Complete Asymptotic Analysis of a Two-Nation Arms Race Model with Piecewise Constant Nonlinearities

Abstract

A discrete time two-nation arms race model involving a piecewise constant nonlinear control function is formulated and studied. By elementary but novel arguments, we are able to give a complete analysis of its asymptotic behavior when the threshold parameter in the control function varies from 0⁺ to ∞. We show that all solutions originated from positive initial values tend to limit one or two cycles. An implication is that when devastating weapons are involved, “terror equilibrium” can be achieved and escalated race avoided. It is hoped that our analysis will provide motivation for further studying of discrete-time equations with piecewise smooth nonlinearities.

1. Introduction

In [1, pages 87–90], a simple dynamical model of a two-nation arms race based on Richardson’s ideas in [2] is explained, and several interesting conclusions are drawn which can be used to explain stable and escalated arms races. Roughly, let 𝐍={0,1,2,…}, and let 𝐴_𝑛 and 𝐵_𝑛 be the amount spent on armaments by two respective countries 𝐴 and 𝐵 in year 𝑛∈𝐍. Assuming, 𝐴 has a fixed amount of distrust of the other country, causing it to retain arms, then𝐴_𝑛=1−𝑟_𝐴𝐴_𝑛−1+𝑠_𝐴𝐵_𝑛−1+𝑢,(1.1) where the constant 𝑠_𝐴 measures country 𝐴’s distrust of country 𝐵 in that it reacts to the way 𝐵 arms itself, 𝑟_𝐴∈(0,1) is a measure of 𝐴’s own economy, and 𝑢 is the basic annual expenditure (e.g., maintenance expense). If we now assume a similar situation for country 𝐵, then we also have𝐵_𝑛=1−𝑟_𝐵𝐵_𝑛−1+𝑠_𝐵𝐴_𝑛−1+𝑣.(1.2) Under the assumption that 𝑟_𝐴=𝑟_𝐵=𝑟 and 𝑠_𝐴=𝑠_𝐵=𝑠, it is shown that if the initial total expenditure 𝐴₋₁+𝐵₋₁ is large and that the distrust factor is also so large that 𝑠>𝑟, then no two countries can sustain exponentially increasing expenditures on arms, and the alternative is war or negotiation. While this model is an oversimplification one, it could help to understand plausible reasons behind World War I (see [2, 3] in which various aspects of arms race modeling are discussed).

The above model cannot explain some of the observations we can make nowadays. Therefore, we need to build various models and analyze their asymptotic behaviors. In this paper, we will build one such model based on the idea that although the distrust factor is the same as in the previous model, the expenditure by the other country in year 𝑛−1 in (1.1) is replaced by 𝑠_𝐴𝑓_𝜆(𝐵_𝑛−1) where𝑓_𝜆𝐵_𝑛−1=1if𝐵^𝑛−10if𝐵_𝑛−1∈(0,𝜆],∈(−∞,0]∪(𝜆,∞),(1.3) and the term 𝐴_𝑛−1 in (1.2) is replaced by a similar one. The “discontinuous” function 𝑓_𝜆 has a clear physical meaning. Indeed, the positive parameter 𝜆 may be treated as a cutoff threshold indicator so that when the competitor is already spending an unreasonable amount of money (such as stocking of hundreds of nuclear missiles that can annihilate our mother earth) or is not spending any, there is no need to add the budget anymore. With this function at hand, we may rewrite our new model as follows:𝑥𝑦_𝑛=𝑎𝑥_𝑛−𝛼+𝑏𝑓_𝜆𝑦_𝑛−1+𝑐,_𝑛=𝑟𝑦_𝑛−𝛽+𝑠𝑓_𝜏𝑥_𝑛−1+𝑡,(1.4) where we have introduced two “delays” 𝛼 and 𝛽 in order to reflect the fact that the expenditure in a previous accounting period may not be recorded precisely, and hence historical expenditure records may be more reliable for use in making future decisions.

Although (1.4) may seem to be a simple model, there are still too many parameters involved. We therefore make further (reasonable) assumptions as follows:𝑐,𝑡=0,𝛼=𝛽=2,𝜏=𝜆>0,𝑟=𝑎∈(0,1),𝑏=𝑠=(1−𝑎).(1.5) The assumption that 𝑐,𝑡=0 means that the fixed expenditures are relatively low in both countries, while we assume that 𝛼=𝛽=2 so as to use the best up-to-date and “reliable” accounting records 𝑥_𝑛−2 and 𝑦_𝑛−2. By choosing 𝑎∈(0,1) and 𝑏=1−𝑎, country 𝐴 is making a decision based on a convex combination of the expenditures 𝑥_𝑛−2 and the blanket-ceiling sum 𝑓_𝜆(𝑦_𝑛−1). If country 𝐵 takes on a similar decision policy, then we end up with𝑦_𝑛=𝑎^∗𝑦_𝑛−2+1−𝑎^∗𝑓_𝜆𝑥_𝑛−1,(1.6) where 𝑎^∗ may or may not differ from 𝑎. The case 𝑎^∗≠𝑎 is only more technically difficult, and therefore, in this paper, we will assume the case 𝑎=𝑎^∗ (which is already nontrivial as we will see) so that both countries play “symmetric” roles in the interactions.

By adopting these assumptions, we then settle on the following dynamical system:𝑥𝑦_𝑛𝑛=𝑎𝑥_𝑛−2+(1−𝑎)𝑓_𝜆𝑦_𝑛−1,=𝑎𝑦_𝑛−2+(1−𝑎)𝑓_𝜆𝑥_𝑛−1,(1.7) for 𝑛∈𝐍, where in this model, 𝑎∈(0,1),𝜆>0. Note that if we let 𝑧=(𝑥,𝑦) and𝐹_𝜆(𝑧)=𝑓_𝜆(𝑦),𝑓_𝜆(𝑥),(1.8) then the above system (1.7) can be written as𝑧_𝑛=𝑎𝑧_𝑛−2+𝑎^𝐹_𝜆𝑧_𝑛−1,𝑛∈𝐍,(1.9) 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 (1.9) determined by 𝑧₋₂ and 𝑧₋₁. Among different 𝑧₋₂ and 𝑧₋₁, those lying in the positive quadrant are of special interests since expenditures are always positive. Therefore, our subsequent interests are basically the asymptotic behaviors of all such solutions determined by 𝑧₋₂ and 𝑧₋₁ with positive components.

We remark that system (1.9) can be regarded as a discrete dynamical system with piecewise smooth nonlinearities. Such systems have not been explored extensively (see, e.g., the discussions on “polymodal” discrete systems in [4], and there are only several recent studies on scalar equations with piecewise smooth nonlinearities [5–9])! Therefore, a complete asymptotic analysis of our equation is essential in the further development of discontinuous (in particular, 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 any 𝑧₋₂,𝑧₋₁ in the quadrant (0,∞)², the solution {𝑧_𝑛}^∞𝑛−2 determined by it also lies in the same quadrant (in the sense that 𝑧_𝑛∈(0,∞)² for 𝑛∈𝐍). 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 the first quadrant in the following manner:𝐴=(0,𝜆]²,𝐵=(𝜆,∞)×(0,𝜆],𝐶=(𝜆,∞)×(𝜆,∞),𝐷=(0,𝜆]×(𝜆,∞),(1.10) where 𝜆 is a fixed positive number, then𝔓={𝐴,𝐵,𝐶,𝐷}(1.11) is a partition of the quadrant (0,∞)².

Note that these subsets depend on 𝜆, but this dependence is not emphasized in the sequel for the sake of convenience.

For any solution {𝑧_𝑛}^∞𝑛=−2 originated from 𝑧₋₂ and 𝑧₋₁ in the above subsets, our main conclusion in this paper is that {𝑧_2𝑛} will tend to some vector 𝑢 and {𝑧_2𝑛+1} will tend to another vector 𝑣 (which may or may not be equal to 𝑢). This implication is important since it says that escalated arms race cannot happen and World War III should not happen if our model is correct!

For the sake of convenience, we record this fact by means of𝑧_𝑛⟶⟨𝑢,𝑣⟩.(1.12) In case 𝑢=𝑣,{𝑧_𝑛} is convergent to 𝑢, and hence we may also write𝑧_𝑛⟶𝑢or𝑧_𝑛⟶⟨𝑢⟩.(1.13)

To arrive at our main conclusion, we note, however, that since 𝑓_𝜆 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 (1.9) 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: suppose that 𝜆=1. If {𝑧_𝑛}^∞𝑛=−2 is a solution of (1.9) with (𝑧₋₂,𝑧₋₁)∈𝐴×𝐵, then as will be seen below, 𝑧_2𝑛→(1,0) and 𝑧_2𝑛+1→(1,1). If we now replace the condition (𝑧₋₂,𝑧₋₁)∈𝐴×𝐵 with the symmetric initial condition (𝑧₋₂,𝑧₋₁)∈𝐴×𝐷, then we will end up with the conclusion that 𝑧_2𝑛→(0,1) and 𝑧_2𝑛+1→(1,1). Such two conclusions will be referred to as dual results. We will see some tables which contain some obvious dual results later.

Next, by (1.8), we may easily see that𝐹_𝜆(𝐴)=𝐤,𝐹_𝜆(𝐵)=𝐢,𝐹_𝜆(𝐶)=𝟎,𝐹_𝜆(𝐷)=𝐣,(1.14) where𝟎=(0,0),𝐢=(1,0),𝐣=(0,1),𝐤=(1,1).(1.15) Therefore, in case {𝑧_𝑘} is a solution of (1.9) such that, say, 𝑧_𝑘∈𝐴 for all large 𝑘, then (1.9) is reduced to𝑧_𝑛=𝑎𝑧_𝑛−2+𝑎^𝐤,(1.16) for all large 𝑛. Hence, linear systems and their related properties will also appear in later discussions. More precisely, two groups of bounding quantities 𝛼_𝑗 and 𝛽_𝑗 are needed:𝛼_𝑗=1+𝜆−1𝑎^𝑗,𝛽_𝑗=𝜆𝑎^𝑗,𝑗∈𝐍.(1.17) Note that they satisfy 𝛼₀=𝛽₀=𝜆 and the recurrence relations 𝛼_𝑗+1=1𝑎𝑎^𝑗+𝑎−1𝑎,𝛽^𝑗+1=1𝑎𝛽^𝑗,𝑗∈𝐍.(1.18) We also need the following two properties of linear systems. Let {𝑥_𝑘}^∞𝑘=−2 be real scalar (or vector) sequences that satisfy𝑥_2𝑘=𝑎𝑥_2𝑘−2+𝑑,𝑘∈𝐍,(1.19)𝑥_2𝑘+1=𝑎𝑥_2𝑘−1+𝑑,𝑘∈𝐍,(1.20) 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 (1.19), then 𝑥_2𝑘=𝑎^𝑘+1𝑥₋₂+1−𝑎^𝑘+11−𝑎𝑑,𝑘∈𝐍.(1.21)(ii)If {𝑥_𝑛}^∞𝑛=−2 is a sequence which satisfies (1.20), then𝑥_2𝑘+1=𝑎^𝑘+1𝑥₋₁+1−𝑎^𝑘+11−𝑎𝑑,𝑘∈𝐍.(1.22)

Finally, we need to consider various ordering arrangements for three or four nonnegative integers 𝑘,𝑝,𝑙,and𝑚. First, the ordering arrangements of three integers 𝑘,𝑝,and𝑙 can be classified into 6 cases: (1) 𝑘=𝑝≤𝑙,(2)𝑙=𝑘<𝑝,(3)𝑝=𝑙<𝑘,(4)𝑘<min{𝑝,𝑙},(5)𝑝<min{𝑘,𝑙},and(6)𝑙<min{𝑘,𝑝}. In fact, let 𝑎,𝑏∈𝑅. Then either 𝑎<𝑏,𝑎=𝑏, or 𝑎>𝑏. Let 𝑎,𝑏,𝑐∈𝑅, then𝑎𝑎𝑎<=>𝑏𝑏𝑏⟹⟹⟹𝑐𝑐𝑐∈∈∈(((−−−∞∞∞,,,𝑎𝑎𝑏))),,,𝑐𝑐𝑐===𝑏𝑎𝑎,,or𝑐𝑐∈∈𝑐∈((𝑏𝑎,,(𝑎𝑎𝑏),)∞,,),𝑐𝑐==𝑎𝑏oorr𝑐𝑐∈∈((𝑎𝑏,,∞∞)),.(1.23) These are equivalent to𝑎<min𝑎{𝑏=,𝑐𝑏}≤,𝑐,𝑏<𝑐=mi𝑎n<{𝑎𝑏,,𝑐},𝑏=𝑐𝑐<<m𝑎,in{𝑎,𝑏},(1.24) by comparing the two sets of statements.

By similar reasoning, there are 12 ordering arrangements for four integers 𝑘,𝑝,𝑙,and𝑚: (1)𝑘𝑘m,i=n(6{𝑝)𝑙,𝑙≤𝑝<}m,mi(n1in1{)𝑙{,𝑝𝑝𝑚,=𝑘},,𝑚𝑚(2}<),m𝑘(7i<)n𝑙{m𝑙=,in𝑘𝑚}{,𝑝<a,n𝑙dm,𝑚(in1}2,{)𝑘(3𝑘,)𝑝=}𝑝,𝑙<(<8m)m𝑙inin={{𝑘𝑚𝑝,,𝑙𝑚=,𝑚}𝑘},<(4)𝑝,𝑝(=9)𝑙𝑚<<mminin{𝑘{,𝑙𝑚,𝑘},,𝑝}(5,)(𝑝10=)𝑚𝑙==𝑚𝑘<<.

Our following plan is quite simple. We will treat our 𝜆 as a bifurcation parameter and distinguish four different cases (i) 𝜆>1, (ii) 𝜆=1, (iii) 0<𝜆<1−𝑎, and (iv) 1−𝑎≤𝜆<1, and consider different 𝑧₋₂,𝑧₋₁ in 𝐴,𝐵,𝐶, or 𝐷 and discuss the precise asymptotic behaviors of the corresponding solutions determined by them.

2. The Case 𝜆>1

This case is relatively simple.

Theorem 2.1. Suppose, 𝜆>1. Let {𝑧_𝑘}^∞𝑘=−2 be any solution of (1.9) with (𝑧₋₂,𝑧₋₁)∈(0,∞)². Then 𝑧_𝑛→𝐤.

Proof. By (1.9), we may see that 𝑥_𝑛≤𝑎𝑥_𝑛−2+𝑎^ and 𝑦_𝑛≤𝑎𝑦_𝑛−2+𝑎^ for 𝑛∈𝐍, then lim_𝑛𝑥_𝑛≤1<𝜆 and lim_𝑛𝑦_𝑛≤1<𝜆. Thus, there exists an integer 𝑚 such that (𝑧_𝑘,𝑧_𝑘+1)∈(0,𝜆]² for all 𝑘≥𝑚. Therefore, 𝑧_𝑘+2=𝑎𝑧_𝑘+𝑎^𝐤 for all 𝑘≥𝑚. In view of (1.21) and (1.22), 𝑧_𝑛→𝐤. The proof is complete.

3. The Case 𝜆=1

In this section, we assume that 𝜆=1. If {𝑧_𝑘}^∞𝑘=−2 is a solution of (1.9) and if 𝑧_𝑘∈𝐴 and 𝑧_𝑘+1∈(0,∞)², then in view of (1.14), 𝐹_𝜆(𝑧_𝑘+1)∈{𝐢,𝐣,𝐤,𝟎}, and hence𝑧_𝑘+2=𝑎𝑧_𝑘+𝑎^𝐹_𝜆𝑧_𝑘+1∈𝐴.(3.1) By similar reasoning, we may consider all other possible cases and collect our findings in tables. To simplify the description of these tables, we first note that {𝛽_𝑗}^∞𝑗=0={1/𝑎^𝑗}^∞𝑗=0 is a strictly increasing and divergent sequence. Therefore, if we let𝐼^(𝑖)=𝛽_𝑖,𝛽_𝑖+1,𝐵^(𝑖)=𝐼^(𝑖)×(0,1],𝐷^(𝑖)=(0,1]×𝐼^(𝑖),𝐶^(𝑖,𝑗)=𝐼^(𝑖)×𝐼^(𝑗),(3.2) then(1,∞)=^∞_𝑖=0𝐼^(𝑖),𝐵=^∞_𝑖=0𝐵^(𝑖),𝐷=^∞_𝑖=0𝐷^(𝑖),𝐶=_{𝑖,𝑗∈𝑁}𝐶^(𝑖,𝑗).(3.3)

First of all, the fact that 𝑧_𝑘∈𝐴 and 𝑧_𝑘+1∈𝐴 implies, 𝑧_𝑘+2∈𝐴 is recorded as the (𝐴,𝐴) entry in Table 1. In this table, we may also find other entries which are self-explanatory.

Table 1 

By similar considerations, we may also obtain Tables 1 and 2.

Table 2 

For instance, let us show the third data row in Table 2. Suppose that 𝑧₋₂∈𝐵^(𝑘) and 𝑧₋₁∈𝐷^(𝑝), where 0≤𝑘≤𝑝. By Table 1, if 0=𝑘≤𝑝, then𝑧𝑧₀=𝑎𝑧₋₂+𝑎^𝐹_𝜆𝑧₋₁=𝑎𝑧₋₂+𝑎^𝐣∈𝑎𝐵⁽⁰⁾+𝑎^𝐣⊆𝐴,₁=𝑎𝑧₋₁+𝑎^𝐹_𝜆𝑧₀=𝑎𝑧₋₁+𝑎^𝐤∈𝑎𝐷^(𝑝)+𝑎^𝐤⊆𝐷.(3.4) If 0<𝑘≤𝑝, then𝑧𝑧₀₁=𝑎𝑧₋₂+𝑎^𝐹_𝜆𝑧₋₁=𝑎𝑧₋₂+𝑎^𝐣∈𝑎𝐵^(𝑘)+𝑎^𝐣⊆𝐵^(𝑘−1),𝑧₂=𝑎𝑧₋₁+𝑎^𝐹_𝜆𝑧₀=𝑎𝑧₋₁+𝑎^𝐢∈𝑎𝐷^(𝑝)+𝑎^𝐢⊆𝐷^(𝑝−1),𝑧₃=𝑎𝑧₀+𝑎^𝐹_𝜆𝑧₁=𝑎𝑧₀+𝑎^𝐣∈𝑎𝐵^(𝑘−1)+𝑎^𝐣⊆𝐵^(𝑘−2),=𝑎𝑧₁+𝑎^𝐹_𝜆𝑧₂=𝑎𝑧₁+𝑎^𝐢∈𝑎𝐷^(𝑝−1)+𝑎^𝐢⊆𝐷^(𝑝−2),(3.5) and by induction,𝑧𝑧_2𝑘=𝑎𝑧_2𝑘−2+𝑎^𝐹_𝜆𝑧_2𝑘−1=𝑎𝑧_2𝑘−2+𝑎^𝐣∈𝑎𝐵⁽⁰⁾+𝑎^𝐣⊆𝐴,_2𝑘+1=𝑎𝑧_2𝑘−1+𝑎^𝐹_𝜆𝑧_2𝑘=𝑎𝑧_2𝑘−1+𝑎^𝐤∈𝑎𝐷^{(𝑝−𝑘)}+𝑎^𝐤⊆𝐷^{(𝑝−𝑘−1)},(3.6) that is, 𝑧_2𝑘∈𝐴 and 𝑧_2𝑘+1∈𝐷.

As another example, let us show the second data row in Table 3. Suppose that (𝑧₋₂,𝑧₋₁)∈𝐶^(𝑘,𝑝)×𝐶^(𝑙,𝑚), where 0≤𝑘<min{𝑙,𝑚,𝑝}, then by Table 1, if 0=𝑘<min{𝑙,𝑚,𝑝},𝑧𝑧₀₁=𝑎𝑧₋₂+𝑎^𝐹_𝜆𝑧₋₁=𝑎𝑧₋₂∈𝑎𝐶^(0,𝑝)⊆𝐷^(𝑝−1),=𝑎𝑧₋₁+𝑎^𝐹_𝜆𝑧₀=𝑎𝑧₋₁+𝑎^𝐣∈𝑎𝐶^(𝑙,𝑚)+𝑎^𝐣⊆𝐼^(𝑙−1)×(1,∞).(3.7) If 0<𝑘<min{𝑙,𝑚,𝑝}, then𝑧𝑧₀₁=𝑎𝑧₋₂+𝑎^𝐹_𝜆𝑧₋₁=𝑎𝑧₋₂∈𝑎𝐶^(𝑘,𝑝)⊆𝐶^{(𝑘−1,𝑝−1)},𝑧₂=𝑎𝑧₋₁+𝑎^𝐹_𝜆𝑧₀=𝑎𝑧₋₁∈𝑎𝐶^(𝑙,𝑚)⊆𝐶^{(𝑙−1,𝑚−1)},𝑧₃=𝑎𝑧₀+𝑎^𝐹_𝜆𝑧₁=𝑎𝑧₀∈𝑎𝐶^{(𝑘−1,𝑝−1)}⊆𝐶^{(𝑘−2,𝑝−2)},=𝑎𝑧₁+𝑎^𝐹_𝜆𝑧₂=𝑎𝑧₁∈𝑎𝐶^{(𝑙−1,𝑚−1)}⊆𝐶^{(𝑙−2,𝑚−2)},(3.8) and by induction,𝑧𝑧_2𝑘2𝑘+1=𝑎𝑧_2𝑘−2+𝑎^𝐹_𝜆𝑧_2𝑘−1=𝑎𝑧_2𝑘−2∈𝑎𝐶^{(0,𝑝−𝑘)}⊆𝐷^{(𝑝−𝑘−1)},=𝑎𝑧_2𝑘−1+𝑎^𝐹_𝜆𝑧_2𝑘=𝑎𝑧_2𝑘−1+𝑎^𝐣∈𝑎𝐶^{(𝑙−𝑘,𝑚−𝑘)}+𝑎^𝐣⊆𝐼^{(𝑙−𝑘−1)}×(1,∞),(3.9) that is, 𝑧_2𝑘∈𝐷^{(𝑝−𝑘−1)} and 𝑧_2𝑘+1∈𝐶^{(𝑙−𝑘−1,𝑡)} for some 𝑡∈𝑁.

Table 3 

Initial condition: (𝑧₋₂,𝑧₋₁)∈𝐶^(𝑘,𝑝)×𝐶^(𝑙,𝑚).

By means of the information obtained so far, let {𝑧_𝑛}^∞𝑛=−2 be a solution of (1.9), we will be able to show the following result.

Theorem 3.1. Suppose that 𝜆=1. Let {𝑧_𝑛}^∞𝑛=−2 be a solution of (1.9) originated from (0,∞)². Then 𝑧_𝑛⟶⟨𝐢⟩,⟨𝐣⟩,⟨𝐤⟩,⟨𝟎,𝐤⟩,⟨𝐤,𝟎⟩,⟨𝐢,𝐤⟩,⟨𝐤,𝐢⟩,⟨𝐣,𝐤⟩or⟨𝐤,𝐣⟩.(3.10)

To this end, let us consider first the case where (𝑧₋₂,𝑧₋₁)∈𝐴×𝐴, then by the (𝐴,𝐴) entry in Table 1, 𝑧₀∈𝐴. By induction, we may then see that 𝑧_𝑘∈𝐴 for all 𝑘≥−2. Hence, by (1.9), we see that𝑧_𝑛=𝑎𝑧_𝑛−2+(1−𝑎)𝐤,𝑛∈𝑁,(3.11) from which we easily obtain𝑧𝑧_2𝑘=𝑎^𝑘+1𝑧₋₂+1−𝑎^𝑘+1𝐤,𝑛∈𝑁,_2𝑘+1=𝑎^𝑘+1𝑧₋₁+1−𝑎^𝑘+1𝐤,𝑛∈𝑁.(3.12) By (1.21) and (1.22), we see that 𝑧_2𝑘,𝑧_2𝑘+1→𝐤 so that 𝑧_𝑛→𝐤. We record this conclusion in the (𝐴,𝐴) entry of Table 4.

Table 4 

Consider another case where 𝑧₋₂∈𝐴 and 𝑧₋₁∈𝐵^(𝑗). Then by the (𝐴,𝐵^(𝑗)) entry of Table 1, we see that 𝑧₀∈𝐴. Since 𝑧₋₁∈𝐵^(𝑗) and 𝑧₀∈𝐴, then by Table 1 again,𝑧₁∈𝑎𝐵^(𝑗)+𝑎^𝐤∈𝐵^(𝑖),(3.13) for some 𝑖∈𝑁. By induction, we may then see that 𝑧_2𝑘∈𝐴 and 𝑧_2𝑘+1∈𝐵 for 𝑘≥−1. Hence, by (1.9), we see that𝑧𝑧_2𝑛=𝑎𝑧_2𝑛−2+𝑎^𝐢,_2𝑛+1=𝑎𝑧_2𝑛−1+𝑎^𝐤,(3.14) for 𝑛∈𝑁. We may then easily see that 𝑧_2𝑛→𝐢 and 𝑧_2𝑛+1→𝐤.

By similar arguments, we may then derive the (𝐴,𝐶^(𝑠,𝑡)),(𝐴,𝐷^(𝑠)),(𝐵^(𝑖),𝐴),(𝐵^(𝑖),𝐵^(𝑠)),(𝐶^(𝑖,𝑗),𝐴),(𝐷^(𝑖),𝐴) and (𝐷^(𝑖),𝐷^(𝑠)) entries in Table 4.

To see why the other entries are correct, we consider a typical case where 𝑧₋₂∈𝐵^(𝑖) and 𝑧₋₁∈𝐷^(𝑠) for some 𝑖,𝑠∈𝑁. Suppose that 0≤𝑖≤𝑠. By Table 2, 𝑧_2𝑖∈𝐴 and 𝑧_2𝑖+1∈𝐷. Hence by the (𝐴,𝐷) entry in Table 4, we see that 𝑧_2𝑘→𝐣 and 𝑧_2𝑘+1→𝐤. While if 0≤𝑠<𝑖, then 𝑧_2𝑠∈𝐵 and 𝑧_2𝑠+1∈𝐴. Hence, by the (𝐵,𝐴) entry in Table 4, we see that 𝑧_2𝑘→𝐤 and 𝑧_2𝑘+1→𝐢.

4. The Case 𝜆∈(0,1−𝑎)

Suppose that 𝜆∈(0,1−𝑎). Then 𝑎𝜆+1−𝑎>𝜆, and {𝛼_𝑗}^∞𝑗=0 is a strictly decreasing sequence which diverges to −∞. Hence, there exists 𝑀∈𝐍 such that 𝛼₀,…,𝛼_𝑀>0 and 𝛼_𝑀+1≤0. Let 𝐷_𝑗=𝛼_𝑗 for 𝑗=0,…,𝑀, and let 𝐷_𝑀+1=0, then(0,𝜆]=^𝑀_𝑗=0𝐷_𝑗+1,𝐷_𝑗.(4.1)

We denote𝐴^(𝑝,𝜆)=𝐵𝐷_{𝑝+1(𝜆,𝑝)},𝐷_𝑝×(0,𝜆],𝐴^(𝜆,𝑝)=(0,𝜆]×𝐷_𝑝+1,𝐷_𝑝,𝐴^(𝑝,𝑞)=𝐷_𝑝+1,𝐷_𝑝×𝐷_𝑞+1,𝐷_𝑞,=𝐶(𝜆,∞_(𝜆,𝑝))×𝐷_𝑝+1,𝐷_𝑝,𝐵^(𝑝,𝜆)=𝐼^(𝑝)×(0,𝜆],𝐵^(𝑝,𝑞)=𝐼^(𝑝)×𝐷_𝑞+1,𝐷_𝑞,𝐷_(𝜆,𝑝)=(𝜆,∞)×𝐼^(𝑝),𝐶^(𝑝,𝜆)=𝐼^(𝑝)×(𝜆,∞),𝐶^(𝑝,𝑞)=𝐼^(𝑝)×𝐼^(𝑞),=(0,𝜆]×𝐼^(𝑝),𝐷^(𝑝,𝜆)=𝐷_𝑝+1,𝐷_𝑝×(𝜆,∞),𝐷^(𝑝,𝑞)=𝐷_𝑝+1,𝐷_𝑝×𝐼^(𝑞),(4.2) then 𝐴=^𝑀_𝑝=0𝐴^(𝑝,𝜆)=^𝑀_𝑝=0𝐴^(𝜆,𝑝)=^𝑀_𝑀𝑝=0_𝑞=0𝐴^(𝑝,𝑞),𝐵=^∞_𝑝=0𝐵^(𝑝,𝜆)=^𝑀_𝑝=0𝐵^(𝜆,𝑝)=^∞_𝑀𝑝=0_𝑞=0𝐶=_∞𝐵^(𝑝,𝑞),_𝑝=0𝐶^(𝑝,𝜆)=^∞_𝑝=0𝐶^(𝜆,𝑝)=^∞_∞𝑝=0_𝑞=0𝐶^(𝑝,𝑞),𝐷=^𝑀_𝑝=0𝐷^(𝑝,𝜆)=^∞_𝑝=0𝐷^(𝜆,𝑝)=^𝑀_∞𝑝=0_𝑞=0𝐷^(𝑝,𝑞).(4.3)

In Table 5, we record the fact that 𝑧_𝛼∈𝐵 and 𝑧_𝛼+1∈𝐵 which implies 𝑧_𝛼+2∈𝐵 as the (𝐵,𝐵) entry, and so forth.

Table 5 

Tables 6 and 7 are similar to Tables 2 and 3.

Table 6 

Table 7 

Initial condition: (𝑧₋₂,𝑧₋₁)∈𝐶^(𝑘,𝑝)×𝐶^(𝑙,𝑚).

For example, let us show the first data row in Table 6. Suppose that 𝑧₋₂∈𝐶^(𝜆,𝑝) and 𝑧₋₁∈𝐵^(𝑚,𝜆) where 0≤𝑝≤𝑚. Then by Table 5, if 0=𝑝≤𝑚,𝑧𝑧₀=𝑎𝑧₋₂+𝑎^𝐹_𝜆𝑧₋₁=𝑎𝑧₋₂+𝑎^𝐢∈𝑎𝐶^(𝜆,0)+𝑎^𝐢⊆𝐵,₁=𝑎𝑧₋₁+𝑎^𝐹_𝜆𝑧₀=𝑎𝑧₋₁+𝑎^𝐢∈𝑎𝐵^(𝑚,𝜆)+𝑎^𝐢⊆𝐵.(4.4) If 0<𝑝≤𝑚, then𝑧𝑧₀₁=𝑎𝑧₋₂+𝑎^𝐹_𝜆𝑧₋₁=𝑎𝑧₋₂+𝑎^𝐢∈𝑎𝐶^(𝜆,𝑝)+𝑎^𝐢⊆𝐶^{(𝜆,𝑝−1)},𝑧₂=𝑎𝑧₋₁+𝑎^𝐹_𝜆𝑧₀=𝑎𝑧₋₁∈𝑎𝐵^(𝑚,𝜆)⊆𝐵^{(𝑚−1,𝜆)},𝑧₃=𝑎𝑧₀+𝑎^𝐹_𝜆𝑧₁=𝑎𝑧₀+𝑎^𝐢∈𝑎𝐶^{(𝜆,𝑝−1)}+𝑎^𝐢⊆𝐶^{(𝜆,𝑝−2)},=𝑎𝑧₁+𝑎^𝐹_𝜆𝑧₂=𝑎𝑧₁∈𝑎𝐵^{(𝑚−1,𝜆)}⊆𝐵^{(𝑚−2,𝜆)},(4.5) and by induction,𝑧𝑧_2𝑝=𝑎𝑧_2𝑝−2+𝑎^𝐹_𝜆𝑧_2𝑝−1=𝑎𝑧_2𝑝−2+𝑎^𝐢∈𝑎𝐶^(𝜆,0)+𝑎^𝐢⊆𝐵,_2𝑝+1=𝑎𝑧_2𝑝−1+𝑎^𝐹_𝜆𝑧_2𝑝=𝑎𝑧_2𝑝−1+𝑎^𝐢∈𝑎𝐵^{(𝑚−𝑝,𝜆)}+𝑎^𝐢⊆𝐵,(4.6) that is, 𝑧_2𝑝∈𝐵 and 𝑧_2𝑝+1∈𝐵.

As another example, let us show the second data row in Table 7. Suppose that (𝑧₋₂,𝑧₋₁)∈𝐶^(𝑘,𝑝)×𝐶^(𝑙,𝑚), where 0≤𝑘<min{𝑙,𝑚,𝑝}. Then by Table 5, if 0=𝑘<min{𝑙,𝑚,𝑝},𝑧₁𝑧₀=𝑎𝑧₋₂+𝑎^𝐹_𝜆𝑧₋₁=𝑎𝑧₋₂∈𝑎𝐶^(0,𝑝)⊆𝐷^{(𝜆,𝑝−1)},=𝑎𝑧₋₁+𝑎^𝐹_𝜆𝑧₀=𝑎𝑧₋₁+𝑎^𝐣∈𝑎𝐶^(𝑙,𝑚)+𝑎^𝐣⊆𝐼^(𝑙−1)×(𝜆,∞).(4.7) If 0<𝑘<min{𝑙,𝑚,𝑝}, then𝑧𝑧₀₁=𝑎𝑧₋₂+𝑎^𝐹_𝜆𝑧₋₁=𝑎𝑧₋₂∈𝑎𝐶^(𝑘,𝑝)⊆𝐶^{(𝑘−1,𝑝−1)},𝑧₂=𝑎𝑧₋₁+𝑎^𝐹_𝜆𝑧₀=𝑎𝑧₋₁∈𝑎𝐶^(𝑙,𝑚)⊆𝐶^{(𝑙−1,𝑚−1)},𝑧₃=𝑎𝑧₀+𝑎^𝐹_𝜆𝑧₁=𝑎𝑧₀∈𝑎𝐶^{(𝑘−1,𝑝−1)}⊆𝐶^{(𝑘−2,𝑝−2)},=𝑎𝑧₁+𝑎^𝐹_𝜆𝑧₂=𝑎𝑧₁∈𝑎𝐶^{(𝑙−1,𝑚−1)}⊆𝐶^{(𝑙−2,𝑚−2)},(4.8) and by induction,𝑧𝑧_2𝑘2𝑘+1=𝑎𝑧_2𝑘−2+𝑎^𝐹_𝜆𝑧_2𝑘−1=𝑎𝑧_2𝑘−2∈𝑎𝐶^{(0,𝑝−𝑘)}⊆𝐷^{(𝜆,𝑝−𝑘−1)},=𝑎𝑧_2𝑘−1+𝑎^𝐹_𝜆𝑧_2𝑘=𝑎𝑧_2𝑘−1+𝑎^𝐣∈𝑎𝐶^{(𝑙−𝑘,𝑝−𝑘)}+𝑎^𝐣⊆𝐼^{(𝑙−𝑘−1)}×(𝜆,∞),(4.9) that is, 𝑧_2𝑘∈𝐷^{(𝜆,𝑝−𝑘−1)} and 𝑧_2𝑘+1∈𝐶^{(𝑙−𝑘−1,𝜆)}.

Theorem 4.1. Suppose that 𝜆∈(0,1−𝑎). Let {𝑧_𝑛}^∞𝑛=−2 be a solution of (1.9) originated from (0,∞)². Then 𝑧_𝑛⟶⟨𝐢⟩,⟨𝐣⟩,⟨𝟎,𝐤⟩or⟨𝐤,𝟎⟩.(4.10)

As in the proof of Theorem 3.1, we may construct Table 8.

Table 8 

For example, the (𝐵,𝐵) entry states that if (𝑧₋₂,𝑧₋₁)∈𝐵×𝐵, then the solution {𝑧_𝑛} of (1.9) originated from it will tend to ⟨𝐢⟩. Indeed, by Table 5, 𝑧₀∈𝐵, and then by induction, 𝑧_𝑘∈𝐵 for all 𝑘≥−2. Hence, by (1.9), we see that𝑧_𝑛=𝑎𝑧_𝑛−2+(1−𝑎)𝐢,𝑛∈𝑁,(4.11) from which we easily obtain𝑧𝑧_2𝑘=𝑎^𝑘+1𝑧₋₂+1−𝑎^𝑘+1𝐢,𝑛∈𝑁,_2𝑘+1=𝑎^𝑘+1𝑧₋₁+1−𝑎^𝑘+1𝐢,𝑛∈𝑁.(4.12) Hence 𝑧_2𝑘,𝑧_2𝑘+1→𝐢 so that 𝑧_𝑛→𝐢. By similar reasoning, we may show the validity of the (𝐴,𝐴),(𝐴,𝐵),(𝐴,𝐶),(𝐴,𝐷), (𝐵,𝐴),(𝐶,𝐴),(𝐷,𝐴), and (𝐷,𝐷) entries.

Next, suppose that (𝑧₋₂,𝑧₋₁)∈𝐵^(𝑝,𝜆)×𝐶^(𝜆,𝑚). Then the solution {𝑧_𝑛} of (1.9) originated from it will tend to ⟨𝟎,𝐤⟩ or ⟨𝐢⟩. Indeed, by Table 6, if 0≤𝑝≤𝑚, then (𝑧_2𝑝,𝑧_2𝑝+1)∈𝐴×𝐶 and by induction, (𝑧_2𝑛+2𝑝,𝑧_{2𝑛+2𝑝+1})∈𝐴×𝐶 for all 𝑛∈𝑁. Hence, by (1.9), we have𝑧𝑧_2𝑛+2𝑝=𝑎^𝑛𝑧_2𝑝,𝑛∈𝑁,_{2𝑛+2𝑝+1}=𝑎^𝑛𝑧_2𝑝+1+(1−𝑎^𝑛)𝐤,𝑛∈𝑁.(4.13) Hence, 𝑧_2𝑛→𝟎,𝑧_2𝑛+1→𝐤. If 0≤𝑚<𝑝, then (𝑧_2𝑚,𝑧_2𝑚+1)∈𝐵×𝐵. By previous argument, we see 𝑧_𝑛→𝐢. By similar reasoning, we may show the correctness of the other entries. The proof is complete.

5. The Case 𝜆∈[1−𝑎,1)

Suppose that 𝜆∈[1−𝑎,1), then 𝑎𝜆+1−𝑎>𝜆. Therefore, we may continue to use the notations described in the previous case 𝜆∈(0,1−𝑎) and proceed as in the previous two sections and derive Tables 9, 10, 11, 12, 13 and 14.

Table 9 

Table 10 

Table 11 

Assumption: (𝑧₋₂,𝑧₋₁)∈𝐴^(𝑘,𝑝)×𝐴^(𝑙,𝑚).

Table 12 

Assumption: (𝑧₋₂,𝑧₋₁)∈𝐶^(𝑘,𝑝)×𝐶^(𝑙,𝑚).

Table 13 

Assumption: (𝑧₋₂,𝑧₋₁)∈𝐷^(𝑘,𝑝)×𝐵^(𝑙,𝑚).

Table 14 

Assumption: (𝑧₋₂,𝑧₋₁)∈𝐵^(𝑝,𝑘)×𝐷^(𝑚,𝑙).

By means of these tables, we may then derive the following result.

Theorem 5.1. Suppose that 𝜆∈[1−𝑎,1). Let {𝑧_𝑛} be a solution of (1.9) originated from (0,∞)². Then 𝑧_𝑛⟶⟨𝐢⟩,⟨𝐣⟩,⟨𝐤,𝟎⟩or⟨𝟎,𝐤⟩.(5.1)

The proof amounts to showing the validity of Table 15.

Table 15 

For example, the (𝐵^(𝑝,𝑘),𝐷^(𝑚,𝑙)) entry states that if (𝑧₋₂,𝑧₋₁)∈𝐵^(𝑝,𝑘)×𝐷^(𝑚,𝑙) where 0≤𝑝<min{𝑘,𝑙,𝑚}, then the solution {𝑧_𝑛} of (1.9) originated from it will tend to ⟨𝐣⟩ or ⟨𝟎,𝐤⟩. Indeed, by Table 14, (𝑧_2𝑝,𝑧_2𝑝+1)∈𝐴^{(𝜆,𝑘−𝑝−1)}×𝐷^{(𝑚−𝑝−1,𝜆)}. Furthermore, by Table 10, if 0≤𝑘−𝑝−1≤𝑚−𝑝−1, then (𝑧_2𝑘,𝑧_2𝑘+1)∈𝐷×𝐷. Hence, by (1.9), we see that𝑧_𝑛=𝑎𝑧_𝑛−2+(1−𝑎)𝐣,𝑛≥2𝑘+2,(5.2) from which we easily obtain𝑧𝑧_2𝑛+2𝑘=𝑎^𝑛𝑧_2𝑘+(1−𝑎^𝑛)𝐣,𝑛∈𝐍,_{2𝑛+2𝑘+1}=𝑎^𝑛𝑧_2𝑘+1+(1−𝑎^𝑛)𝐣,𝑛∈𝐍.(5.3) Hence, 𝑧_𝑛→𝐣.

If 0≤𝑚−𝑝−1<𝑘−𝑝−1, then (𝑧_2𝑚,𝑧_2𝑚+1)∈𝐴×𝐶. By Table 9 again, (𝑧_2𝑛,𝑧_2𝑛+1)∈𝐴×𝐶 for all 𝑛≥𝑚. Hence, by (1.9), we see that𝑧𝑧_2𝑛=𝑎𝑧_2𝑛−2,𝑛>𝑚,_2𝑛+1=𝑎𝑧_2𝑛−1+(1−𝑎)𝐤,𝑛>𝑚,(5.4) from which we easily obtain𝑧𝑧_2𝑛+2𝑚=𝑎^𝑛𝑧_2𝑚,𝑛∈𝐍,_{2𝑛+2𝑚+1}=𝑎^𝑛𝑧_2𝑚+1+(1−𝑎^𝑛)𝐤,𝑛∈𝐍.(5.5) Hence, 𝑧_2𝑛→𝟎,𝑧_2𝑛→𝐤. By similar reasoning, we may derive the other entries of Table 15. The proof is complete.

6. Conclusions and Remarks

We have discussed a simple two-nation arms race model with a positive threshold 𝜆 hidden in a nonlinear piecewise constant control function. Treating 𝜆 as a bifurcation parameter which varies from 0⁺ to +∞, we have discussed the limiting behaviors of all possible solutions of (1.9) originated from the positive orthant (0,∞)².

Let 𝟎=(0,0),𝐢=(1,0),𝐣=(0,1), and 𝐤=(1,1).

(i)For 𝜆>1, all solutions of (1.9) originated from the positive orthant tend to ⟨𝐤⟩.(ii)For 𝜆=1, all such solutions must either be tending to ⟨𝐢⟩,⟨𝐣⟩, or ⟨𝐤⟩ or to ⟨𝟎,𝐤⟩,⟨𝐤,𝟎⟩,⟨𝐢,𝐤⟩,⟨𝐤,𝐢⟩,⟨𝐣,𝐤⟩, or ⟨𝐤,𝐣⟩.(iii) For 0<𝜆<1, all such solutions must either be tending to the steady states ⟨𝐢⟩ or ⟨𝐣⟩, or to ⟨𝐤,𝟎⟩ or ⟨𝟎,𝐤⟩.

Recall that a sequence {𝑧_𝑛}^∞𝑛=−2 is asymptotically 𝜔-periodic if it can be expressed as the sum of two sequences {𝑝_𝑛}^∞𝑛=−2 and {𝑞_𝑛}^∞𝑛=−2, where 𝑝_𝑛→0 and {𝑞_𝑛}^∞𝑛=−2 is periodic with prime period 𝜔. Therefore, as a direct consequence of our investigations, all solutions of (1.9) originated from the positive orthants must either tend to limit 1-cycles ⟨𝐢⟩,⟨𝐣⟩, or ⟨𝐤⟩ or to limit 2-cycles ⟨𝟎,𝐤⟩,⟨𝐢,𝐤⟩, or ⟨𝐤,𝐣⟩. Such a conclusion meets our expectation of a “terror equilibrium” in nuclear arms races.

We conclude our investigations with the following remarks. One may object that the above east-west view of conflicts has less bite nowadays and that this way of thinking of the problem is far too simple. Indeed, there is now a second-generation literature that incorporates real strategic thinking, with sound foundation in decision theory and game theory (see, for instance, [10–12]). Yet to the best of our knowledge, there is no complete mathematical analysis similar to those described above. We hope that our results will be useful in furthering the mathematical investigation of arms race models based on more recent and realistic social models.

Acknowledgment

The first author is supported by the National Science Foundation (no. 11161049).