Nonstationary Fronts in the Singularly Perturbed Power-Society Model

Abstract

The theory of contrasting structures in singularly perturbed boundary problems for nonlinear parabolic partial differential equations is applied to the research of formation of steady state distributions of power within the nonlinear “power-society” model. The interpretations of the solutions to the equation are presented in terms of applied model. The possibility theorem for the problem of getting the solution having some preassigned properties by means of parametric control is proved.

1. Introduction

Since the work [1], the theory of contrasting structures has become one of the most booming areas of research of the singularly perturbed differential equations [2–4].

The contrasting structures having the form of nonstationary fronts for parabolic partial differential equations were studied in [5]. The theory was applied to propagation of magnetic fronts in spiral galaxies [6–8]. Here we consider the nonstationary fronts in the Mikhailov “power-society” model [9–12] and the possibility to control them.

In the most general case the “power-society” model has the form of a Neumann boundary value problem for nonlinear parabolic integrodifferential equation. In the absence of some political mechanisms the model is reduced to singularly perturbed parabolic Neumann boundary value problem: 𝜕𝑝𝜕𝑡=𝜀²𝜕²𝑝𝜕𝑥²+𝐹𝑝,𝑥,𝜕𝑝𝜕𝑥_𝑥=0=𝜕𝑝𝜕𝑥_𝑥=1𝑝()=0,𝑥,0=𝑝⁰(𝑥). (1) Here 𝜀≪1 is a small positive parameter. Within the “power-society” model this parameter is small if the hierarchy is long or if society is strong.

The steady-state problem has the form 𝜀²𝜕²𝑝𝜕𝑥²+𝐹𝑝,𝑥=0,𝜕𝑝𝜕𝑥_𝑥=0=𝜕𝑝𝜕𝑥_𝑥=1=0. (2)

Let the following conditions hold [2–4].(1) The function 𝐹(𝑝,𝑥) has continuous partial derivatives for 0≤𝑥≤1 and 𝑝∈(−∞,+∞).(2) The degenerate equation 𝐹(𝑝,𝑥)=0 has three roots 𝑝=𝜑₁(𝑥), 𝑝=𝜑₂(𝑥), and 𝑝=𝜑₃(𝑥) such that 𝜑₁(𝑥)<𝜑₂(𝑥)<𝜑₃(𝑥), 0≤𝑥≤1.(3) The following inequalities 𝜕𝐹𝜕𝑝_{𝑝=𝜑1(𝑥)}<0,𝜕𝐹𝜕𝑝_{𝑝=𝜑2(𝑥)}>0,𝜕𝐹𝜕𝑝_{𝑝=𝜑3(𝑥)}<0 (3) take place.(4) The equation Φ(𝑥)=^𝜑3_𝜑^(𝑥)_1(𝑥)𝐹𝑝,𝑥𝑑𝑝=0 (4) has isolated root 𝑥=𝑥₀ on the interval 0<𝑥<1.

Under these conditions,(i) if Φ^′(𝑥₀)<0, then the solution 𝑝(𝑥,𝜀) of the problem (2) exists such that lim_𝜀→0𝑝()=𝜑𝑥,𝜀³(𝑥),0<𝑥<𝑥₀,𝜑₁(𝑥),𝑥₀<𝑥<1, (5) and it is an asymptotically stable stationary solution of problem (1);(ii) if Φ^′(𝑥₀)>0, solution 𝑝(𝑥,𝜀) of problem (2) exists such that lim_𝜀→0𝑝()=𝜑𝑥,𝜀¹(𝑥),0<𝑥<𝑥₀,𝜑₃(𝑥),𝑥₀<𝑥<1, (6) and it is an asymptotically stable stationary solution of problem (1).

The solutions that satisfy (5) or (6) are called the step-like contrasting structures or stationary fronts. There are also other stable stationary solutions of the problem (1). In particular, under Conditions 1–3 the existence of two more solutions, one of which is close to 𝜑₁(𝑥): lim_𝜀→0𝑝(𝑥,𝜀)=𝜑₁(𝑥),0<𝑥<𝑥₀ (7) and the other one is close to 𝜑₃(𝑥): lim_𝜀→0𝑝(𝑥,𝜀)=𝜑₃(𝑥),0<𝑥<𝑥₀, (8) is guaranteed.

There is an important problem of correspondence between a set of initial functions and a set of steady stationary solutions: given initial function 𝑝⁰(𝑥), what steady-state solution will we have at 𝑡→+∞? And there is the inverse problem: if one of the steady states is more desirable than others, which conditions on 𝑝⁰(𝑥) guarantee approach to this desirable steady state?

At last, when studying mathematical models of particular processes there is the following question which arises: if the existing 𝑝⁰(𝑥) does not correspond to the desirably steady state, is it possible to change the right-hand part of (1) so that the solution would evolve to the desirable steady state?

This work is aimed at considering these problems for the “power-society” model which describes the dynamics of the power distribution in a hierarchy.

We base our study on the theory of contrasting structures [2–5], especially on the Butuzov-Nedelko theorem [13]. Some other issues related to nonstationary fronts were studied in [14–16].

2. Nonstationary Fronts and Interpretation in the Nonlinear Singularly Perturbed “Power-Society” Model

This section deals with mathematical modeling of the processes of power dynamics in the hierarchical structures. The model was firstly introduced by Mikhailov, 1994, and the books by Samarskii and Mikhailov 1997 and Mikhailov 2005 should also be mentioned.

Here the hierarchy is a ranked set of instances. Each instance has a particular set of powers. The amount of powers changes with time, and we call such variability the power dynamics. We suppose that there exists a numerical variable which specifies the amount of powers of a particular instance. The power dynamics appear through (a) the self-streamlining of the hierarchy and (b) the influence of the society.

Let us denote the rank of the instance in the hierarchy by 𝑥 so that 𝑥=0 at the top of the hierarchy and 𝑥=1 at the bottom. Denote by 𝑝(𝑥,𝑡) the amount of powers of instance at time 𝑡.

The equation of the “power-society” model [1, 2] has the form (1), and 𝐹(𝑝,𝑥) is called the reaction of a civil society. The paper [1] has shown that if 𝐹(𝑝,𝑥)=−𝑘₁(𝑝−𝑝₀(𝑥)) (where 𝑘₁=const>0 and the function 𝑝₀(𝑥) is the attractive power profile), then the solution 𝑝=𝑝₀(𝑥) of the stationary degenerated equation 𝐹(𝑝,𝑥)=0 is stable. This means that the solution 𝑝(𝑥,𝑡,𝜀) of (1) tends to 𝑝₀(𝑥) when 𝑡→+∞, 0<𝑥<1. So for sufficiently large values of 𝑡 the power profile is close to 𝑝₀(𝑥).

It was very important in the paper [1] that only one attractive profile is supposed to exist. Here we consider the case of two stable power profiles 𝜑₁(𝑥) and 𝜑₃(𝑥), and each of them is attractive. We call 𝜑₁(𝑥) the participatory profile and 𝜑₃(𝑥) the iron-hand profile. Both of them are stable due to inequalities (3).

Henceforth we consider the function 𝐹(𝑝,𝑥) having the cubic nonlinearity. So we consider the equation 𝜕𝑝𝜕𝑡=𝜀²𝜕²𝑝𝜕𝑥²−𝑘₁(𝑥)𝑝−𝜑₁(𝑥)𝑝−𝜑₂(𝑥)𝑝−𝜑₃(𝑥), (9) with boundary value conditions 𝜕𝑝𝜕𝑥_𝑥=0=𝜕𝑝𝜕𝑥_𝑥=1=0. (10) The following conditions are supposed to hold.

Condition 1. The functions 𝑘₁(𝑥), 𝜑_𝑖(𝑥), 𝑖=1,2,3, have the continuous derivatives for 0≤𝑥≤1.

Condition 2. Inequalities 𝑘₁(𝑥)>0, 𝜑₁(𝑥)<𝜑₂(𝑥)<𝜑₃(𝑥) hold true for 0≤𝑥≤1.

We also notice that though 𝜑₁(𝑥)>0 because of the politological meaning of the function 𝜑_𝑖(𝑥), this condition must not be required from the mathematical point of view.

Making the substitution (see [17]) 𝜑𝑝=𝑞³(𝑥)−𝜑₁(𝑥)2+𝜑₃(𝑥)+𝜑₁(𝑥)2, (11) we obtain the equation for the function 𝑢(𝑥,𝑡,𝜀): 𝜕𝑞𝜕𝑡=𝜀²𝜕²𝑞𝜕𝑥²−𝛾(𝑥)𝑞²−1𝑞−𝜑(𝑥)+𝜀²𝜕𝑞Φ𝜕𝑥¹(𝑥)+𝜀²Φ₂𝑞,𝑥. (12) Here 𝛾(𝑥)=𝑘₁(𝑥)𝜑₃(𝑥)−𝜑₁(𝑥)2²𝜑(𝑥)=>0,2𝜑₂(𝑥)−𝜑₃(𝑥)−𝜑₁(𝑥)𝜑₃(𝑥)−𝜑₁(𝑥) (13) (notice that −1<𝜑(𝑥)<1), Φ₁(𝑥)=2𝜑^′₃(𝑥)−𝜑^′₁(𝑥)𝜑₃(𝑥)−𝜑₁(𝑥),Φ₂𝜑𝑞,𝑥=𝑞^3′′(𝑥)−𝜑₁^′′(𝑥)𝜑₃(𝑥)−𝜑₁(𝑥)+𝜑₃^′′(𝑥)+𝜑₁^′′(𝑥)𝜑₃(𝑥)−𝜑₁(𝑥). (14) Function 𝑢(𝑥,𝑡,𝜀) satisfies boundary conditions 𝜑₃(0)−𝜑₁(0)𝜕𝑞𝜕𝑥_𝑥=0+𝜑^′₃(0)−𝜑^′₁(0)𝑞_𝑥=0+𝜑^′₃(0)+𝜑^′₁(0)𝜑=0,₃(1)−𝜑₁(1)𝜕𝑞𝜕𝑥_𝑥=1+𝜑^′₃(1)−𝜑^′₁(1)𝑞_𝑥=1+𝜑^′₃(1)+𝜑^′₁(1)=0. (15) Consider stationary (𝜕/𝜕𝑡=0) equation related to (12): 𝜀²𝑞^′′=𝛾(𝑥)𝑞²−1𝑞−𝜑(𝑥)−𝜀²𝜕𝑞Φ𝜕𝑥¹(𝑥)−𝜀²Φ₂𝑞,𝑥. (16) Using the boundary functions method [18] we construct the asymptotic contrast solution of the problem (16) and (15). The first-order asymptotic expansion has the form 𝑞(𝑥,𝑡,𝜀)=𝑞₀(𝑥)+𝜀𝑞₁(𝑥)+Π₀𝑞(𝜏)+𝜀Π₁𝑞(𝜏)+𝜀𝑄𝑞𝜏₀,𝜏₁, (17) where 𝑞₀(𝑥) and 𝑞₁(𝑥) are the regular terms of asymptotic expansion, Π₀𝑞(𝜏) and Π₁𝑞(𝜏) are zero- and first-order transition layer functions, 𝜏=(𝑥−𝑥_∗)/𝜀 is a stretched variable, 𝑥_∗=𝑥_∗(𝜀) is a transition point in a small vicinity of which the transition layer is localized, and 𝑄𝑞(𝜏₀,𝜏₁) is function describing the boundary layers near the points 𝑥=0, 𝑥=1 and 𝜏₀=𝑥/𝜀, 𝜏₁=(1−𝑥)/𝜀. The transition point has the following asymptotic form: 𝑥_∗=𝑥₀+𝜀𝑥₁+𝜀²𝑥₂+⋯. (18) Using the boundary functions method procedure [3, 4] we obtain that the principal term 𝑥₀ of the expansion (18) can be found from the equation 𝜑𝑥₀=0. (19) The full principal order function 𝑞(𝜏)=𝑞₀(𝑥₀)+Π₀𝑞(𝜏) can be found from equation 𝑞^′′=𝛾𝑥₀𝑞𝑞²,𝑞()𝑞()−1−∞=1,+∞=−1. (20)(21) From (20) and (21) we have 𝑞=1−exp2𝛾𝑥₀𝜏1+exp2𝛾𝑥₀𝜏. (22) So the principal term of the stationary power profile has the form 𝑝_st(𝑥,𝜀)=1−exp2𝛾𝑥₀𝑥−𝑥₀/𝜀1+exp2𝛾𝑥₀𝑥−𝑥₀×𝜑/𝜀₃(𝑥)−𝜑₁(𝑥)2+𝜑₃(𝑥)+𝜑₁(𝑥)2. (23) The power profile 𝑝_st(𝑥,𝜀) is close to the iron-hand profile 𝜑₃(𝑥) when 0≤𝑥<𝑥₀ and to the participatory profile 𝜑₁(𝑥) when 𝑥₀<𝑥≤1. In the vicinity of the transition point 𝑥₀ we have 𝜕𝑝/𝜕𝑥≅𝜀⁻¹. We call such power profiles the contrast power profiles.

Equation (19) can be written in the form 𝜑₂𝑥₀=𝜑₃𝑥₀+𝜑₁𝑥₀2. (24) We call the function ℎ₁(𝑥)=𝜑₂(𝑥)−𝜑₁(𝑥) the participatory domain's width and function ℎ₃(𝑥)=𝜑₃(𝑥)−𝜑₂(𝑥) the iron-hand domain's width. Then (20) can be interpreted in the following way: at the transition point 𝑥₀ of the stationary contrast power profile (SCPP) the participatory domain's width is equal to the iron-hand domain's width ℎ₁(𝑥₀)=ℎ₃(𝑥₀).

The stability of contrast structures of (9) was investigated by Bozhevol’nov and Nefëdov [5] and Vasil’eva et al. [6]. In terms of the “power-society” model the stability result can be interpreted as follows.

SCPP, which are close to the iron-hand profile at the top ranks of the hierarchy (𝑝≈𝜑₃(𝑥) when 0≤𝑥<𝑥₀) and to the participatory profile at the bottom ranks (𝑝≈𝜑₁(𝑥) when 𝑥₀<𝑥≤1), are stable if the iron-hand domain's width is greater than the participatory domain's width at the top ranks of the hierarchy (ℎ₃(𝑥)>ℎ₁(𝑥) when 0≤𝑥<𝑥₀) and less at the bottom ranks (ℎ₃(𝑥)<ℎ₁(𝑥) when 𝑥₀<𝑥≤1). If ℎ₃(𝑥)<ℎ₁(𝑥) when 0≤𝑥<𝑥₀ and ℎ₃(𝑥)>ℎ₁(𝑥) when 𝑥₀<𝑥≤1 then the SCPP is unstable.

Remark 1. Similar statement holds for the so-called confederative SCPP which are close to the participatory profile at the top ranks of the hierarchy and to the iron-hand profile at the bottom ranks (𝑝≈𝜑₁(𝑥) when 0≤𝑥<𝑥₀ and 𝑝≈𝜑₃(𝑥) when 𝑥₀<𝑥≤1). They are stable if ℎ₃(𝑥)<ℎ₁(𝑥) when 0≤𝑥<𝑥₀ and ℎ₃(𝑥)>ℎ₁(𝑥) when 𝑥₀<𝑥≤1 and unstable if ℎ₃(𝑥)>ℎ₁(𝑥) when 0≤𝑥<𝑥₀ and ℎ₃(𝑥)<ℎ₁(𝑥) when 𝑥₀<𝑥≤1.

Consider again nonstationary equation (9). Suppose that at time 𝑡=𝑡₀ contrasting structure has appeared with the transition layer at the vicinity of the point 𝑥=𝜉. Then for 𝑡>𝑡₀ the solution is a nonstationary contrast structure: 𝑝≈𝜑₃(𝑥) when 𝑥<𝑅(𝑡,𝜀) and 𝑝≈𝜑₁(𝑥) when 𝑥>𝑅(𝑡,𝜀), where the transition point 𝑅(𝑡,𝜀) depends on time. We call such power profile the nonstationary contrast power profile (NCPP).

Let us construct the asymptotic NCPP.

Like in Section 3, make the substitution (6) and consider (7). It was shown in [7] that the principal term of the nonstationary contrast structure looks similar to one of the stationary contrast structure (18): 𝑞=1−exp2𝛾(𝑅)(𝑥−𝑅)/𝜀1+exp2𝛾(𝑅)(𝑥−𝑅)/𝜀, (25) where the function 𝑅=𝑅(𝑡,𝜀) can be found from the equation 𝑑𝑅𝑑𝑡=−𝜀2𝛾(𝑅)𝜑(𝑅). (26) So the principal term of NCPP has the form 𝑝(𝑥,𝑡,𝜀)=1−exp2𝛾(𝑅)(𝑥−𝑅)/𝜀1+exp(𝑅)×𝜑2𝛾)(𝑥−𝑅/𝜀₃(𝑥)−𝜑₁(𝑥)2+𝜑₃(𝑥)+𝜑₁(𝑥)2. (27) Power profile 𝑝(𝑥,𝑡,𝜀) is close to the iron-hand profile 𝜑₃(𝑥) when 0≤𝑥<𝑅 and to the participatory profile 𝜑₁(𝑥) when 𝑅<𝑥≤1. The value of 𝑑𝑅/𝑑𝑡 represents the speed of the transition layer. In terms of the “power-society” model the expression for 𝑑𝑅/𝑑𝑡 has the form 𝑑𝑅𝑑𝑡=−𝜀𝑘₁(𝑅)22𝜑₂(𝑅)−𝜑₃(𝑅)−𝜑₁(𝑅) (28) or 𝑑𝑅𝑑𝑡=𝜀𝑘₁(𝑅)2ℎ₃(𝑅)−ℎ₁(𝑅). (29) Consider now some important cases of using formula (29); see also [17].

2.1. Attraction to the “Iron-Hand” Profile (1)

Let the “iron-hand” domain’s width be larger than participatory domain's width: ℎ₃(𝑥)>ℎ₁(𝑥) for any 𝑥∈[0,1]. This means that the iron-hand profile looks more attractive from the society's point of view. Then SCPP do not exist because (24) has no roots. After appearing at time 𝑡₀ the contrast structure begins to move according to formula (29). Evidently 𝑑𝑅/𝑑𝑡>0, and after small time of order 𝜀⁻¹ transition point 𝑅(𝑡,𝜀) comes to the right end of the segment [0,1]. So the power profile appears close to the iron-hand profile for any 𝑥∈[0,1]. Notice that if at time 𝑡=0 function 𝑝(𝑥,0,𝜀) is entirely in the participatory domain then for any 𝑡 the power profile is close to the participatory profile even if ℎ₃(𝑥)>ℎ₁(𝑥). For appearing the power profile close to the iron-hand profile function 𝑝(𝑥,0,𝜀) must be located in the iron-hand domain on at least one point in the interval (0,1). This statement is based on the theorem proved by Bozhevol’nov and Nefedov [5].

2.2. Attraction to the “Iron-Hand” Profile (2)

Let point 𝑥₀∈(0,1) exist such that ℎ₁(𝑥₀)=ℎ₃(𝑥₀), ℎ₁(𝑥)>ℎ₃(𝑥) when 𝑥<𝑥₀ and ℎ₁(𝑥)<ℎ₃(𝑥) when 𝑥>𝑥₀. Then unstable SCPP exist having transition layer in the vicinity of the point 𝑥₀. Let function 𝑝(𝑥,0,𝜀) be in the iron-hand domain for 𝑥<𝜉 and in the participatory domain for 𝑥>𝜉 where 𝑥₀<𝜉<1. Then the power profile 𝑝(𝑥,𝑡,𝜀) is attracted to the iron-hand profile for 𝑥<𝜉 and to the participatory profile for 𝑥>𝜉. After appearing at time 𝑡₀ the contrast structure begins to move according to formula (29) and initial condition 𝑅(0,𝜀)=𝜉. As 𝜉>𝑥₀ then 𝑑𝑅/𝑑𝑡>0. So after small time of order 𝜀⁻¹ transition point 𝑅(𝑡,𝜀) comes to the right end of the segment [0,1]. So the power profile appears close to the iron-hand profile for any 𝑥∈[0,1].

2.3. Attraction to the Participatory Profile (1)

Let ℎ₃(𝑥)<ℎ₁(𝑥) for any 𝑥∈[0,1]. This means that the participatory profile looks more attractive from the society's point of view. Then SCPP do not exist because (24) has no roots. After appearing at time 𝑡₀ the contrast structure begins to move according to formula (29). Evidently 𝑑𝑅/𝑑𝑡<0 and after small time of order 𝜀⁻¹ transition point 𝑅(𝑡,𝜀) comes to the left end of the segment [0,1]. So the power profile appears close to the participatory profile for any 𝑥∈[0,1]. Notice that if at time 𝑡=0 function 𝑝(𝑥,0,𝜀) is entirely in the iron-hand domain then for any 𝑡 the power profile is close to the iron-hand profile even if ℎ₃(𝑥)>ℎ₁(𝑥). For appearing the power profile close to the participatory profile function 𝑝(𝑥,0,𝜀) must be smooth and located in the participatory domain on at least one point in the interval (0,1). This statement is based on the theorem proved by Bozhevol’nov and Nefedov [5].

2.4. Attraction to the Participatory Profile (2)

Let point 𝑥₀∈(0,1) exist such that ℎ₁(𝑥₀)=ℎ₃(𝑥₀), ℎ₁(𝑥)>ℎ₃(𝑥) when 𝑥<𝑥₀ and ℎ₁(𝑥)<ℎ₃(𝑥) when 𝑥>𝑥₀. Then unstable SCPP exist having transition layer in the vicinity of the point 𝑥₀. Let function 𝑝(𝑥,0,𝜀) be in the iron-hand domain for 𝑥<𝜉 and in the participatory domain for 𝑥>𝜉 where 𝑥₀<𝜉<1. Then according to (3) power profile 𝑝(𝑥,𝑡,𝜀) is attracted to the iron-hand profile for 𝑥<𝜉 and to the participatory profile for 𝑥>𝜉. After appearing at time 𝑡₀ the contrast structure begins to move according to formula (26) and initial condition 𝑅(0,𝜀)=𝜉. As 𝜉<𝑥₀ then 𝑑𝑅/𝑑𝑡<0. So after small time of order 𝜀⁻¹ transition point 𝑅(𝑡,𝜀) comes to the left end of the segment [0,1]. So the power profile appears close to the participatory profile.

2.5. Attraction to SCPP

Let point 𝑥₀∈(0,1) exist such that ℎ₁(𝑥₀)=ℎ₃(𝑥₀), ℎ₁(𝑥)<ℎ₃(𝑥) when 𝑥<𝑥₀ and ℎ₁(𝑥)>ℎ₃(𝑥) if 𝑥>𝑥₀. Then the stable SCPP exist having transition layer in the vicinity of the point 𝑥₀. If there is 𝜉∈(0,1) such that the initial function 𝑝⁰(𝑥) satisfies 𝑝⁰(𝑥)>𝜑₂(𝑥), 0<𝑥<𝜉, and 𝑝⁰(𝑥)<𝜑₂(𝑥), 𝜉<𝑥<1, then the contrasting structure appears after a short time 𝑡₀ such that 𝑝(𝑥)≈𝜑₃(𝑥) for 0<𝑥<𝜉 and 𝑝(𝑥)≈𝜑₁(𝑥) for 𝜉<𝑥<1. Then the contrasting structure begins to move according to formula (29) and initial condition 𝑅(𝑡₀,𝜀)=𝜉. So if 𝜉<𝑥₀ then 𝑑𝑅/𝑑𝑡>0 and if 𝜉>𝑥₀ then 𝑑𝑅/𝑑𝑡<0. So when 𝑡→∞, the transition point 𝑅(𝑡,𝜀) tends to the stationary transition point 𝑥₀. Thus the SCPP having transition layer in the vicinity of point 𝑥₀ appears.

3. Parametric Optimization

The total amount of power of the hierarchy is 𝑃(𝑡,𝜀)=∫¹₀𝑝(𝑥,𝑡,𝜀). It was shown in [19] that there exists the optimal value 𝑃₀ of the total power which provides a maximum of steady-state consumption per capita (in frame of the “power-society-economics” model [19]). So we should introduce the control parameter into the “power-society” model to make it controllable. So the problem would be to find the value of the control parameter under which 𝑃(𝑡,𝜀)→𝑃₀, when 𝑡→∞, 𝜀→0.

Generally speaking, the model could be formulated such that the control is considered to be a function of time or 𝑥. In any case, the control describes the exogenous impact on the political system, such as a political pressure through media and political institutions. We restrict ourselves to the parametric control.

Definition 2. The value 𝑃₀ is called the asymptotically achievable amount of the total power if there exists an admissible value of control parameter 𝑢 such that the steady-state total power 𝑃_𝑢(𝜀) satisfies 𝑃_𝑢(𝜀)→𝑃₀ when 𝑡→∞, 𝜀→0.

So consider the “power-society” model with nonlinear reaction of civil society: 𝑓𝑝,𝑥,𝑡,𝑢=−𝑘₁(𝑥)𝑝−𝜑₁(𝑥)×𝑝−𝜑₂(𝑥)+𝛾𝑢𝑝−𝜑₃(𝑥). (30) Here 𝑘₁(𝑥)>0, the functions 𝑘₁(𝑥), 𝜑₁(𝑥), 𝜑₂(𝑥), and 𝜑₃(𝑥) have continuous derivatives, and 𝛾 is a constant. Thus the lowest and the biggest roots 𝜑₁(𝑥), 𝜑₃(𝑥) of the degenerate equation 𝐹(𝑝,𝑥,𝑢)=0 do not depend on the control, but there is an impact from the control to the “middle” root 𝜑₂(𝑥). So the model has the form 𝜕𝑝𝜕𝑡=𝜀²𝜕²𝑝𝜕𝑥²−𝑘₁(𝑥)𝑝−𝜑₁(𝑥)×𝑝−𝜑₂(𝑥)+𝛾𝑢𝑝−𝜑₃(𝑥),𝜕𝑝𝜕𝑥_𝑥=0=𝜕𝑝𝜕𝑥_𝑥=1=0,𝑝_𝑡=0=𝑝⁰(𝑥). (31)(32) The initial function 𝑝⁰(𝑥) is supposed to be smooth and satisfying 𝜑₁(𝑥)<𝑝⁰(𝑥)<𝜑₃(𝑥). In other words, the initial distribution of power is between the iron-hand and participatory profiles.

The steady-state equation for (31) has the form 𝜀²𝜕²𝑝𝜕𝑥²=𝑘₁(𝑥)𝑝−𝜑₁(𝑥)𝑝−𝜑₂(𝑥)+𝛾𝑢𝑝−𝜑₃(𝑥),𝜕𝑝𝜕𝑥_𝑥=0=𝜕𝑝𝜕𝑥_𝑥=1=0. (33)

Let us stress here that the control influences the relation between the width of the iron-hand domain and the width of the participatory domain.

Let the following conditions be fulfilled.

Condition 3. We have −1≤𝑢≤1.

Condition 4. We have 𝜑₁(𝑥)<𝜑₂(𝑥)−𝛾<𝜑₂(𝑥)+𝛾<𝜑₃(𝑥) for 0≤𝑥≤1.

Condition 5. We have 𝐻^′(𝑥)<0 for 0≤𝑥≤1, and here 𝐻(𝑥)=𝜑₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥).

Conditions 3 and 4 introduce the normalization of the control such that for any admissible control 𝑢∈[−1;1] the root 𝜑₂(𝑥)+𝛾𝑢 is between the 𝜑₁(𝑥) and 𝜑₃(𝑥). The steady-state solution has no more than one transition point due to Condition 5.

If the control parameter 𝑢 is increased, the root of the equation 𝜑₁(𝑥)+𝜑₃(𝑥)−2(𝜑₂(𝑥)+𝛾𝑢)=0 will move to the left. This means greater support to the participation ideas. If it exists. Analogically, the less the value of 𝑢 is, the more to the right the root of this equation is.

Consider the following problem. Let the desirable (optimal) value of total power be 𝑃₀. Is there a value of control parameter 𝑢, under which the steady-state solution (33) is such that 𝑃_𝑢(𝜀)=∫¹₀𝑝(𝑥,𝜀)𝑑𝑥→𝑃₀ when 𝜀→0?

From the practical point of view, such a formulation of the problem can be justified in the following way. We know from the “power-society-economics” model that the optimal value of the total power is some 𝑃₀, so we should try to tune the political system to provide this optimal value of power for the steady-state regime.

Several cases should be distinguished.

Let us start our consideration from the situation in which both equations 𝜑₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥)𝜑+𝛾=0,₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥)−𝛾=0 (34)(35) have roots in the interval (0;1). Let us denote these roots by 𝑎 and 𝑏, respectively. Here we have 𝑎<𝑏 in view of Condition 5.

The points 𝑥=𝑎 and 𝑥=𝑏 are the main asymptotic terms for the boundaries of the range within which the transition point of the stationary front is located.

Therefore, the value 𝑃_𝑎𝑠(𝑢)=lim_𝜀→0𝑃(𝑢,𝜀)=lim_𝜀→0¹₀𝑝(𝑥,𝑢,𝜀)𝑑𝑥 (36) satisfies the inequality ^𝑎₀𝜑₃(𝑥)𝑑𝑥+¹_𝑎𝜑₁(𝑥)𝑑𝑥≤𝑃_𝑎𝑠(𝑢)≤^𝑏₀𝜑₃(𝑥)𝑑𝑥+¹_𝑏𝜑₁(𝑥)𝑑𝑥. (37)

Thus, in this case, 𝑃₀ is asymptotically achievable, if inequality ^𝑎₀𝜑₃(𝑥)𝑑𝑥+¹_𝑎𝜑₁(𝑥)𝑑𝑥≤𝑃₀≤^𝑏₀𝜑₃(𝑥)𝑑𝑥+¹_𝑏𝜑₁(𝑥)𝑑𝑥 (38) holds true.

The steady-state problem (32) and (33) has also solutions without transition layers: the iron-hand profile and the participatory one. So the values of total power 𝑃₀=¹₀𝜑₃(𝑥)𝑑𝑥,𝑃₀=¹₀𝜑₁(𝑥)𝑑𝑥, (39) are also asymptotically achievable.

So, if both (34) and (35) have roots in the interval (0;1) then the set of asymptotically achievable values comprises the closed interval (38) and two isolated values (39): one of them is to the left of this closed interval, and the other one is to the right of it.

Now let us consider the situation in which (34) has a root 𝑥=𝑎∈(0;1) and (35) has no roots on (0;1).

In other words, 𝜑₂(𝑥)−𝛾<(𝜑₁(𝑥)+𝜑₃(𝑥))/2 for any 𝑥. So the values of 𝑃₀ in the closed interval ^𝑎₀𝜑₃(𝑥)𝑑𝑥+¹_𝑎𝜑₁(𝑥)𝑑𝑥≤𝑃₀≤¹₀𝜑₃(𝑥)𝑑𝑥 (40) are asymptotically achievable.

Let now (35) have the root 𝑥=𝑏∈(0;1), and let (34) have no roots in the interval. That is, at 𝑢=1, for any 𝑥, the middle root 𝜑₂(𝑥)+𝛾 is larger than half-sum of 𝜑₁(𝑥) and 𝜑₃(𝑥). Then the asymptotically achievable values are given by the inequality ¹₀𝜑₁(𝑥)𝑑𝑥≤𝑃₀≤^𝑏₀𝜑₃(𝑥)𝑑𝑥+¹_𝑏𝜑₁(𝑥)𝑑𝑥. (41) At last, consider the case in which neither of the equations has a root in the interval (0;1). Then the following three subcases are possible: 𝜑₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥)𝜑+𝛾<0,₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥)𝜑−𝛾<0forany𝑥,₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥)𝜑+𝛾>0,₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥)𝜑−𝛾>0forany𝑥,₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥)𝜑+𝛾>0,₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥)−𝛾<0forany𝑥. (42)(43)(44)

It can be easily shown (see [2], e.g.) that in subcases (42) and (43) a steady-state front does not exist for any control parameter. So only the values of 𝑃₀ given by (39) are asymptotically achievable.

In the subcase (44) for any given 𝑥₀∈(0;1), such 𝑢 exists that the problem (32) and (33) has the stationary front with the transition point in the 𝜀-vicinity of 𝑥₀. Therefore, any 𝑃₀ from closed interval ¹₀𝜑₁(𝑥)𝑑𝑥≤𝑃₀≤¹₀𝜑₃(𝑥)𝑑𝑥 (45) is asymptotically achievable.

The above speculations can be summarized as follows.

Theorem 3. Consider the problem (32) and (33) with parametric control. Let the following conditions be satisfied: (1)𝑘₁(𝑥), 𝜑₁(𝑥), 𝜑₂(𝑥), 𝜑₃(𝑥)∈𝐶¹[0,1],(2)−1≤𝑢≤1,(3)𝜑₁(𝑥)<𝜑₂(𝑥)−𝛾<𝜑₂(𝑥)+𝛾<𝜑₃(𝑥), (4)𝐻^′(𝑥)<0, where 𝐻(𝑥)=𝜑₁(𝑥)+𝜑₃(𝑥)−2𝜑₂(𝑥). Then the set of asymptotically achievable values is not empty.

After the provided analysis of a steady-state problem (32) and (33), we go back to the initial parabolic partial problem (31) and (32).

Let some value 𝑃₀ be asymptotically achievable in the corresponding stationary problem. It means that there is a value of parametric control 𝑢, at which the problem (31), (32) has the steady-state solution for which the total power 𝑃_𝑢(𝜀) of the hierarchy asymptotically tends to 𝑃₀ when 𝜀→0. However for the 𝑃_𝑢(𝑡,𝜀)=∫¹₀𝑝_𝑢(𝑥,𝑡,𝜀)𝑑𝑥 we have 𝑃_𝑢(𝑡,𝜀)→𝑃_𝑢(𝜀) just for some class of initial functions 𝑝⁰(𝑥).

Thus, there is a problem to determine the class of initial power distributions for which, under the found value parametric control, the solution of the parabolic partial problem converges to the proper steady-state solution at 𝑡→∞.

The answer is given by the following theorem.

Theorem 4. (1) Let all the conditions of Theorem 3 be satisfied;
(2) let 𝑃₀ be an asymptotically achievable value of total power (denote by 𝑢=𝑢₀ the corresponding value of parametric control);
(3) let the point 𝑥₀∈(0;1) exist such that 𝑃₀=^𝑥0₀𝜑₃(𝑥)𝑑𝑥+¹_𝑥0𝜑₁(𝑥)𝑑𝑥; (46)
(4) let the points 𝑥₁∈(0;𝑥₀), 𝑥₂∈(𝑥₀;1) exist such that 𝑝⁰𝑥₁>𝜑₂𝑥₁+𝛾𝑢₀,𝑝⁰𝑥₂<𝜑₂𝑥₂+𝛾𝑢₀. (47) Then the solution 𝑝_𝑢0(𝑥,𝑡,𝜀) of the parabolic partial problem (31) and (32) is such that the total power 𝑄_𝑢0(𝑡,𝜀)=¹₀𝑝_𝑢0(𝑥,𝑡,𝜀)𝑑𝑥 (48) converges to 𝑃₀∶lim_𝜀→∞lim_𝑡→∞𝑃𝑡,𝑢₀,𝜀=𝑃₀. (49)

Proof. It is easy to see that under these conditions the Butuzov-Nedelko theorem is fulfilled [13]. Therefore, at the chosen value of control 𝑢=𝑢₀, the solution 𝑝_𝑢0(𝑥,𝑡,𝜀) of the problem (31) and (32) has a passage to the limit lim_𝑡→∞𝑝_𝑢0(𝑥,𝑡,𝜀)=𝑝_st,𝑢0(𝑥,𝜀), (50) where 𝑝_st(𝑥,𝜀) is the steady state solution for which lim_𝑡→∞𝑝_st,𝑢0()=𝜑𝑥,𝜀³(𝑥),𝑥<𝑥₀,𝜑₁(𝑥),𝑥>𝑥₀. (51) By integrating (50) from 𝑥=0 to 𝑥=1 we get lim_𝑡→∞¹₀𝑝_𝑢0(𝑥,𝑡,𝜀)𝑑𝑥=¹₀𝑝_st,𝑢0(𝑥,𝜀)𝑑𝑥. (52) Passing to a limit 𝜀→0, we get lim_𝜀→0lim_𝑡→∞¹₀𝑝_𝑢0(𝑥,𝑡,𝜀)𝑑𝑥=lim_𝜀→0¹₀𝑝_st,𝑢0()=𝑥,𝜀𝑑𝑥_𝑥00𝜑₃(𝑥)𝑑𝑥+¹_𝑥0𝜑₁(𝑥)𝑑𝑥=𝑃₀. (53) Thus lim_𝜀→0lim_𝑡→∞𝑃(𝑡,𝑢₀,𝜀)=𝑃₀. Theorem 4 is proved.

4. Conclusion

It is shown that the theory of contrasting structures in singularly perturbed boundary value problems allows for investigating the properties of nonstationary fronts in the singularly perturbed “power-society” model. Depending on the initial condition, these fronts evolve to one of the asymptotically stable steady-state distributions of power within a government hierarchy.

There are some reasons to introduce a concept of desirable steady-state total amount of power of the hierarchy. The possibility theorem is proved for the problem of getting this amount by means of parametric control. The results can be used in investigating governing hierarchical systems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was partially supported by Russian Foundation for Basic Research (project 12-06-00205).