Limit 2-Cycles for a Discrete-Time Bang-Bang Control Model

Abstract

A discrete-time periodic model with bang-bang feedback control is investigated. It is shown that each solution tends to one of four different types of limit 2-cycles. Furthermore, the accompanying initial regions for each type of solutions can be determined. When a threshold parameter is introduced in the bang-bang function, our results form a complete bifurcation analysis of our control model. Hence, our model can be used in the design of a control system where the state variable fluctuates between two state values with decaying perturbation.

1. Introduction

Discrete-time control systems of the form 𝐱_𝑛=𝐴_𝑛𝐱_𝑛−1+𝐺𝑛,𝐮_𝑛−1, (1.1) with 𝐱_𝑛∈𝐑^𝑛 and 𝐮_𝑛∈𝐑^𝑚, are of great importance in engineering (see, e.g., any text books on discrete-time signals and systems).

Indeed, such a system consists of a linear part which is easily produced by design and a nonlinear part which allows nonlinear feedback controls of the form 𝐮_𝑛=𝐐𝐱_𝑛, (1.2) commonly seen in engineering designs.

In a commonly seen situation, 𝐱_𝑛 and 𝐮_𝑛 belong to 𝐑¹, while 𝐮_𝑛 takes on two fixed values (on-off values) depending on whether the state variable is above or below a certain value (as commonly seen in thermostat control). In some cases, it is desirable to see that the state value 𝑥_𝑛 fluctuates between two fixed values with decaying perturbations as time goes by (an example will be provided at the end of this paper). Here, the important question is whether we can design such a control system that fulfils our objectives.

In this note, we will show that a very simple feedback system of the form 𝑥_𝑛=𝑎_𝑛𝑥_𝑛−2+𝑏_𝑛𝑓_𝜆𝑥_𝑛−1+𝑑_𝑛,𝑛∈𝐍={0,1,2,…}, (1.3) can achieve such a goal provided that:

(i) we take {𝑎_𝑛}^∞_𝑛=0,{𝑏_𝑛}^∞_𝑛=0,{𝑑_𝑛}^∞_𝑛=0 to be 2-periodic sequences with 𝑎₀,𝑎₁∈(0,1),𝑏₀,𝑏₁∈(0,+∞),𝑑₀,𝑑₁∈𝐑,

(ii) while the control function 𝑓_𝜆 is taken to be the step (activation) or bang bang function [1] defined by 𝑓_𝜆(𝑢)=1,if𝑢≤𝜆,−1,if𝑢>𝜆, (1.4) where 𝜆 may be regarded as a threshold parameter.

Remarks: (i) Note that in case 𝜆=0, our function 𝑓₀ is reduced to the well-known Heaviside function 𝐻(𝑢)=1,if𝑢≤0,−1,if𝑢>0. (1.5) These bang bang controllers are indeed used in daily control mechanisms; for example, a water heater that maintains desired temperature by turning the applied power on and off based on temperature feedback is an example application.

(ii) As for the sequences {𝑎_𝑛}^∞_𝑛=0,{𝑏_𝑛}^∞_𝑛=0 and {𝑑_𝑛}^∞_𝑛=0, we have assumed that they are periodic with a prime period 𝜔. We could have considered more general periodic sequences since a large number of environmental parameters are generated in periodic manners, and such structural nature should be reflected in the choice of our sequences. However, in the early stage of our study, it is quite reasonable to assume that they have a common prime period 2 (instead of various prime periods).

(iii) Finally, we have selected 𝑎₀,𝑎₁∈(0,1). A simple reason is that without the feedback control and forcing sequence {𝑑_𝑛}, our system is a stable one and which can easily be realized in practice.

(iv) Equation (1.3) has a second-order delay in the open loop part and a first-order delay in the control function. It may equally be well to choose a system that has a first-order delay in the open loop part and a second-order delay in the control function. Such a model will be handled in another paper.

(v) The simple prototype studied here is representative of a much wider class of discrete-time periodic systems with piecewise constant feedback controls [2–10], and hence we hope that our results will lead to much more general ones for complex systems involving such discontinuous controls.

Clearly, given any initial state value pair (𝑥₋₂,𝑥₋₁) in 𝐑², we can generate through (1.3) a unique real sequence {𝑥_𝑛}^∞_𝑛=−2. Such a (state) sequence is called a solution of (1.3) originated from (𝑥₋₂,𝑥₋₁). What is interesting is that by elementary analysis, we can show that for any value of the threshold parameter 𝜆, there are at most four possible types of limiting behaviors for solutions of (1.3), and we can determine exactly the range of the parameter values and the exact “initial region” from which each type of solutions originates from (see the concluding section for more details).

To this end, we first note that by the transformation 𝑢_𝑛=𝑥_𝑛−𝜆, (1.3) is equivalent to 𝑢_𝑛=𝑎_𝑛𝑢_𝑛−2+𝑏_𝑛𝐻𝑢_𝑛−1+𝑐_𝑛,𝑛∈𝐍, (1.6) where 𝑐_𝑛=𝑑_𝑛+(𝑎_𝑛−1)𝜆. Next, by means of the identification 𝑢_2𝑛=𝑦_𝑛 and 𝑢_2𝑛+1=𝑧_𝑛 for 𝑛∈{−1,0,…}, we note further that (1.6) is equivalent to the following two-dimensional autonomous dynamical system: 𝑦_𝑛=𝑎₀𝑦_𝑛−1+𝑏₀𝐻𝑧_𝑛−1+𝑐₀,𝑧_𝑛=𝑎₁𝑧_𝑛−1+𝑏₁𝐻𝑦_𝑛+𝑐₁,𝑛∈𝐍, (1.7) which is a special case of the system (1.1). By such a transformation, we are then considering the subsequences {𝑢_2𝑛} and {𝑢_2𝑛+1} consisting of even and odd terms of the solution sequence {𝑢_𝑛} of (1.6). Therefore, all the asymptotic properties of (1.6) can be obtained from those of (1.7).

To study the asymptotic properties of (1.7), we first note that its solution is of the form {(𝑦_𝑛,𝑧_𝑛)}^∞_𝑛=−1 where (𝑦₋₁,𝑧₋₁) is now a point in the real plane. By considering all possible initial data pairs (𝑦₋₁,𝑧₋₁)∈𝐑², we will be able to show that every solution of (1.7) tends to one of four vectors. To describe these four vectors, we set 𝜉^±_𝑖=𝑐_𝑖±𝑏_𝑖1−𝑎_𝑖,𝑖=0,1. (1.8) Then, the four vectors are 𝜉⁻₀,𝜉⁻₁,𝜉⁺₀,𝜉⁻₁,𝜉⁺₀,𝜉⁺₁,𝜉⁻₀,𝜉⁺₁, (1.9) and since 𝑏₀,𝑏₁>0, we see that 𝜉⁻₀<𝜉⁺₀ and 𝜉⁻₁<𝜉⁺₁,and hence they form the corners of a rectangle.

Depending on the relative location of the origin (0,0) with respect to this rectangle, we may then distinguish eleven exhaustive (but not mutually distinct; see Section 3 in the following) cases:(i)0>max{𝜉⁺₀,𝜉⁺₁}, (ii)0<min{𝜉⁻₀,𝜉⁻₁}, (iii)min{𝜉⁺₀,𝜉⁺₁}<0<max{𝜉⁺₀,𝜉⁺₁}, (iv)min{𝜉⁻₀,𝜉⁻₁}<0<max{𝜉⁻₀,𝜉⁻₁}; (v)max{𝜉⁺₀,𝜉⁺₁}=0, (vi)min{𝜉⁻₀,𝜉⁻₁}=0; (vii)0=min{𝜉⁺₀,𝜉⁺₁}=𝜉⁺₀<𝜉⁺₁, (viii)0=min{𝜉⁺₀,𝜉⁺₁}=𝜉⁺₁<𝜉⁺₀; (ix)0=max{𝜉⁻₀,𝜉⁻₁}=𝜉⁻₁>𝜉⁻₀, (x)0=max{𝜉⁻₀,𝜉⁻₁}=𝜉⁻₀>𝜉⁻₁; (xi)max{𝜉⁻₀,𝜉⁻₁}<0<min{𝜉⁺₀,𝜉⁺₁}.

For each case, we intend to show that solutions of (1.7) originated from different parts of the plane will tend to one of the four vectors in (1.9). To facilitate description of the various parts of the plane, we introduce the following notations: 𝐴^±_𝑖,𝑗=−1−𝑎^𝑗_𝑖𝑎^𝑗_𝑖𝜉^±_𝑖𝑅,𝑗∈𝐍,𝑖=0,1,₊=(0,+∞),𝑅⁻=(−∞,0]. (1.10)

2. Main Results

Cases (i), (ii), (iii), and (iv)
First of all, the first four cases 0>max{𝜉⁺₀,𝜉⁺₁},0<min{𝜉⁻₀,𝜉⁻₁},min{𝜉⁺₀,𝜉⁺₁}<0<max{𝜉⁺₀,𝜉⁺₁}, and min{𝜉⁻₀,𝜉⁻₁}<0<max{𝜉⁻₀,𝜉⁻₁} are relatively easy. Indeed, suppose 0>max{𝜉⁺₀,𝜉⁺₁}. Let (𝑦,𝑧)={(𝑦_𝑛,𝑧_𝑛)}^∞_𝑛=−1 be a solution of (1.7). By (1.7), 𝑦_𝑛≤𝑎₀𝑦_𝑛−1+𝑏₀+𝑐₀,𝑧_𝑛≤𝑎₁𝑧_𝑛−1+𝑏₁+𝑐₁. Then, limsup_𝑛𝑦_𝑛≤𝑏₀+𝑐₀1−𝑎₀=𝜉⁺₀<0,limsup_𝑛𝑧_𝑛≤𝑏₁+𝑐₁1−𝑎₁=𝜉⁺₁<0. (2.1)(2.2) Therefore, there exists an 𝑚₀∈𝐍 such that 𝑦_𝑛,𝑧_𝑛∈𝑅⁻ for all 𝑛≥𝑚₀. By (1.7) again, 𝑦_𝑛=𝑎₀𝑦_𝑛−1+𝑏₀+𝑐₀ and 𝑧_𝑛=𝑎₁𝑧_𝑛−1+𝑏₁+𝑐₁ for 𝑛>𝑚₀. Then, 𝑎₀,𝑎₁∈(0,1) imply lim_𝑛𝑦_𝑛,𝑧_𝑛=𝑐₀+𝑏₀1−𝑎₀,𝑐₁+𝑏₁1−𝑎₁=𝜉⁺₀,𝜉⁺₁. (2.3)In summary, suppose max{𝜉⁺₀,𝜉⁺₁}<0 and suppose (𝑦₋₁,𝑧₋₁)∈𝐑², then the solution {(𝑦_𝑛,𝑧_𝑛)} originated from (𝑦₋₁,𝑧₋₁) will tend to (𝜉⁺₀,𝜉⁺₁). We record this result as the first data row in Table 1.

Table 1 

By symmetric arguments, the second data row is also correct. To see the validity of the third data row, we first note that min{𝜉⁺₀,𝜉⁺₁}<0<max{𝜉⁺₀,𝜉⁺₁} if and only if 𝜉⁺₀<0<𝜉⁺₁ or 𝜉⁺₁<0<𝜉⁺₀. If 𝜉⁺₀<0<𝜉⁺₁ holds, then by (1.7), 𝑦_𝑛≤𝑎₀𝑦_𝑛−1+𝑏₀+𝑐₀ for all 𝑛∈𝐍. Hence, limsup_𝑛𝑦_𝑛≤(𝑏₀+𝑐₀)/(1−𝑎₀)=𝜉⁺₀<0. Therefore, there exists an 𝑚₀∈𝐍 such that 𝑦_𝑛<0 for 𝑛≥𝑚₀. Thus, 𝑧_𝑛=𝑎₁𝑧_𝑛−1+𝑏₁+𝑐₁ for 𝑛>𝑚₀. Then lim_𝑛𝑧_𝑛=𝜉⁺₁>0. Therefore, there exists an 𝑚₁≥𝑚₀ such that 𝑧_𝑛>0 for all 𝑛>𝑚₁. Then, by (1.7) again, 𝑦_𝑛=𝑎₀𝑦_𝑛−1−𝑏₀+𝑐₀ for all 𝑛>𝑚₁+1, and hence lim_𝑛𝑦_𝑛=𝜉⁻₀. The case where 𝜉⁺₁<0<𝜉⁺₀ is similarly proved. Finally, the fourth data row is established by arguments symmetric to those for the third row.

Case (v)
Next, we assume that 0=max{𝜉⁺₀,𝜉⁺₁}. Since 𝑎_𝑖∈(0,1),𝑏_𝑖∈(0,+∞),and𝑐_𝑖∈𝐑 for 𝑖=0,1, then 𝜉⁺_𝑖=(𝑐_𝑖+𝑏_𝑖)/(1−𝑎_𝑖)>(𝑐_𝑖−𝑏_𝑖)/(1−𝑎_𝑖)=𝜉⁻_𝑖 for 𝑖=0,1. We see that max{𝜉⁻₀,𝜉⁻₁}<0, and 𝐴⁻_𝑖,0=0,lim_𝑗𝐴⁻_𝑖,𝑗=lim_𝑗(−(1−𝑎^𝑗_𝑖)/(𝑎^𝑗_𝑖))𝜉⁻_𝑖=+∞ for 𝑖=0,1. Therefore, 𝑅⁺=⋃^+∞_𝑗=0(𝐴⁻_𝑖,𝑗,𝐴⁻_𝑖,𝑗+1] for 𝑖=0,1. Furthermore, if 𝜉⁺₀<𝜉⁺₁=0, then lim_𝑗𝐴⁺_0,𝑗=+∞,𝑅⁺=⋃^+∞_𝑗=0(𝐴⁺_0,𝑗,𝐴⁺_0,𝑗+1], and if 𝜉⁺₁<𝜉⁺₀=0, then lim_𝑗𝐴⁺_1,𝑗=+∞,𝑅⁺=⋃^+∞_𝑗=0(𝐴⁺_1,𝑗,𝐴⁺_1,𝑗+1]. We need to consider three cases: (i) 𝜉⁺₀<𝜉⁺₁=0, (ii) 𝜉⁺₁<𝜉⁺₀=0, and (iii) 𝜉⁺₀=𝜉⁺₁=0. By arguments similar to those used in the derivation of Table 1, we may derive Table 2.

Table 2 

max{𝜉⁺₀,𝜉⁺₁}=0.

For instance, suppose 𝜉⁺₀<𝜉⁺₁=0. Let (𝑦₋₁,𝑧₋₁)∈𝑅⁻×𝑅⁻. Then, by (1.7), we have 𝑦₀=𝑎₀𝑦₋₁+𝑏₀+𝑐₀<𝑎₀𝑦₋₁<0,𝑧₀=𝑎₁𝑧₋₁+𝑏₁+𝑐₁=𝑎₁𝑧₋₁<0, and by induction, we may easily see that 𝑦_𝑛,𝑧_𝑛∈𝑅⁻ for all 𝑛∈𝐍. Thus, 𝑦_𝑛=𝑎₀𝑦_𝑛−1+𝑏₀+𝑐₀,𝑧_𝑛=𝑎₁𝑧_𝑛−1+𝑏₁+𝑐₁, and hence lim_𝑛(𝑦_𝑛,𝑧_𝑛)=(𝜉⁺₀,𝜉⁺₁). As another example, let (𝑦₋₁,𝑧₋₁)∈𝑅⁺×𝑅⁻, then 𝑦₋₁∈(𝐴⁺_0,𝑘,𝐴⁺_0,𝑘+1] for some 𝑘∈𝐍. By (1.7) and induction, we may easily see that (𝑦_𝑘,𝑧_𝑘)∈𝑅⁻×𝑅⁻. Our conclusion comes from the previous case. As a further example, let (𝑦₋₁,𝑧₋₁)∈(𝐴⁻_0,𝑘,𝐴⁻_0,𝑘+1]×(𝐴⁻_1,𝑠,𝐴⁻_1,𝑠+1]⊂𝑅⁺×𝑅⁺, where 0≤𝑘≤𝑠, then by (1.7) and induction, we may easily see that (𝑦_𝑘,𝑧_𝑘)∈𝑅⁻×𝑅⁺. Our conclusion now follows from the fourth data row.

Case (vi)
This case is a dual of the Case (v). Indeed, assume that 0=min{𝜉⁻₀,𝜉⁻₁}. Then, min{𝜉⁺₀,𝜉⁺₁}>0 and 𝐴⁺_𝑖,0=0,lim_𝑗𝐴⁺_𝑖,𝑗=lim_𝑗(−(1−𝑎^𝑗_𝑖)/(𝑎^𝑗_𝑖))𝜉⁺₀=−∞ for 𝑖=0,1. Thus, 𝑅⁻=⋃^+∞_𝑗=0(𝐴⁺_𝑖,𝑗+1,𝐴⁺_𝑖,𝑗] for 𝑖=0,1. Furthermore, if 0=𝜉⁻₁<𝜉⁻₀, then lim_𝑗𝐴⁻_0,𝑗=−∞,𝑅⁻=⋃^+∞_𝑗=0(𝐴⁻_0,𝑗+1,𝐴⁻_0,𝑗], and if 0=𝜉⁻₀<𝜉⁻₁, then lim_𝑗𝐴⁻_1,𝑗=−∞,𝑅⁻=⋃^+∞_𝑗=0(𝐴⁻_1,𝑗+1,𝐴⁻_1,𝑗]. We need to consider three cases: (i) 0=𝜉⁻₁<𝜉⁻₀, (ii) 0=𝜉⁻₀<𝜉⁻₁, and (iii) 𝜉⁻₀=𝜉⁻₁=0. By arguments similar to those in the previous case, we may obtain the asymptotic behaviors of (1.7) summarized in Table 3.

Table 3 

min{𝜉⁻₀,𝜉⁻₁}=0.

Cases (vii) and (viii)
By arguments similar to those described previously, the corresponding asymptotic behaviors of (1.7) can be summarized in Tables 4 and 5.

Table 4 

0=min{𝜉⁺₀,𝜉⁺₁}=𝜉⁺₀<𝜉⁺₁.

Table 5 

0=min{𝜉⁺₀,𝜉⁺₁}=𝜉⁺₁<𝜉⁺₀.

Case (ix) and (x)
By arguments similar to those described previously, the corresponding asymptotic behaviors of (1.7) can be summarized in Tables 6 and 7.

Table 6 

0=max{𝜉⁻₀,𝜉⁻₁}=𝜉⁻₁>𝜉⁻₀.

Table 7 

0=max{𝜉⁻₀,𝜉⁻₁}=𝜉⁻₀>𝜉⁻₁.

Case (xi)
By arguments similar to those described previously, the corresponding asymptotic behaviors of (1.7) can be summarized in Table 8.

Table 8 

max{𝜉⁻₀,𝜉⁻₁}<0<min{𝜉⁺₁,𝜉⁺₀}.

3. Remarks

We remark that the different Cases (i)–(xi) discussed above may not be mutually distinct. For instance, the conditions min{𝜉⁺₀,𝜉⁺₁}<0<max{𝜉⁺₀,𝜉⁺₁} and min{𝜉⁻₀,𝜉⁻₁}<0<max{𝜉⁻₀,𝜉⁻₁} are not mutually exclusive. However, the corresponding conclusions in Table 1 show that they are compatible (and hence should not cause any problem).

Next, we turn our attention to our original equation (1.3). By 𝑐_𝑖=𝑑_𝑖+(𝑎_𝑖−1)𝜆, we may see that 𝜉^±_𝑖=𝑐_𝑖±𝑏_𝑖1−𝑎_𝑖𝑑=−𝜆+^𝑖±𝑏_𝑖1−𝑎_𝑖,𝑖=0,1. (3.1) Therefore, the results in the previous section for the system (1.7) can easily be translated into results for (1.3). For instance, by Table 1, we may see that when 0>max{𝜉⁺₀,𝜉⁺₁},thatis,𝜆>max{(𝑏₀+𝑑₀)/(1−𝑎₀),(𝑏₁+𝑑₁)/(1−𝑎₁)}, a solution {𝑥_𝑛}^∞_𝑛=−2 with (𝑥₋₂,𝑥₋₁)∈𝐑² will satisfy lim_𝑛𝑥_2𝑛=𝑏₀+𝑑₀1−𝑎₀,lim_𝑛𝑥_2𝑛+1=𝑏₁+𝑑₁1−𝑎₁. (3.2) As another example, the condition 𝜉⁺₀<𝜉⁺₁=0 is equivalent to (𝑏₀+𝑑₀)/(1−𝑎₀)<(𝑏₁+𝑑₁)/(1−𝑎₁)=𝜆. Let {𝑥_𝑛}^∞_𝑛=−2 be a solution of (1.3) with (𝑥₋₂,𝑥₋₁)∈𝑅⁻×𝑅⁺. Then, by Table 2, we may see that lim_𝑛𝑥_2𝑛=−𝑏₀+𝑑₀1−𝑎₀,lim_𝑛𝑥_2𝑛+1=𝑏₁+𝑑₁1−𝑎₁. (3.3)

By arguments similar to those just described, the corresponding asymptotic behaviors of solutions {𝑥_𝑛} of (1.3) can be summarized as follow:(i)if 𝜆<min{(𝑑₀−𝑏₀)/(1−𝑎₀),(𝑑₁−𝑏₁)/(1−𝑎₁)}, then 𝑥_2𝑛,𝑥_2𝑛+1→𝑑₀−𝑏₀1−𝑎₀,𝑑₁−𝑏₁1−𝑎₁, (3.4)(ii)if 𝜆=min{(𝑑₀−𝑏₀)/(1−𝑎₀),(𝑑₁−𝑏₁)/(1−𝑎₁)}, then 𝑥_2𝑛,𝑥_2𝑛+1→𝑑₀−𝑏₀1−𝑎₀,𝑑₁−𝑏₁1−𝑎₁,𝑑₀+𝑏₀1−𝑎₀,𝑑₁−𝑏₁1−𝑎₁𝑑or⁰−𝑏₀1−𝑎₀,𝑑₁+𝑏₁1−𝑎₁, (3.5)(iii)if min{(𝑑₀−𝑏₀)/(1−𝑎₀),(𝑑₁−𝑏₁)/(1−𝑎₁)}<𝜆<max{(𝑑₀+𝑏₀)/(1−𝑎₀),(𝑑₁+𝑏₁)/(1−𝑎₁)}, then 𝑥_2𝑛,𝑥_2𝑛+1→𝑑₀−𝑏₀1−𝑎₀,𝑑₁+𝑏₁1−𝑎₁𝑑or⁰+𝑏₀1−𝑎₀,𝑑₁−𝑏₁1−𝑎₁, (3.6)(iv)if 𝜆=max{(𝑑₀+𝑏₀)/(1−𝑎₀),(𝑑₁+𝑏₁)/(1−𝑎₁)}, then 𝑥_2𝑛,𝑥_2𝑛+1→𝑑₀+𝑏₀1−𝑎₀,𝑑₁+𝑏₁1−𝑎₁,𝑑₀+𝑏₀1−𝑎₀,𝑑₁−𝑏₁1−𝑎₁𝑑or⁰−𝑏₀1−𝑎₀,𝑑₁+𝑏₁1−𝑎₁, (3.7)(v)if 𝜆>max{(𝑑₀+𝑏₀)/(1−𝑎₀),(𝑑₁+𝑏₁)/(1−𝑎₁)}, then 𝑥_2𝑛,𝑥_2𝑛+1→𝑑₀+𝑏₀1−𝑎₀,𝑑₁+𝑏₁1−𝑎₁. (3.8)

We remark that the precise initial regions of each type of solutions in the above statements can be inferred from our previous tables. Such repetitions, however, need not to be spelled out in detail for obvious reasons. Instead, based on the statements made above, it is more important to point out that our original motivation can be fulfilled.(i)Equation (1.3) possesses exactly four 2-periodic solutions {𝜉^±₀,𝜉^±₁} with 𝜉^±_𝑖=−𝜆+(𝑑_𝑖±𝑏_𝑖)/(1−𝑎_𝑖)). Every other solution tends to one of these four solutions "according to the information given in the previous section."

As an example, consider a plant which is supposed to produce a type of products with capacity 𝑥_𝑛, where 𝑛 now denotes economic stages. Suppose that the stages reflect booms and busts experienced by an economy characterized by alternating periods of economic growth and contraction. Then, during busts, the plant should be managed in a fashion so as to produce at low capacity and during booms at high capacity. Suppose that it is estimated that 𝜉⁻₀=1 unit capacity is demanded during busts and 𝜉⁻₁=10 unit capacity during booms. Then, an automated plant of the form (1.7) may be built to fit the estimated demands: 𝑦_𝑛=12𝑦_𝑛−1+32𝐻𝑧_𝑛−1+2,𝑧_𝑛=13𝑧_𝑛−1+23𝐻𝑦_𝑛+223,𝑛∈𝐍, (3.9) where the “structural” parameters 𝑎₀=1/2,𝑎₁=1/3,𝑏₀=3/2,𝑏₁=2/3,𝑐₀=2,and𝑐₁=22/3 are chosen since they, as may be checked easily, guarantee that the capacities 𝑦_𝑛 and 𝑧_𝑛 will tend to 1 and 10, respectively. In fact, for all (𝑦₋₁,𝑧₋₁)∈𝐑², by (1.7), we have 𝑦_𝑛=(1/2)𝑦_𝑛−1+(3/2)𝐻(𝑧_𝑛−1)+2≥(1/2)𝑦_𝑛−1−(3/2)+2=(1/2)𝑦_𝑛−1+1/2, and 𝑧_𝑛=(1/3)𝑧_𝑛−1+(2/3)𝐻(𝑦_𝑛)+(22/3)≥(1/3)𝑧_𝑛−1−(2/3)+(22/3)=(1/3)𝑧_𝑛−1+(20/3)  for𝑛∈𝐍. Thus, liminf_𝑛𝑦_𝑛≥1andliminf_𝑛𝑧_𝑛≥10. Therefore, there is 𝑛^′∈𝐍 such that 𝑦_𝑛,𝑧_𝑛∈𝑅⁺ for𝑛≥𝑛^′. Then, 𝑦_𝑛=12𝑦_𝑛−1+12,𝑧_𝑛=13𝑧_𝑛−1+203,𝑛>𝑛^′. (3.10) We get lim_𝑛𝑦_𝑛=1=(𝑐₀−𝑏₀)/(1−𝑎₀)=𝜉⁻₀ and lim_𝑛𝑧_𝑛=10=(𝑐₁−𝑏₁)/(1−𝑎₁)=𝜉⁻₁.

Acknowledgment

Project supported by the National Natural Science Foundation of China (11161049). Mathematics Subject Classifications: 39A11, 39A23, 92B20.