer to PDF] . 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 (39), complex but manageable analytical results can be obtained. These are left for further studies.
We remark that the precise region from which each type of solution originates can also be given (by implementing a simple computer program).
Our proofs also indicate that more general multiple neuron recurrent McCulloch-Pitts type neural networks possess similar behavior. However, the derivations may involve more delicate combinatorial arguments and are left for further studies.
Acknowledgment
This study is supported by the Natural Science Foundation of China (11161049).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
[1] H. Zhu, L. Huang, "Asymptotic behavior of solutions for a class of delay difference equation," Annals of Differential Equations , vol. 21, no. 1, pp. 99-105, 2005.
[2] Y. Chen, "All solutions of a class of difference equations are truncated periodic," Applied Mathematics Letters , vol. 15, no. 8, pp. 975-979, 2002.
Appendix
We let [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] We assume [figure omitted; refer to PDF] . Set [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] We assume [figure omitted; refer to PDF] . Set [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] there exist integers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .
Set [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Similarly, since [figure omitted; refer to PDF] there exist integers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .
Set [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] Furthermore, since [figure omitted; refer to PDF] then if we let [figure omitted; refer to PDF] then [figure omitted; refer to PDF] Furthermore, [figure omitted; refer to PDF] We also note that [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF]
Lemma A.1.
A solution [figure omitted; refer to PDF] of (7) with [figure omitted; refer to PDF] [figure omitted; refer to PDF] or [figure omitted; refer to PDF] tends towards the limit 2-cycle [figure omitted; refer to PDF] [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .
The proof of Lemma A.1 is the same as those discussed in the proofs of Theorems 6 and 7.
Lemma A.2.
Suppose [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then,
(i) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;
(ii) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;
(iii): [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ;
(iv) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .
Lemma A.3.
Suppose [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then,
(i) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;
(ii) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;
(iii): [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ;
(iv) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .
Lemma A.4.
Suppose [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then [figure omitted; refer to PDF]
Lemma A.5.
Suppose [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then [figure omitted; refer to PDF]
Lemma A.6.
Suppose [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] then [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then [figure omitted; refer to PDF]
Lemma A.7.
Suppose [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a solution of (7) with [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] then [figure omitted; refer to PDF] If [figure omitted; refer to PDF] then [figure omitted; refer to PDF]
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Abstract
A nonlinear recurrence involving a piecewise constant McCulloch-Pitts function and 2 k -periodic coefficient sequences is investigated. By allowing the threshold parameter to vary from 0 to ∞ , we work out a complete bifurcation analysis for the asymptotic behaviors of the corresponding solutions. Among other things, we show that each solution tends towards one of four different limits. Furthermore, the accompanying initial regions for each type of solutions can be determined. It is hoped that our analysis will provide motivation for further results for recurrent McCulloch-Pitts type neural networks.
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