Bifurcation in a Discrete-Time Piecewise Constant Dynamical System

Abstract

The study of recurrent neural networks with piecewise constant transition or control functions has attracted much attention recently because they can be used to simulate many physical phenomena. A recurrent and discontinuous two-state dynamical system involving a nonnegative bifurcation parameter is studied. By elementary but novel arguments, we are able to give a complete analysis on its asymptotic behavior when the parameter varies from 0 to ∞. It is hoped that our analysis will provide motivation for further results on large-scale recurrent McCulloch-Pitts-type neural networks and piecewise continuous discrete-time dynamical systems.

1. Introduction

It is generally accepted that the McCulloch-Pitts model of a neural network can be used as the components of computer-like systems. That is, where neural networks are commonly used to learn something, a McCulloch-Pitts neuron is constructed to do a particular job. Although the same job can be done by means of traditional Boolean components, it is interesting to see how it all works using components which are closer to “biological” components. One particular important component can be described by means of the step (activation) function defined by 𝑓_𝜎(𝑢)=1if𝑢≤𝜎−1if𝑢>𝜎 (1) with a nonnegative (threshold) real parameter 𝜎. Roughly, a neuron may receive an inhibitory (indicated by a −1 sign) value if the input signal has strength exceeding the biological threshold 𝜎 and otherwise it remains intact with an excitory (indicated by a +1 sign) value.

Therefore, if we let 𝑥_𝑛 be the state value of a neural unit during the time period 𝑛, then the following recurrence relation: 𝑥_𝑛=𝐹𝑓_𝜎𝑥_𝑛−1,𝑥_𝑛−2 (2) may be used to describe a one-neuron McCulloch-Pitts system where the state value is updated from the two most recent state values. There are now many studies that are concerned with such neural networks.

For a neural network system that contains two or more neural units, things are much more complicated, and a variety of models can be designed. In usual practice, we can build a mathematical model and use it to generate simulation results, and from these results, we may understand the general properties of our models. What is now more difficult but important is to make mathematical conclusions that may provide a full understanding of all the properties observed.

In this paper, we consider a two-neuron dynamical neural network system: 𝑥_𝑛=𝑎𝑥_𝑛−2+(1−𝑎)𝑓_𝜎𝑦_𝑛−1,𝑦_𝑛=𝑎𝑦_𝑛−2+(1−𝑎)𝑓_𝜎𝑥_𝑛−1 (3) for 𝑛∈𝐍={0,1,2,…}, where 𝑎∈(0,1), and 𝑓_𝜎∶𝐑→𝐑 is the function in. Note that if we make the change of variables 𝑥_𝑛=1−𝑎𝑏𝑢_𝑛,𝑦_𝑛=1−𝑎𝑏𝑣_𝑛, (4) where 𝑏 is an arbitrary positive number, then (3) becomes the “more general” system: 𝑢_𝑛=𝑎𝑢_𝑛−2+𝑏𝑓_{𝑏𝜎/(1−𝑎)}𝑣_𝑛,𝑣_𝑛=𝑎𝑣_𝑛−2+𝑏𝑓_{𝑏𝜎/(1−𝑎)}𝑢_𝑛, (5) and hence all properties of (5) can be inferred from those of (3).

In this paper, we are concerned with the asymptotic behavior of (3) when the nonnegative parameter 𝜎 is treated as a bifurcation variable and allowed to vary from 0 to ∞ (cf. [1, 2]). What is interesting is that a complete asymptotic and bifurcation analysis can be obtained. With such success, it is expected that more general discontinuous recurrent McCulloch-Pitts-type neural networks [3] can be dealt with to some extent in a similar manner.

To simplify matters, note that if we let 𝑧=(𝑥,𝑦) and 𝐹_𝜎(𝑧)=𝑓_𝜎𝑦,𝑓_𝜎(𝑥), (6) then the above system (3) can be written as 𝑧_𝑛=𝑎𝑧_𝑛−2+𝑎^′𝐹_𝜎𝑧_𝑛−1,𝑛∈𝐍, (7) where we write 𝑧_𝑛=(𝑥_𝑛,𝑦_𝑛) and 𝑎^′=1−𝑎 for the sake of convenience. The above vector equation is a three-term recurrence relation. Hence for given 𝑧₋₂ and 𝑧₋₁ in the plane, a unique sequence {𝑧_𝑘}^∞_𝑘=−2 can be calculated from it. Such a sequence is called a solution of (7) determined by 𝑧₋₂ and 𝑧₋₁.

Depending on the locations of 𝑧₋₂ and 𝑧₋₁, it is clear that the behavior of the corresponding solution may differ. For this reason, it is convenient to distinguish various parts of 𝑅². Before doing so, it is convenient to adopt several simplifying conventions. First, we set 𝑅⁻_𝜎=(−∞,𝜎],𝑅⁺_𝜎=(𝜎,∞). (8) Next, if 𝐼 and 𝐽 are real intervals, their cross-product 𝐼×𝐽 will be denoted by 𝐼𝐽 instead, and we will assume that this product receives the priority attention in a mathematical expression. By means of these convections, 𝑅⁺_𝜎𝑅⁺_𝜎=(𝜎,∞)²,𝑅⁺_𝜎𝑅⁻_𝜎=()×(],𝑅𝜎,∞−∞,𝜎_−𝜎𝑅⁻_𝜎=(−∞,𝜎]²,𝑅⁻_𝜎𝑅⁺_𝜎=(−∞,𝜎]×(𝜎,∞). (9) Clearly, {𝑅⁺_𝜎𝑅⁺_𝜎,𝑅⁺_𝜎𝑅⁻_𝜎,𝑅⁻_𝜎𝑅⁺_𝜎,𝑅⁻_𝜎𝑅⁻_𝜎} is a partition of 𝐑². Other subsets of the plane will be introduced in the subsequent sections.

For solutions originated from 𝑧₋₂ and 𝑧₋₁ in the above subsets, we will show that they are all “asymptotically periodic.” More precisely, we say that a positive integer 𝜂 is a period of a scalar or vector sequence 𝑣={𝑣_𝑛}^∞_𝑛=𝛼 if 𝑣_𝜂+𝑛=𝑣_𝑛 for all 𝑛≥𝛼 and that 𝜏 is the least or prime period of {𝑣_𝑛}^∞_𝑛=𝛼 if 𝜏 is the least among all periods of {𝑣_𝑛}^∞_𝑛=𝛼. The sequence {𝑣_𝑛}^∞_𝑛=𝛼 is said to be 𝜏-periodic if 𝜏 is the least period of {𝑣_𝑛}^∞_𝑛=𝛼. The sequence {𝑣_𝑛}^∞_𝑛=𝛼 is said to be asymptotically periodic if there exist real numbers 𝑣⁽⁰⁾,𝑣⁽¹⁾,…,𝑣^(𝜔−1), where 𝜔 is a positive integer such that lim_𝑛→∞𝑣_{𝜔𝑛+𝑖}=𝑣^(𝑖),𝑖=0,1,…,𝜔−1. (10) In case {𝑣⁽⁰⁾,𝑣⁽¹⁾,…,𝑣^(𝜔−1),𝑣⁽⁰⁾,𝑣⁽¹⁾,…,𝑣^(𝜔−1),…} is also an 𝜔-periodic sequence, we say that 𝑣 is an asymptotically 𝜔-periodic sequence tending to the limit 𝜔-cycle (this term is introduced since the underlying concept is similar to that of the limit cycle in the theory of ordinary differential equations) ⟨𝑣⁽⁰⁾,𝑣⁽¹⁾,…,𝑣^(𝜔−1)⟩. For the sake of convenience, for such an asymptotically 𝜔-periodic sequence 𝑣, we write 𝑣_𝑛⟶𝑣⁽⁰⁾,𝑣⁽¹⁾,…,𝑣^(𝜔−1). (11) Note that in case 𝑣 is asymptotically 1-periodic, then it is a sequence convergent to 𝑣⁽⁰⁾ for some 𝑣⁽⁰⁾. The converse is also true. In such a case, we simply write 𝑣_𝑛→𝑣⁽⁰⁾ instead of 𝑣_𝑛→⟨𝑣⁽⁰⁾⟩.

Having these terminologies at hand, our main issue is to show that each solution of (3) originated from 𝐑² is either asymptotically 1- or 2-periodic. Note, however, that since 𝑓_𝜎 is a discontinuous function, the standard theories that employ continuous arguments cannot be applied to yield asymptotic criteria. Fortunately, we may resort to elementary arguments as to be seen below.

Before doing so, let us make a few remarks. First note that our system (7) is autonomous (time invariant) and also symmetric in the sense that under two sets of “symmetric initial conditions,” the behaviors of the corresponding solutions are also “symmetric.” This statement can be made more precise in mathematical terms. However, a simple example is sufficient to illustrate this: if {𝑧_𝑛}^∞_𝑛=−2 is a solution of (3) with (𝑧₋₂,𝑧₋₁)∈𝑅⁺_𝜎𝑅⁺_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎, then as will be seen below, 𝑧_2𝑛→(1,−1) and 𝑧_2𝑛+1→(−1,1). If we now replace the condition (𝑧₋₂,𝑧₋₁)∈𝑅⁺_𝜎𝑅⁺_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎 with the symmetric initial condition (𝑧₋₂,𝑧₋₁)∈𝑅⁻_𝜎𝑅⁻_𝜎×𝑅⁺_𝜎𝑅⁺_𝜎, then we will end up with the conclusion that 𝑧_2𝑛→(−1,1) and 𝑧_2𝑛+1→(1,−1). Such two conclusions will be referred to as dual results, and the principle of proof for either one can be applied to that of the other.

In the sequel, we will first distinguish three different cases (i) 0≤𝜎<1, (ii) 𝜎=1, and (iii) 𝜎>1 and then consider different 𝑧₋₂, 𝑧₋₁ in 𝑅⁺_𝜎𝑅⁺_𝜎,𝑅⁺_𝜎𝑅⁻_𝜎,𝑅⁻_𝜎𝑅⁺_𝜎, or 𝑅⁻_𝜎𝑅⁻_𝜎 and the (asymptotic) behaviors of the corresponding solutions determined by them.

We will need the following simple but useful results. First, let {𝑥_𝑘}^∞_𝑘=−2 be real scalar (or vector) sequences that satisfy 𝑥_2𝑘=𝑎𝑥_2𝑘−2𝑥+𝑑,𝑛∈𝐍,_2𝑘+1=𝑎𝑥_2𝑘−1+𝑑,𝑘∈𝐍, (12)(13) where 𝑎∈(0,1), and 𝑑 is a real number (resp., a real vector).(i)If {𝑥_𝑛}^∞_𝑛=−2 is a sequence which satisfies (12), then 𝑥_2𝑘=𝑎^𝑘+1𝑥₋₂+1−𝑎^𝑘+11−𝑎𝑑,𝑘∈𝐍. (14)(ii)If {𝑥_𝑛}^∞_𝑛=−2 is a sequence which satisfies (13), then 𝑥_2𝑘+1=𝑎^𝑘+1𝑥₋₁+1−𝑎^𝑘+11−𝑎𝑑,𝑘∈𝐍. (15)

Second, the function 𝐹_𝜎 in (6) satisfies 𝐹_𝜎𝑅⁺_𝜎𝑅⁺_𝜎=−^→𝐤,𝐹_𝜎𝑅⁺_𝜎𝑅⁻_𝜎=^→𝐢,𝐹_𝜎𝑅⁻_𝜎𝑅⁺_𝜎=−^→𝐢,𝐹_𝜎𝑅⁻_𝜎𝑅⁻_𝜎=^→𝐤, (16) where ^→𝐢=(1,−1) and ^→𝐤=(1,1).

Third, we need to consider various ordering arrangements for three nonnegative integers 𝑘, 𝑝, and 𝑙 or four nonnegative integers 𝑘, 𝑝, 𝑙, and 𝑚. First, the ordering arrangements of three integers 𝑘, 𝑝, and 𝑙 can be classified into 6 cases: ()()()()(){).A1𝑘=𝑝≤𝑙A2𝑙=𝑘<𝑝A3𝑝=𝑙<𝑘A4𝑘<min𝑝,𝑙A5𝑝<min𝑘,𝑙}(A6𝑙<min𝑘,𝑝 (17)

Indeed, let 𝑎,𝑏,𝑐∈𝐑. Then, since either 𝑎<𝑏, 𝑎=𝑏 or 𝑎>𝑏, we see that (),()(),()(),(),()().𝑎<𝑏⇒𝑐∈−∞,𝑎𝑐=𝑎,𝑐∈𝑎,𝑏,𝑐=𝑏or𝑐∈𝑏,∞𝑎=𝑏⇒𝑐∈−∞,𝑎,𝑐=𝑎or𝑐∈𝑎,∞𝑎>𝑏⇒𝑐∈−∞,𝑏𝑐=𝑏,𝑐∈𝑏,𝑎,𝑐=𝑎or𝑐∈𝑎,∞ (18) These are equivalent to {}{}{},𝑎=𝑏≤𝑐,𝑐=𝑎<𝑏,𝑏=𝑐<𝑎,𝑎<min𝑏,𝑐,𝑏<min𝑎,𝑐,𝑐<min𝑎,𝑏 (19) by comparing the two sets of statements (18) and (19).

By similar reasoning, there are 12 ordering arrangements for four nonnegative integers 𝑘, 𝑝, 𝑙, and 𝑚: (){)(){){}()()())())(){).B1𝑘=𝑝≤min𝑙,𝑚}(B2𝑘<min𝑝,𝑙,𝑚B3𝑝<min𝑘,𝑙,𝑚}(B4𝑝=𝑙<min𝑘,𝑚B5𝑝=𝑙=𝑚<𝑘B6𝑙<min𝑝,𝑘,𝑚B7𝑙=𝑚<min𝑘,𝑝(B8𝑙=𝑚=𝑘<𝑝B9𝑚<min𝑙,𝑘,𝑝(B10𝑚=𝑘<min𝑙,𝑝B11𝑝=𝑚<min𝑙,𝑘}(B12𝑘=𝑙<min𝑝,𝑚 (20)

In the sequel, if 𝛼,𝛽∈𝐑 and 𝐼,𝐽 are real intervals, then we adopt the convention that {}{},{}×,{}×.𝛼𝐼=𝛼𝑥∣𝑥∈𝐼,𝛼+𝐼=𝛼+𝑥∣𝑥∈𝐼𝛼𝐼𝐽=𝛼𝑥∣𝑥∈𝐼𝛼𝑦∣𝑦∈𝐽𝛼,𝛽+𝐼𝐽=𝛼+𝑥∣𝑥∈𝐼𝛽+𝑦∣𝑦∈𝐽 (21) On the other hand, since 𝑎∈(0,1),𝜎>0, we have ()𝜎<𝜎+1−𝑎𝑎<𝜎+1−𝑎²𝑎²<⋯<𝜎+1−𝑎^𝑘𝑎^𝑘⟶+∞. (22) If we let 𝐽^(𝑘)=𝜎+1−𝑎^𝑘𝑎^𝑘,𝜎+1−𝑎^𝑘+1𝑎^𝑘+1𝐽,𝑘∈𝐍,₍₋₁₎=𝑎𝐽⁽⁰⁾−𝑎^′=(𝑎𝜎−(1−𝑎),𝜎], (23) then 𝑎𝐽^(𝑘)−𝑎^′=𝐽^(𝑘−1)𝐽,𝑘≥0,₍₋₁₎⊂𝑅⁻_𝜎,𝑅⁺_𝜎=^∞_𝑘=0𝐽^(𝑘),𝑅⁺_𝜎𝑅⁺_𝜎=^∞_𝑘,𝑠=0𝐽^(𝑘)𝐽^(𝑠)=^∞_𝑘=0𝐽^(𝑘)𝑅⁺_𝜎=^∞_𝑠=0𝑅⁺_𝜎𝐽^(𝑠). (24)(25)

2. The Case 𝜎=1

In this section, we assume that 𝜎=1.

Let {𝑧_𝑛}^∞_𝑛=−2 be a solution of (3). Let us consider first the case where (𝑧_𝑛,𝑧_𝑛+1)∈𝑅⁻₁𝑅⁻₁×𝑅⁻₁𝑅⁻₁. Then by (7) and (16), we see that 𝑧_𝑛+2=𝑎𝑧_𝑛+𝑎^′→𝐤,𝑎𝑅⁻₁𝑅⁻₁+𝑎^′𝐹₁𝑅⁻₁𝑅⁻₁=𝑎𝑅⁻₁𝑅⁻₁+𝑎^′→𝐤]×(+()=(]×(]=𝑎((−∞,1−∞,1])1−𝑎)(1,1−∞,1−∞,1=𝑅⁻₁𝑅⁻₁. (26) Hence, we may see further that 𝑧_𝑛+2∈𝑅⁻₁𝑅⁻₁. Similarly, since 𝑎𝐽^(𝑘)𝑅⁻₁+𝑎^′𝐹₁𝑅⁻₁𝐽^(𝑠)=𝑎𝐽^(𝑘)𝑅⁻₁+𝑎^′−^→𝐢=𝑎𝐽^(𝑘)×(−∞,1]+(1−𝑎)(−1,1)=𝐽^(𝑘−1)×(−∞,𝜎]=𝐽^(𝑘−1)𝑅⁻₁, (27) we see that if (𝑧_𝑛,𝑧_𝑛+1)∈𝐽^(𝑘)𝑅⁻₁×𝑅⁻₁𝐽^(𝑠), then 𝑧_𝑛+2∈𝐽^(𝑘−1)𝑅⁻₁. By similar considerations, we may build a “directed graph” represented by Table 1.

Table 1 

In this table, we record the fact that 𝑎𝑅⁻₁𝑅⁻₁+𝑎^′𝐹₁(𝑅⁻₁𝑅⁻₁)⊆𝑅⁻₁𝑅⁻₁ as the (𝑅⁻₁𝑅⁻₁,𝑅⁻₁𝑅⁻₁) entry and so forth.

Next, we let {𝑧_𝑛}^∞_𝑛=−2 be a solution of (3) again. If (𝑧₋₂,𝑧₋₁)∈𝑅⁻₁𝑅⁻₁×𝑅⁻₁𝑅⁻₁, then from Table 1 we see that 𝑧₀∈𝑅⁻₁𝑅⁻₁. Now that (𝑧₋₁,𝑧₀)∈𝑅⁻₁𝑅⁻₁×𝑅⁻₁𝑅⁻₁, we may use Table 1 to infer that 𝑧₁∈𝑅⁻₁𝑅⁻₁ again. By induction, we see that 𝑧_𝑛∈𝑅⁻₁𝑅⁻₁ for all 𝑛≥−2. By (7), we then see that 𝑧_𝑛=𝑎𝑧_𝑛−2+𝑎^′→𝐤,𝑛∈𝐍, (28) and hence by (14) and (15), we see that 𝑧_𝑛→^→𝐤 as 𝑛→∞. If (𝑧₋₂,𝑧₋₁)∈𝑅⁻₁𝑅⁺₁×𝑅⁻₁𝑅⁻₁, then from the (𝑅⁻₁𝑅⁺₁,𝑅⁻₁𝑅⁻₁) entry of Table 1 we see that 𝑧₀∈𝑅⁻₁𝑅⁺₁. Now that (𝑧₋₁,𝑧₀)∈𝑅⁻₁𝑅⁻₁×𝑅⁻₁𝑅⁺₁, by Table 1 again, we see that 𝑧₁=𝑎𝑧₋₁+𝑎^′−^→𝐢∈𝑅⁻₁𝑅⁻₁. (29) By induction, we see that 𝑧_2𝑘∈𝑅⁻₁𝑅⁺₁ and 𝑧_2𝑘+1∈𝑅⁻₁𝑅⁻₁ for 𝑘≥−1. By (7), we may see further that 𝑧_2𝑛=𝑎𝑧_2𝑛−2+𝑎^′→𝐤,𝑧_2𝑛+1=𝑎𝑧_2𝑛−1−𝑎^′→𝐢, (30) for 𝑛∈𝐍, and hence by (14) and (15), 𝑧_2𝑛→^→𝐤 and 𝑧_2𝑛+1→−^→𝐢 as 𝑛→∞. By considering 𝑧₋₂ and 𝑧₋₁ in different parts of the plane, we may apply the same principle to obtain Table 2.

Table 2 

In this table, there are seven indeterminate cases. Let us go through two cases. First, let (𝑧₋₂,𝑧₋₁)∈𝑅⁺₁𝑅⁻₁×𝑅⁻₁𝑅⁺₁. Then in view of (25), (𝑧₋₂,𝑧₋₁)∈𝐽^(𝑘)𝑅⁻₁×𝑅⁻₁𝐽^(𝑠) for some 𝑘,𝑠∈𝐍. There are then two subcases: (i) 0≤𝑘≤𝑠 or (ii) 0≤𝑠<𝑘. In the former case, by Table 1, 𝑧₀=𝑎𝑧₋₂+𝑎^′𝐹₁𝑧₋₁∈𝐽^(𝑘−1)𝑅⁻₁, (31) and (hence) 𝑧₁=𝑎𝑧₋₁+𝑎^′𝐹₁𝑧₀∈𝑅⁻₁𝐽^(𝑠−1), (32) and by induction, 𝑧_2𝑘∈𝐽⁽⁻¹⁾𝑅⁻₁⊆𝑅⁻₁𝑅⁻₁,𝑧_2𝑘+1∈𝑅⁻₁𝑅⁺₁. (33) In the latter case, we may similarly show that (𝑧_2𝑠,𝑧_2𝑠+1)∈𝑅⁺₁𝑅⁻₁×𝑅⁻₁𝑅⁻₁. By considering (𝑧₋₂,𝑧₋₁) in different parts of the plane, we may then construct the following three self-explanatory Tables 3, 4, and 5.

Table 3 

Table 4 

Table 5 

As a consequence, if (𝑧₋₂,𝑧₋₁)∈𝐽^(𝑘)𝑅⁻₁×𝑅⁻₁𝐽^(𝑠) and 0≤𝑘≤𝑠, then the solution of (3) originated from 𝑧₋₂ and 𝑧₋₁ will satisfy (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁻₁𝑅⁻₁×𝑅⁻₁𝑅⁺₁. Then by Table 2, 𝑧_𝑛→⟨−^→𝐢,^→𝐤⟩.

Next, let (𝑧₋₂,𝑧₋₁)∈𝑅⁺₁𝑅⁺₁×𝑅⁺₁𝑅⁺₁. Then by (25), (𝑧₋₂,𝑧₋₁)∈𝐽^(𝑘)𝐽^(𝑝)×𝐽^(𝑙)𝐽^(𝑚) for some 𝑘,𝑝,𝑙,𝑚∈𝐍.As mentioned before, there are twelve different ordering arrangements B1–B12 for these integers. In case B1 holds, we may make use of Table 1 repeatedly to show that (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁻₁𝑅⁻₁×𝑅⁺₁𝑅⁺₁ (which is recorded in self-explanatory Table 6), and then by Table 2, we finally see that 𝑧_𝑛→⟨−^→𝐤,^→𝐤⟩.

Table 6 

In conclusion, we have shown the following result.

Theorem 1. Suppose that 𝜎=1. Then, a solution of (3) must either be asymptotically 1-periodic tending to the limit cycles ^→𝐢,−^→𝐢,𝑜𝑟^→𝐤, (34) or asymptotically 2-periodic tending to the limit cycles ^→𝐢,^→𝐤,−^→𝐢,^→𝐤,𝑜𝑟−^→𝐤,^→𝐤. (35)

We remark that, as can be seen from the above exhaustive arguments, the “region of attraction” of each limit cycle can be given, here if 𝑆 is the set of all solutions of (3) that tend to the limit cycle 𝐺, then {(𝑧₋₂,𝑧₋₁)∈𝑅²×𝑅²|{𝑧_𝑘}^∞_𝑘=−2∈𝑆} is called the region of attraction of 𝐺.

Example 2. Consider the region of attraction of the limit cycle ⟨^→𝐢,^→𝐤⟩. By Table 2, we first see that solutions with initial values in 𝑅⁻₁𝑅⁻₁×𝑅⁺₁𝑅⁻₁ or 𝑅⁺₁𝑅⁻₁×𝑅⁻₁𝑅⁻₁ tend to the limit cycle ⟨^→𝐢,^→𝐤⟩ (and hence ⟨^→𝐤,^→𝐢⟩); and solutions with initial values in the seven indeterminate cases may or may not tend to the limit cycle ⟨^→𝐢,^→𝐤⟩.
Therefore, we next consider Table 3. If (𝑧₋₂,𝑧₋₁)∈𝐽^(𝑘)𝑅⁻₁×𝑅⁻₁𝐽^(𝑠) where 0≤𝑠<𝑘, then (𝑧_2𝑠,𝑧_2𝑠+1)∈𝑅⁺₁𝑅⁻₁×𝑅⁻₁𝑅⁻₁ and hence 𝑧_𝑛→⟨^→𝐤,^→𝐢⟩. If (𝑧₋₂,𝑧₋₁)∈𝑅⁻₁𝐽^(𝑘)×𝐽^(𝑠)𝑅⁻₁ where 0≤𝑘≤𝑠, then (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁻₁𝑅⁻₁×𝑅⁺₁𝑅⁻₁ and hence 𝑧_𝑛→⟨^→𝐢,^→𝐤⟩. Furthermore, no other cases in Tables 3, 4, and 5 can lead to (𝑧_𝑚,𝑧_𝑚+1)∈𝑅⁺₁𝑅⁻₁×𝑅⁺₁𝑅⁻₁,𝑅⁻₁𝑅⁺₁×𝑅⁻₁𝑅⁺₁, or 𝑅⁺₁𝑅⁺₁×𝑅⁻₁𝑅⁻₁.
Next, we consider Table 6. By direct examination, there is no case which leads to (𝑥_𝑚,𝑥_𝑚+1)∈𝑅⁺₁𝑅⁻₁×𝑅⁻₁𝑅⁻₁ or 𝑅⁻₁𝑅⁻₁×𝑅⁺₁𝑅⁻₁. There is no case, except possibly B10, that leads to (𝑥_𝑚,𝑥_𝑚+1)∈𝐽^(𝑘)𝑅⁻₁×𝑅⁻₁𝐽^(𝑠) with 0≤𝑠<𝑘, or (𝑧₋₂,𝑧₋₁)∈𝑅⁻₁𝐽^(𝑘)×𝐽^(𝑠)𝑅⁻₁ with 0≤𝑘≤𝑠. However, if B10 holds and 𝑝−𝑘−1≤𝑙−𝑘−1, then (𝑧₋₂,𝑧₋₁)∈𝐽^(𝑘)𝐽^(𝑝)×𝐽^(𝑙)𝐽^(𝑚) implies (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁻₁𝐽^{(𝑝−𝑘−1)}×𝐽^{(𝑙−𝑘−1)}𝑅⁺₁, which in turn implies (by Table 3) that (𝑧_2𝑝,𝑧_2𝑝+1)∈𝑅⁻₁𝑅⁻₁×𝑅⁺₁𝑅⁻₁. Hence, 𝑧_𝑛→⟨^→𝐢,^→𝐤⟩.
In summary, the region of attraction of ⟨^→𝐢,^→𝐤⟩ is the union of the following sets: 𝑅⁻₁𝑅⁻₁×𝑅⁺₁𝑅⁻₁,𝑅⁺₁𝑅⁻₁×𝑅⁻₁𝑅⁻₁,𝑅⁻₁𝐽^(𝑘)×𝐽^(𝑠)𝑅⁻₁(),𝐽0≤𝑘≤𝑠_(𝑘)𝑅⁺₁×𝑅⁻₁𝐽^(𝑠)(),𝐽0≤𝑠<𝑘_(𝑘)𝐽^(𝑃)×𝐽^(𝑙)𝐽^(𝑚),.𝑚=𝑘<min𝑙,𝑝𝑝−𝑘−1≤𝑙−𝑘−1 (36)

3. The Case 𝜎>1

In this section, we assume that 𝜎>1.

Lemma 3. Suppose that 𝜎>1. Let {𝑧_𝑛}^∞_𝑛=−2 be a solution of (3) originated from 𝑧₋₂,𝑧₋₁∈𝑅⁺_𝜎𝑅⁺_𝜎. Then, there exists 𝑚∈{−1,0,…} such that 𝑧_𝑚∈𝑅⁺_𝜎𝑅⁺_𝜎 and 𝑧_𝑚+1∈𝐑²⧵𝑅⁺_𝜎𝑅⁺_𝜎.

Proof. Let {𝑧_𝑛}^∞_𝑛=−2 be a solution of (3) originated from 𝑧₋₂,𝑧₋₁∈𝑅⁺_𝜎𝑅⁺_𝜎. Assume to the contrary that 𝑧_𝑛∈𝑅⁺_𝜎𝑅⁺_𝜎 for all 𝑛∈𝐍. Then, 𝑧_𝑛=𝑎𝑧_𝑛−2+𝑎^′−^→𝐤,𝑛∈𝐍. (37) By (14) and (15), we see that 𝑧_𝑛→−^→𝐤 as 𝑛→∞. But 𝜎>1 implies −^→𝐤 is in the interior of 𝑅⁻_𝜎𝑅⁻_𝜎 which is disjointed from 𝑅⁺_𝜎𝑅⁺_𝜎. A contradiction is, thus, obtained.

In view of Lemma 3, we only need to investigate the asymptotic behaviors of solutions {𝑧_𝑛}^∞_𝑛=−2 of (3) such that 𝑧₋₂ and 𝑧₋₁ are not simultaneously in 𝑅⁺_𝜎𝑅⁺_𝜎. To this end, we first note that ()𝜎<𝜎±1−𝑎𝑎<𝜎±1−𝑎²𝑎²<⋯<𝜎±1−𝑎^𝑘𝑎^𝑘⟶+∞. (38) Hence, if we let 𝐼^(𝑘)=𝜎−1−𝑎^𝑘𝑎^𝑘,𝜎−1−𝑎^𝑘+1𝑎^𝑘+1𝐼,𝑘∈𝐍,₍₋₁₎=𝑎𝐼⁽⁰⁾+𝑎^′=(𝑎𝜎+(1−𝑎),𝜎], (39) then 𝑎𝐼^(𝑘)+𝑎^′=𝐼^(𝑘−1)𝑅,𝑘∈𝐍,_+𝜎=^∞_𝑘=0𝐽^(𝑘)=^∞_𝑘=0𝐼^(𝑘),𝑅⁺_𝜎𝑅⁻_𝜎=^∞_𝑘=0𝐼^(𝑘)𝑅⁻_𝜎=^∞_𝑘=0𝐽^(𝑘)𝑅⁻_𝜎,𝑅⁻_𝜎𝑅⁺_𝜎=^∞_𝑘=0𝑅⁻_𝜎𝐼^(𝑘)=^∞_𝑘=0𝑅⁻_𝜎𝐽^(𝑘),𝑅⁺_𝜎𝑅⁺_𝜎=^∞_𝑘,𝑠=0𝐼^(𝑘)𝐽^(𝑠)=^∞_𝑘,𝑠=0𝐽^(𝑘)𝐼^(𝑠)=^∞_𝑘,𝑠=0𝐼^(𝑘)𝐼^(𝑠)=^∞_𝑘,𝑠=0𝐽^(𝑘)𝐽^(𝑠),𝐼⁽⁻¹⁾,𝐽⁽⁻¹⁾⊂𝑅⁻_𝜎. (40) As in the previous section, we may build a table (consisting of two separate parts Tables 7 and 8) which plays the same role as Table 1.

Table 7 

Table 8 

Then, we may construct Tables 9–14 which play the same role as Tables 3–6.

Table 9 

Table 10 

Table 11 

Table 12 

Table 13 

Table 14 

Then by Lemma 3 and these tables, we may show the following result.

Lemma 4. Suppose that 𝜎>1. Let {𝑧_𝑛}^∞_𝑛=−2 be a solution of (3). Then, there exists 𝑚∈{−2,−1,…} such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎.

Proof. In view of Lemma 3, we may assume that 𝑧₋₂∈𝐑² and 𝑧₋₁∈𝐑²⧵𝑅⁺_𝜎𝑅⁺_𝜎, or, 𝑧₋₂∈𝐑²⧵𝑅⁺_𝜎𝑅⁺_𝜎 and 𝑧₋₁∈𝑅⁺_𝜎𝑅⁺_𝜎.
Consider first the case where 𝑧₋₂∈𝑅⁺_𝜎𝑅⁻_𝜎 and 𝑧₋₁∈𝑅⁻_𝜎𝑅⁺_𝜎. Then, 𝑧₋₂∈𝐽^(𝑘)𝑅⁻_𝜎 and 𝑧₋₁∈𝑅⁻_𝜎𝐽^(𝑠) for some 𝑘,𝑠∈𝐍. By Table 11, if 0≤𝑘≤𝑠, then (𝑧_2𝑘−1,𝑧_2𝑘)∈𝑅⁻_𝜎𝐽^{(𝑠−𝑘)}×𝑅⁻_𝜎𝑅⁻_𝜎. Since 𝐽^{(𝑠−𝑘)}⊂(𝜎,+∞), there exists some 𝑡∈𝐍 such that (𝑧_2𝑘−1,𝑧_2𝑘)∈𝑅⁻_𝜎𝐼^(𝑡)×𝑅⁻_𝜎𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎. If 0≤𝑠<𝑘, then (𝑧_2𝑠,𝑧_2𝑠+1)∈𝐽^{(𝑘−𝑠−1)}𝑅⁻_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎. Since 𝐽^{(𝑘−𝑠−1)}⊂(𝜎,+∞), there exists some 𝑡∈𝐍 such that (𝑧_2𝑠,𝑧_2𝑠+1)∈𝐼^(𝑡)𝑅⁻_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎.
Consider next the case where 𝑧₋₂∈𝑅⁻_𝜎𝑅⁻_𝜎 and 𝑧₋₁∈𝑅⁺_𝜎𝑅⁺_𝜎. Then, 𝑧₋₁∈𝐼^(𝑠)𝐼^(𝑘) for some 𝑘,𝑠∈𝐍. By Table 12 if 0≤𝑘≤𝑠, then (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁻_𝜎𝑅⁻_𝜎×𝐼^{(𝑠−𝑘−1)}𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎. If 0≤𝑠<𝑘, then (𝑧_2𝑠,𝑧_2𝑠+1)∈𝑅⁻_𝜎𝑅⁻_𝜎×𝑅⁻_𝜎𝐼^{(𝑘−𝑠−1)}. By the last entry in Table 9, we see that there is some 𝑚 such that (𝑧_𝑚,𝑧_𝑚+1)∈𝑅⁻_𝜎𝑅⁻_𝜎.
Consider next the case where 𝑧₋₂∈𝑅⁺_𝜎𝑅⁻_𝜎 and 𝑧₋₁∈𝑅⁺_𝜎𝑅⁺_𝜎. Then, 𝑧₋₂∈𝐽^(𝑘)𝑅⁻_𝜎 and 𝑧₋₁∈𝐼^(𝑙)𝐽^(𝑝) for some 𝑘,𝑙,𝑝∈𝐍. By Table 13, if A1 holds, that is, 0≤𝑘=𝑝≤𝑙, then (𝑧_2𝑘−1,𝑧_2𝑘)∈𝐼^{(𝑙−𝑘)}𝐽^{(𝑝−𝑘)}×𝑅⁻_𝜎𝑅⁻_𝜎. Then, there exists 𝑡∈𝐍 such that (𝑧_2𝑘−1,𝑧_2𝑘)∈𝐼^{(𝑙−𝑘)}𝐼^(𝑡)×𝑅⁻_𝜎𝑅⁻_𝜎. By Table 12 again, if 0≤𝑡≤𝑙−𝑘, then (𝑧_{2𝑡+2𝑘+1},𝑧_{2𝑡+2𝑘+2})∈𝐼^{(𝑙−𝑘−𝑡−1)}𝑅⁻_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎; if 0≤𝑙−𝑘<𝑡, then (𝑧_2𝑙+1,𝑧_2𝑙+2)∈𝑅⁻_𝜎𝐼^{(𝑡−𝑙+𝑘−1)}×𝑅⁻_𝜎𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎. If A5 holds, that is, 0≤𝑝<min{𝑘,𝑙}, then (𝑧_2𝑝,𝑧_2𝑝+1)∈𝐽^{(𝑘−𝑝−1)}𝑅⁻_𝜎×𝐼^{(𝑙−𝑝−1)}𝑅⁻_𝜎. Then, there exists 𝑡∈𝐍 such that (𝑧_2𝑝,𝑧_2𝑝+1)∈𝐼^(𝑡)𝑅⁻_𝜎×𝐼^{(𝑙−𝑝−1)}𝑅⁻_𝜎. By Table 10 again, if 0≤𝑡≤𝑙−𝑝−1, then (𝑧_{2𝑡+2𝑝+2},𝑧_{2𝑡+2𝑝+3})∈𝑅⁻_𝜎𝑅⁻_𝜎×𝐼^{(𝑙−𝑝−1−𝑡−1)}𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎; if 0≤𝑙−𝑝−1<𝑡, then (𝑧_2𝑙,𝑧_2𝑙+1)∈𝑅⁻_𝜎𝐼^{(𝑡−𝑙+𝑝)}×𝑅⁻_𝜎𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎. The other cases are similarly proved.
Consider next the case where 𝑧₋₂∈𝑅⁺_𝜎𝑅⁺_𝜎 and 𝑧₋₁∈𝑅⁺_𝜎𝑅⁻_𝜎. Then, 𝑧₋₂∈𝐼^(𝑙)𝐽^(𝑝) and 𝑧₋₁∈𝐽^(𝑘)𝑅⁻_𝜎 for some 𝑘,𝑙,𝑝∈𝐍. Next, we need to consider six cases: (i) 0=𝑙<𝑝, (ii) 0<𝑙<𝑝, (iii) 0=𝑙=𝑝, (iv) 0<𝑙=𝑝, (v) 0=𝑝<𝑙, and (vi) 0<𝑝<𝑙. Suppose that 0<𝑙<𝑝 holds, by Table 14, then (𝑧₋₁,𝑧₀)∈𝐽^(𝑘)𝑅⁻_𝜎×𝐼^(𝑙−1)𝐽^(𝑝−1). Furthermore, we need to consider six cases: A1–A6. Suppose that A3 holds, that is, 𝑝−1=𝑙−1<𝑘, then by Table 13, (𝑧_2𝑙−1,𝑧_2𝑙)∈𝐽^{(𝑘−𝑙)}𝑅⁻_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎. Then there exists 𝑡∈𝐍 such that (𝑧_2𝑙−1,𝑧_2𝑙)∈𝐼^(𝑡)𝑅⁻_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎. By the third entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎. Suppose that A6 holds, that is, 𝑙−1<min{𝑘,𝑝−1}, then by Table 13, (𝑧_2𝑙−1,𝑧_2𝑙)∈𝐽^{(𝑘−𝑙)}𝑅⁻_𝜎×𝑅⁻_𝜎𝐽^{(𝑝−𝑙−1)}. If 𝑘−𝑙≤𝑝−𝑙−1, then by Table 11, (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁻_𝜎𝐽^{(𝑝−𝑘−𝑙)}×𝑅⁻_𝜎𝑅⁻_𝜎. Then, there exists 𝑡∈𝐍 such that (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁻_𝜎𝐼^(𝑡)×𝑅⁻_𝜎𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎; if 𝑝−𝑙−1<𝑘−𝑙, then by Table 11, (𝑧_2𝑝−1,𝑧_2𝑝)∈𝐽^{(𝑘−𝑝)}𝑅⁻_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎. Then there exists some 𝑡∈𝐍 such that (𝑧_2𝑝−1,𝑧_2𝑝)∈𝐼^(𝑡)𝑅⁻_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎. By the last entry in Table 9, we see that there is some 𝑚 such that 𝑧_𝑚,𝑧_𝑚+1∈𝑅⁻_𝜎𝑅⁻_𝜎. The other cases are similarly proved.
The proof is complete.

Theorem 5. Suppose that 𝜎>1. Then, every solution of (3) tends to ^→𝐤.
Indeed, let {𝑧_𝑛}^∞_𝑛=−2 be a solution of (3). In view of Lemma 4, one may assume that 𝑧₋₂,𝑧₋₁∈𝑅⁻_𝜎𝑅⁻_𝜎. Then by the (𝑅⁻_𝜎𝑅⁻_𝜎,𝑅⁻_𝜎𝑅⁻_𝜎) entry in Table 7, one may sees that 𝑧₀∈𝑅⁻_𝜎𝑅⁻_𝜎 and by induction, 𝑧_𝑛∈𝑅⁻_𝜎𝑅⁻_𝜎 for all 𝑛≥−2. By (14) and (15), then sees that 𝑧_𝑛→^→𝐤 as required.

4. The Case 0≤𝜎<1

Suppose that 0≤𝜎<1. Then, ()𝜎>𝜎−1−𝑎𝑎>𝜎−1−𝑎²𝑎²>⋯>𝜎−1−𝑎^𝑘𝑎^𝑘⟶−∞. (41) If we let 𝐾^(𝑘)=𝜎−1−𝑎^𝑘+1𝑎^𝑘+1,𝜎−1−𝑎^𝑘𝑎^𝑘𝐾,𝑘∈𝐍,₍₋₁₎=𝑎𝐾⁽⁰⁾+𝑎^′=(𝜎,𝑎𝜎+(1−𝑎)], (42) then 𝑎𝐾^(𝑘)+𝑎^′=𝐾^(𝑘−1)𝑅,𝑘∈𝐍,_+𝜎=^∞_𝑘=0𝐽^(𝑘),𝑅⁻_𝜎=^∞_𝑘=0𝐾^(𝑘),𝑅⁻_𝜎𝑅⁻_𝜎=^∞_𝑘=0𝐾^(𝑘)𝑅⁻_𝜎=^∞_𝑘=0𝑅⁻_𝜎𝐾^(𝑘)=^∞_𝑘,𝑠=0𝐾^(𝑘)𝐾^(𝑠),𝑅⁺_𝜎𝑅⁺_𝜎=^∞_𝑘=0𝐽^(𝑘)𝑅⁺_𝜎=^∞_𝑘=0𝑅⁺_𝜎𝐽^(𝑘)=^∞_𝑘,𝑠=0𝐽^(𝑘)𝐽^(𝑠),𝑅⁺_𝜎𝑅⁻_𝜎=^∞_𝑘=0𝐽^(𝑘)𝑅⁻_𝜎=^∞_𝑘=0𝑅⁺_𝜎𝐾^(𝑘)=^∞_𝑘,𝑠=0𝐽^(𝑘)𝐾^(𝑠),𝑅⁻_𝜎𝑅⁺_𝜎=^∞_𝑘=0𝐾^(𝑘)𝑅⁺_𝜎=^∞_𝑘=0𝑅⁻_𝜎𝐽^(𝑘)=^∞_𝑘,𝑠=0𝐾^(𝑘)𝐽^(𝑠). (4) Let {𝑧_𝑛}^∞_𝑛=−2 be a solution of (3). Let us consider first the case where 𝑧_𝑛,𝑧_𝑛+1∈𝑅⁺_𝜎𝑅⁻_𝜎. Then by (7) and (16), we see that 𝑧_𝑛+2=𝑎𝑧_𝑛+𝑎^′→𝐢. (44) Note that (in view of our assumption that 0≤𝜎<1) 𝑎𝑅⁺_𝜎𝑅⁻_𝜎+𝑎^′𝐹_𝜎𝑅⁺_𝜎𝑅⁻_𝜎=𝑎𝑅⁺_𝜎𝑅⁻_𝜎+𝑎^′→𝐢)×(+()=(())×((=𝑎((𝜎,∞−∞,𝜎])1−𝑎)(1,−1𝑎𝜎+1−𝑎,∞−∞,𝑎𝜎−1−𝑎)]⊆𝑅⁺_𝜎𝑅⁻_𝜎. (45) Hence, we may see further that 𝑧_𝑛+2∈𝑅⁺_𝜎𝑅⁻_𝜎. Similarly, since 𝑎𝑅⁺_𝜎𝑅⁺_𝜎+𝑎^′𝐹_𝜎𝑅⁻_𝜎𝑅⁻_𝜎=𝑎𝑅⁺_𝜎𝑅⁺_𝜎+𝑎^′→𝐤)×(+()=(())×(())=𝑎((𝜎,∞𝜎,∞))1−𝑎)(1,1𝑎𝜎+1−𝑎,∞𝑎𝜎+1−𝑎,∞⊆𝑅⁺_𝜎𝑅⁺_𝜎,𝑎𝑅⁻_𝜎𝑅⁻_𝜎+𝑎^′𝐹_𝜎𝑅⁺_𝜎𝑅⁺_𝜎=𝑎𝑅⁻_𝜎𝑅⁻_𝜎−𝑎^′→𝐤]×(−()=((×((=𝑎((−∞,𝜎−∞,𝜎])1−𝑎)(1,1−∞,𝑎𝜎−1−𝑎)]−∞,𝑎𝜎−1−𝑎)]⊆𝑅⁻_𝜎𝑅⁻_𝜎, (46) we see that if (𝑧_𝑛,𝑧_𝑛+1)∈𝑅⁺_𝜎𝑅⁺_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎, then 𝑧_𝑛+2∈𝑅⁺_𝜎𝑅⁺_𝜎 and 𝑧_𝑛+3∈𝑅⁻_𝜎𝑅⁻_𝜎. By similar considerations, we may build a table (consisting of two separate parts Tables 15 and 16) which plays the same role as Table 1.

Table 15 

Table 16 

Next, we let {𝑧_𝑛}^∞_𝑛=−2 be a solution of (3) again. If (𝑧₋₂,𝑧₋₁)∈𝑅⁺_𝜎𝑅⁺_𝜎×𝑅⁻_𝜎𝑅⁻_𝜎, then by the (𝑅⁺_𝜎𝐽^(𝑘),𝑅⁻_𝜎𝐼^(𝑠)) entry of Table 15, we see that 𝑧₀∈𝑅⁺_𝜎𝑅⁺_𝜎. Now that (𝑧₋₁,𝑧₀)∈𝑅⁻_𝜎𝑅⁻_𝜎×𝑅⁺_𝜎𝑅⁺_𝜎, we may use the (𝐾^(𝑘)𝑅⁻_𝜎,𝐽^(𝑠)𝑅⁺_𝜎) entry of Table 16 to infer 𝑧₁∈𝑅⁻_𝜎𝑅⁻_𝜎 again. By induction, we see that 𝑧_2𝑛∈𝑅⁺_𝜎𝑅⁺_𝜎,𝑧_2𝑛+1∈𝑅⁻_𝜎𝑅⁻_𝜎 for all 𝑛≥−2. By (7), we then see that 𝑧_2𝑛=𝑎𝑧_𝑛−2+𝑎^′→𝐤𝑧,𝑛∈𝐍,_2𝑛+1=𝑎𝑧_2𝑛−1−𝑎^′→𝐤,𝑛∈𝐍, (47) and hence by (14) and (15), we see that 𝑧_2𝑛→^→𝐤,𝑧_2𝑛+1→−^→𝐤 as 𝑛→∞. By considering 𝑧₋₂ and 𝑧₋₁ in different parts of the plane, we may apply the same principle to obtain Table 17.

Table 17 

In this table, there are twelve indeterminate cases. Let us go through one case. Let (𝑧₋₂,𝑧₋₁)∈𝑅⁻_𝜎𝑅⁻_𝜎×𝑅⁺_𝜎𝑅⁻_𝜎. Then, (𝑧₋₂,𝑧₋₁)∈𝐾^(𝑘)𝑅⁻_𝜎×𝑅⁺_𝜎𝐾^(𝑠) for some 𝑘,𝑠∈𝐍. There are then two subcases: (i) 0≤𝑘≤𝑠 or (ii) 0≤𝑠<𝑘. In the former case, by Table 15, 𝑧₀=𝑎𝑧₋₂+𝑎^′𝐹_𝜎𝑧₋₁∈𝐾^(𝑘−1)𝑅⁻_𝜎, (48) and (hence) 𝑧₁=𝑎𝑧₋₁+𝑎^′𝐹_𝜎𝑧₀∈𝑅⁺_𝜎𝐾^(𝑠−1), (49) and so forth, and finally, 𝑧_2𝑘∈𝐾⁽⁻¹⁾𝑅⁻_𝜎⊆𝑅⁺_𝜎𝑅⁻_𝜎,𝑧_2𝑘+1∈𝑅⁺_𝜎𝑅⁻_𝜎. (50) In the latter case, we may similarly show that (𝑧_2𝑠,𝑧_2𝑠+1)∈𝑅⁻_𝜎𝑅⁻_𝜎×𝑅⁺_𝜎𝑅⁺_𝜎. By considering (𝑧₋₂,𝑧₋₁) in different parts, we may then construct four self-explanatory Tables 18, 19, 20, and 21.

Table 18 

Table 19 

Table 20 

Table 21 

As a consequence, if (𝑧₋₂,𝑧₋₁)∈𝐾^(𝑘)𝑅⁻×𝑅⁺𝐾^(𝑠) and 0≤𝑘≤𝑠, then the solution of (3) originated from 𝑧₋₂ and 𝑧₋₁ will satisfy (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁺_𝜎𝑅⁻_𝜎×𝑅⁺_𝜎𝑅⁻_𝜎. Then by Table 17, 𝑧_𝑛→⟨^→𝐢⟩.

Next, let (𝑧₋₂,𝑧₋₁)∈𝑅⁻_𝜎𝑅⁺_𝜎×𝑅⁺_𝜎𝑅⁻_𝜎. Then, (𝑧₋₂,𝑧₋₁)∈𝐾^(𝑘)𝐽^(𝑝)×𝐽^(𝑙)𝐾^(𝑚) for some 𝑘,𝑝,𝑙,𝑚∈𝐍.As mentioned before, there are twelve different ordering arrangements B1–B12 for the nonnegative integers 𝑘,𝑝,𝑙,and𝑚. In case B2 holds, we may make use of Table 22 repeatedly to show that (𝑧_2𝑘,𝑧_2𝑘+1)∈𝑅⁺_𝜎𝐽^{(𝑝−𝑘−1)}×𝐽^{(𝑙−𝑘−1)}𝑅⁻_𝜎 (which is recorded in self-explanatory Tables 22 and 23). If 0≤𝑝−𝑘−1≤𝑙−𝑘−1, then by Table 19, (𝑧_2𝑝,𝑧_2𝑝+1)∈𝑅⁺_𝜎𝑅⁻_𝜎×𝑅⁺_𝜎𝑅⁻_𝜎. By Table 17 again, 𝑧_𝑛→⟨^→𝐢⟩.

Table 22 

Table 23 

In conclusion, we have shown the following result.

Theorem 6. Suppose that 0≤𝜎<1. Then, a solution of (3) must be either asymptotically 1-periodic tending to the limit cycles ⟨^→𝐢⟩ or ⟨−^→𝐢⟩ or asymptotically 2-periodic tending to the limit cycle ⟨^→𝐤,−^→𝐤⟩.

We remark that, as can be seen from the above exhaustive arguments and Example 2, the region of attraction of each limit cycle can be given precisely (by implementing a simple computer program, if necessary).

5. Conclusions

In neural network terminologies, we have discussed a simple two-neuron recurrent McCulloch-Pitts-type neural network with a nonnegative threshold 𝜎. For 𝜎>1, all solutions tend to the limiting state (1,1); for 𝜎=1, all solutions must either be tending to the limiting states (−1,1),(1,−1) or (1,1), or asymptotically 2-periodic tending to the limit cycles ⟨(1,−1),(1,1)⟩,⟨(−1,1),(1,1)⟩, or ⟨(−1,−1),(1,1)⟩; and for 0≤𝜎<1, all solutions must either be tending to the steady states (−1,1), or (1,−1) or asymptotically 2-periodic tending to the limit cycle ⟨(1,1),(−1,−1)⟩. In all cases, the corresponding region of attraction of each limit cycle can be given.

Our proofs show that more general multiple-neuron recurrent McCulloch-Pitts-type neural networks possess similar behaviors. However, the derivations may involve more delicate graph theoretic arguments and are better left for other studies.