A new method to control chaos in an economic system

Applied Mathematics and Computation 217 (2010) 2370–2380

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A new method to control chaos in an economic system Jianguo Du a,b,*, Tingwen Huang c, Zhaohan Sheng b, Haibin Zhang a,b a b c

School of Business Administration, Jiangsu University, Zhenjiang 212013, China School of Management Science and Engineering, Nanjing University, Nanjing 210093, China Texas A&M University at Qatar, Doha, P.O. Box 5825, Qatar

a r t i c l e

i n f o

Keywords: Duopoly games Chaos control Phase space compression Limiter method Performance measure

a b s t r a c t In this paper, the method to control chaos by using phase space compression is applied to economic systems. Because of economic significance of state variable in economic dynamical systems, the values of state variables are positive due to capacity constraints and financial constraints, we can control chaos by adding upper bound or lower bound to state variables in economic dynamical systems, which is different from the chaos stabilization in engineering or physics systems. The knowledge about system dynamics and the exact variety of parameters are not needed in the application of this control method, so it is very convenient to apply this method. Two kinds of chaos in the dynamic duopoly output systems are stabilized in a neighborhood of an unstable fixed point by using the chaos controlling method. The results show that performance of the system is improved by controlling chaos. In practice, owing to capacity constraints, financial constraints and cautious responses to uncertainty in the world, the firm often restrains the output, advertisement expenses, research cost etc. to confine the range of these variables’ fluctuation. This shows that the decision maker uses this method unconsciously in practice. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Since Ott et al. [1] introduced the OGY control method, researchers are increasingly interested in controlling chaos of the nonlinear systems. In recent years, a lot of studies have been devoted to this topic, such as Boccaletti and Grebogi [2], He and Westerhoff [3], Holyst and Urbanowicz [4], Li et al. [5], Kopel [6], Liu and Wang [7], Sheng et al. [8], Song et al. [9], Stoop and Wagner [10], Wagner and Stoop [11], Wieland and Westerhoff [12], Xiang et al. [13] and Zhang and Shen [14,15]. As we know, economic systems are nonlinear and may display chaotic behavior (see [12,16–30]). When some systems’ dynamics are chaotic, some players’ performance decreases contrasted with the equilibrium. Generally, these players should adapt some certain methods to control the chaos. At present, most of methods for chaos control are designed for the physics systems, which are mainly used in the natural science and engineering. For the chaos control in the economic systems, there only are a few introductions. Ahmed et al. [31] and Agiza [32] use OGY method to control the chaos in economic systems. Kass [33] associates the chaotic targeting method with the OGY method to stabilize chaos in a dynamical macroeconomic model. Kopel [6] uses the chaotic targeting method to control chaos of a monopoly output adjustment model. Holyst and Urbanowicz [4] uses delayed feedback control method (DFC) to control chaos in a duopoly investment model. Wieland and Westerhoff [12] apply the OGY method and DFC separately to stabilizing chaos in an exchange rate dynamic model. In fact, through the control parameter’s perturbation, the original OGY method forces the unstable orbits to become stable on the stable manifolds of hyperbolic fixed points. It requires the unstable fixed points are dependant to the control

* Corresponding author at: School of Business Administration, Jiangsu University, Zhenjiang 212013, China. E-mail address: [email protected] (J. Du). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.07.036

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parameters. Furthermore, the stable manifolds lie in the points. While such information may be identified from observations in natural science applications, chaos control in an economic context is often seen as rather critical. The chaotic targeting method also requires knowledge of the map and its fixed point, so this method is not convenient in its practical using. DFC avoids fancy data processing used in the OGY method, and its application is very straightforward. Moreover, DFC represents a self-adaptive behavior of economic entities. DFC may be the only chaos control method satisfying with the basic economic features except the limiters methods introduced by Wieland and Westerhoff [12]. The limiter method is explored by Wagner and Stoop [11], Zhang and Shen [14,15], Stoop and Wagner [10]. Zhang and Shen [14,15] call it as phase space compression. One advantage of the limiter method is that it does not add complexity to the system by increasing the size of the system’s state space [3]. Another advantage is that stabilization may be achieved by infrequent interventions. The limiter method has realized the control of chaos and hyper-chaos through limiting of the strange attractor’s space in chaos and hyper-chaos system. Actually, this is to restrain the system variable values in a subset. In most economic systems, some state variables can be controlled in certain area by the object, such as the output, the research investment, the advertisement cost, price and so on. All these can help us to control chaos in some economic systems using the limiter method. He and Westerhoff [3] studied chaos control of economic systems using limiter method. They discussed commodity markets by using price limiters. They have achieved chaos control through giving price upper limiter or lower limiter. It is worthy to be noticed that the economic model studied by He and Westerhoff [3] is one-dimension. In two-dimensions, especially in imperfect competitive market, the problems need to be studied furthermore, such as whether the players should carry out the control and whether it needs all players to take control actions in order to control chaos successfully. The remainder of this paper is organized as follows. Section 2 presents the limiter method in multi-dimensional economic systems. In Section 3, we introduce a dynamic output game with bounded rationality and examine the dynamics of the model by using stability and bifurcation analysis. Section 3 also gives performance indices measuring the performance of economic systems and shows the results from numerical simulations. In Section 4, we use the control method to control the chaos of output model and give the comparison for the players’ income before the control and the income after the control. The final section concludes the paper.

2. The limiter method in economic systems Wagner and Stoop [11], Zhang and Shen [14,15], and Stoop and Wagner [10] put forward the limiter method (phase space compression) to control chaos in physics systems. We will introduce this method’s applications in certain economic systems. Consider the following discrete-time dynamic system:

X i ðt þ 1Þ ¼ M i ðX 1 ðtÞ; . . . ; X N ðtÞÞ;

i ¼ 1; . . . ; N:

ð1Þ

In (1), t = 1, 2, . . ., is the discrete-time variable. Xi is a state variable, and Mi is a mapping of RN ´ R, i = 1, 2, . . ., N. Let the chaotic solution be in certain space for system (1), the trajectory of the solution is expressed by strange attractors in phase space. From the characteristics of strange attractors, we know that the trajectory of the solution in system (1) will be limited in certain space V with boundary. Then we can choose a nonempty subset W of V, and W  V, and the solution of system (1) will be limited in space W. That is, the orbit evolvement can be controlled according to the following equation

X i ðt þ 1Þ ¼ minfX 1i ; maxfX 0i ; M i ðX 1 ðtÞ; . . . ; X N ðtÞÞgg; ðX 01 ; X 02 ; . . . ; X 0N Þ;

i ¼ 1; 2; . . . ; N:

ð2Þ

ðX 11 ; X 12 ; . . . ; X 1N Þ

In (2), 2 Wo. As we know, the positive Lyapunov exponent k value is the essential characteristics of chaos or hyper-chaos in the nonlinear system. The positive k value reflects the divergence or expansion of nonlinear system’s orbits on certain direction or torus. Through compressing the system orbits space and limiting the orbit’s free divergence or expansion, the above method can realize the control of chaos and hyper-chaos state if positive k becomes negative in the controlling course. If we use the above method in certain economic systems, we define an upper limit X 1i and lower limit X 0i for the value of variables. We compare the variable’s values generated by the decision-making rule in (1) at time t + 1 with the upper limit and lower limit. If the value is between the upper value and lower value, this value is the state variable’s value at time t + 1. If it is smaller than the lower value, the state variable’s value is the lower limit; if it is bigger than the upper limit, state variable’s is the upper limit at time t + 1. Because this method is to control chaos through the restrict values with the upper limit and lower limit, we call it upper and lower limiter method. In some economic systems, the value of the economic variables will not approach to infinite because of finite economic resources. In addition, there are many economic variables are non-negative, such as interest rate, output, advertisement cost, price, research investment and sales income. Therefore, the value of variables will not approach to negative infinite. That is to say, we can control chaos only by adding the upper limit and lower limit of state variables. We call the method of adding the lower limit to economic variables lower limiter method, while we call the method of adding the upper limit to economic variables upper limiter method. The lower limiter method controls orbits of system according to the following equation:

X i ðt þ 1Þ ¼ maxfX 0i ; Mi ðX 1 ðtÞ; . . . ; X N ðtÞÞg;

i ¼ 1; 2; . . . ; N:

ð3Þ

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The upper limiter method controls orbits of system according to the following equation:

X i ðt þ 1Þ ¼ minfX 1i ; M i ðX 1 ðtÞ; . . . ; xN ðtÞÞg;

i ¼ 1; 2; . . . ; N:

ð4Þ

No matter the upper and lower limiter method, or the upper limiter method or the lower limiter method, actually these methods control chaos through limiting the value of control variables. Although there are many varieties in multi-dimension situation, we call all these methods limiter method. Why do these forms of limiter method control chaos? The reason is similar to the phase space compression which restrains the expansion of attractors. Unfortunately, it cannot be proved strictly. The knowledge about system dynamics and the exact variety of parameters are not needed in the application of the limiter method, so it is very convenient to use this method. In economic competition, different player may use different control forms, such as some players use upper limit method, the others may add lower limit method. Therefore, there are some varieties in the application of limiter method. For two-dimension game model, one player uses the limiter method to restrain the free expansion of orbits, and this will lead to orbits of another player who is not able to expand freely. This action is able to achieve the goal of chaos control. It has great significance in economic competition. 3. The output duopoly game with bounded rationality Suppose that two firms produce the homogeneous product, and there is a duopoly competition in the market. Then one firm is labeled by i = 1, and the other i = 2. The strategy space is the choice of the output, and the decision-making takes place in the discrete-time periods t = 0, 1, 2, . . .: qi(t) represents the output of the ith firm during the period t, and let the profit be their payment. The price p in period t is determined by the total supply Q(t) = q1(t) + q2(t) through a demand function p = p(Q) = a  bQ. Here, a and b are positive constants, and a is the highest price in the market. C i ðqi Þ ¼ ci þ di qi þ ei q2i represents the cost function. Because of the economic scale, the cost function Ci(qi) climbs with the increase of the product output. We can assume: C 0i ðqi Þ > 0; C 00i ðqi Þ P 0; i ¼ 1; 2. According to the assumption, we can get di > 0, ei > 0. Due to the existence of the fixed cost, Ci(0) > 0. That is ci > 0. In order to make the two-players’ game on the rails, the marginal profit of the ith firm (i = 1, 2) is less than the highest price of the homogeneous product in the market. Therefore, di + 2eiqi < a, i = 1, 2. At last, the after-tax profit of the ith firm at the period t(t = 0, 1, 2, . . .) is:

Li ðq1 ðtÞ; q2 ðtÞÞ ¼ ½qi ðtÞða  bQðtÞÞ  ðc þ di qi ðtÞ þ ei q2i ðtÞÞð1  sÞ;

i ¼ 1; 2;

ð5Þ

where s is the tax rate of business income tax, and 0 6 s < 1, s = 0 represent pre-tax profit. The marginal profit of the ith firm at the period t is:

@Li ðq1 ; q2 Þ ¼ ½a  bQ ðtÞ  bqi ðtÞ  di  2ei qi ðtÞð1  sÞ; @qi

i ¼ 1; 2:

ð6Þ

In order to get the maximum profit, every firm carries out the output decision-making. Suppose that the two producers do not have complete knowledge of the market, hence they adjust the last period’s output adaptively based on a local estimate of the marginal profit [34,35]. For example, if the firm thinks the marginal profit of the period t is positive, it will increase the output of the period t + 1; whereas if the marginal profit is negative, it will decrease the output. Therefore, we assume that the output of the period t + 1 has the following form [34,35]:

qi ðt þ 1Þ ¼ qi ðtÞ þ ai qi ðtÞ

@Li ðq1 ; q2 Þ ; @qi

i ¼ 1; 2;

ð7Þ

where ai is a positive parameter, and it presents the relative speed which is adjusted by the output of the ith firm. We take (6) into (7), then we get the following form:

qi ðt þ 1Þ ¼ qi ðtÞ þ ai qi ðtÞ½a  bQðtÞ  ðb þ 2ei Þqi ðtÞ  di ð1  sÞ;

i ¼ 1; 2:

ð8Þ

3.1. The dynamics analysis of the output decision-making model In order to make the solution of the output model with economical significance, we study the non-negative stable state solution of the model in this paper. The equilibrium solution of the dynamics system (8) is the following algebraic non-negative solution:



q1 ða  bQ  ðb þ 2e1 Þq1  d1 Þ ¼ 0;

ð9Þ

q2 ða  bQ  ð2b þ 2e2 Þq2  d2 Þ ¼ 0: From (9), we can get four fixed points:

 E0 ¼ ð0; 0Þ;

E1 ¼

 a  d1 ;0 ; 2b þ 2e1

  a  d2 ; E2 ¼ 0; 2b þ 2e2

1 Þð2bþ2e2 Þbðad2 Þ 2 Þð2bþ2e1 Þbðad1 Þ where q1 ¼ ðad ; q2 ¼ ðad . 3b2 þ4be þ4be þ4e e 3b2 þ4be þ4be þ4e e 1

2

1 2

1

2

1 2

E ¼ ðq1 ; q2 Þ;

J. Du et al. / Applied Mathematics and Computation 217 (2010) 2370–2380

Fig. 1. Partial numerical values simulation of the system (8).

Fig. 2. The graph of the i = 1st firm, i = 2nd firm’s aggregate profits, aggregate sales revenues variety with the change of a1.

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Since E1 is on the boundary of the decision set J = {(q1, q2)jq1 P 0, q2 P 0}, it is called boundary equilibrium. E* is the unique Nash equilibrium provided that:

8 < d1 þ bðd1 d2 Þ < a; bþ2e2

: d2 þ bðd2 d1 Þ < a:

ð10Þ

bþ2e1

According to the Jacobian matrix of system (8), we can draw the conclusion that the boundary equilibria E0, E1 and E2 are unstable equilibrium points, while the Nash equilibrium point E* is local stable (detailed deduction may see [34,35]). We can show the stability of the Nash equilibrium point E* through the numerical simulations, and show the system achieves chaos through period doubling bifurcation. When the values of the parameters are a2 = 0.1, a = 10, b = 1, c1 = 1.1, c2 = 1.1, d1 = 1, d2 = 1, e1 = 1, e2 = 1.1, s = 0.3, Fig. 1(a) shows that the bifurcation of the system (8) with the change of a1 and the corresponding Lyanpunov exponents. From Fig. 1(a), we can see, when the value of a1 is small enough, the output of the player converges at the equilibrium point E*. The Nash equilibrium point E* will become unstable when a1 > 0.3842, and the chaos occurs after the period doubling bifurcation. Take the corresponding output (6), the profit displays bifurcation diagram (see Fig. 1(b)). In Fig. 1(b), L1 represents the profit of the ith = 1st firm, and L2 represents the profit of the ith = 2nd firm. We can see the change of profit is the same as the output of the player with the variance of determinant parameters. That shows if the firm does the decision-making more carefully, the output and the profit will be more stable. 3.2. Performance measures In order to measure the system’s performance when observable chaos occurs in the model and compare the performance before and after the controlling in Section 4, we introduce performance measures frequently used in the business and accounting literature, namely aggregate profits, aggregate sales revenues in Kopel [6]. At the same time we develop the performance measures from one dimension to two dimensions. Let Ri(t) = qi(t)(a  bQ(t)), i = 1, 2, denote the turnover of the ith firm, and Q0 the initial output of the producers. We then introduce the following performance indices:

Fig. 3. i = 1st firm use the upper and lower limiter method to make the output, cost, and profit stable to two-period orbit.

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I. Aggregate profits:

LTi ðQ 0 ; a1 ; a2 Þ ¼

T X

Li ðq1 ðtÞ; q2 ðtÞÞ;

i ¼ 1; 2;

ð11Þ

t¼0

where LTi ðQ 0 ; a1 ; a2 Þ; i ¼ 1; 2 stands for the aggregate after-tax profits of the ith firm, the qi(t), i = 1, 2 is determined by the decision rule (8), and Li(q1(t), q2(t)), i = 1, 2 by (5). The time horizon is denoted by T. II. Aggregate sales revenues:

RTi ðQ 0 ; a1 ; a2 Þ ¼

T X

Ri ðq1 ðtÞ; q2 ðtÞÞ;

i ¼ 1; 2:

ð12Þ

t¼0

Using the two performance indices, now we evaluate the performance of the system in chaos. The study indicated that the system is chaotic when the firm’s speed for the adjustment of output is big enough, as Fig. 1 shows. The firm makes (1.8, 1.8) as initial output, and consider the aggregate sales revenues and aggregate profits at the first 100 periods (T = 100). When chaos displays in the system, the aggregate profits and sales revenues vibrate. As Fig. 2(a) and (b), when the high adjustment speed of the i = 1st firm is the only factor for system chaos, the i = 1st firm’s aggregate profits and sales revenues will decrease a lot compared with the equilibrium, while the i = 2nd firm’s total profits and sales income will increase a lot; vice versa from the symmetry of model (8). In addition, from Fig. 2(c) and (d), we observe when chaos displays in the system due to very large speed adjustment of the output, i = 1st firm’s, i = 2nd firm’s total profits and sales income will decrease a lot compared with the low-periodic state. Whatever the situation is, the player whose profits decrease will find out the reasons and take corresponding measure. Next section we will discuss the chaos control in the output dynamical model. 4. Chaos control in the dynamic output model In the output duopoly game, players can control the output to certain degree, so we can take the limiter method to control chaos in the model.

Fig. 4. i = 1st firm use the upper and lower limiter method to make the output, cost, and profit from stable to one-period orbit.

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Taking the upper limit q1i and lower limit q0i for the output at same time, we obtain the control system as follows:

   qi ðt þ 1Þ ¼ min q1i ; max q0i ; qi ðtÞ þ ai qi ðtÞða  bQ ðtÞ  ðb þ 2ei Þqi ðtÞ  di Þð1  rÞ :

Taking the upper limit

q1i

for the output, we obtain

  qi ðt þ 1Þ ¼ min q1i ; qi ðtÞ þ ai qi ðtÞða  bQ ðtÞ  ðb þ 2ei Þqi ðtÞ  di Þð1  rÞ : Taking lower limit

q0i

ð13Þ

ð14Þ

for the output, we obtain

  qi ðt þ 1Þ ¼ max q0i ; qi ðtÞ þ ai qi ðtÞða  bQðtÞ  ðb þ 2ei Þqi ðtÞ  di Þð1  rÞ :

ð15Þ

In (13)–(15), i = 1, 2. When chaos displays in the system due to one player’s fast response to the uncertain world, we can assume that chaos caused by i = 1st firm in system (8). From Fig. 1, when the values of parameters are: a = 10, b = 1, a1 = 0.52, a2 = 0.1, c1 = 1.1, c2 = 1, d1 = 1, d2 = 1, e1 = 1, e2 = 1.1, s = 0.3, the output, and profits display chaos. At the same time, the biggest Lyapunov exponent is 0.3421, bigger than zero. From Fig. 2(a) and (b), i = 1st firm’s aggregate profits and aggregate sales revenues will decrease a lot compared with the equilibrium, while i = 2nd firm’s aggregate profits and aggregate sales revenues will increase a lot. So only will i = 1st firm take some control measures. In this case, we can makeq02 ¼ 0; q12 ¼ 1. Using the upper and lower limiter method, the different orbits can be controlled through choosing the proper upper and lower limit. Let q01 ¼ 1:76; q11 ¼ 2:1 in system (13). The result of numerical value simulation is seen as Fig. 3, and the orbit which passes the initial point (2, 2) changes from stable to two-period orbit quickly. If the value of upper limit q11 is not bigger than 1.8228 or the value of lower limit q01 is not smaller than 1.8228, namely, Nash equilibrium output of i = 1st firm, the control orbit approaches to one-period orbit. For example, the value of q01 increases to 1.83, the control will be convergent to one-period orbit, seen as Fig. 4. If we decrease the lower limit q01 , and increase upper limit q11 , the control orbit is convergent to higher period orbit. The upper and lower limiter method can restrain from excess adjustment of the output since a1 is too big, and the margin profit is bigger or smaller than zero. Using the upper limiter method, the different orbits can be controlled through choosing the proper upper limit. We take q11 ¼ 2; q12 ¼ 1 into the control system (14). The result of numerical value simulation is seen as Fig. 5, and the orbit which has passed the initial control point (2, 2) becomes stable to two-period orbit quickly. If the value of upper limit q11 is not big-

Fig. 5. i = 1st firm use the upper limit control method to make the output, cost, and profit change from stable to two-period orbit.

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ger than 1.8228, namely, Nash equilibrium output of i = 1st firm, the orbit tends to one-period orbit. If we increase upper limit q11 , the orbit is convergent to higher period orbit. The upper limiter method can restrict the action to increase the output at time t + 1 in excess due to the marginal profit bigger than zero, and the margin profit is bigger than zero. Consequently, it also avoid the too downward output at time t + 2 caused by too high output at time t + 1 which makes the margin profit be much smaller than zero. Therefore, the upper limiter method can be used to suppress the chaos. Using the lower limiter method, the different orbits can be controlled through choosing the proper lower limit. Let q01 ¼ 1:77 in system (15). The result of numerical value simulation is seen as Fig. 6, and the orbit which passes the initial point (2, 2) changes from stable to two-period orbit quickly. If the value of lower limit q01 is not bigger than 1.8228, i.e. i = 1st firm’s output at Nash equilibrium point, the orbit may be convergent to one-period orbit. If we lower limit q01 properly, the orbit approaches to higher period orbit. The lower limiter method can restrict to make output at time t + 1 too small since a1 is so big, and the margin profit is smaller than zero. Therefore, it also avoid too much output at time t + 2 caused by so large output at time t + 1 which makes the margin profit be much bigger than zero. Therefore, lower limit method can be used to control chaos. The above control results have no relation with the initial output of the system, and the time to start control. Therefore, in decision-making, the player can reduce the negative impact caused by fast response to the market through setting up a limiter for the output consciously, and finally it can make the output and profit become stable. Next we will discuss the variety of i = 1st firm’s each economic index before and after the control. From Table 1, we can see when i = 1st firm uses the control by using the limiter method, there are quite improvements for the aggregate profits and aggregate sales revenues no matter the aim of control is one-period orbit or two-period orbits. The numerical value simulation shows that, no matter the control aim is one-period orbit or two-period orbits, and no matter the method is upper and lower limiter method, as long as the value of the upper or lower limit locates in the neighborhood of the equilibrium point, the aggregate profits and aggregate sales revenues will be improved significantly and there are few differences in improvement. The experiment result shows that, although the control can be on high-period orbit through proper values of upper limit or lower limit, it cannot make the performance of i = 1st firm improve significantly. Sometimes, these two economic indices decline instead. Therefore, when i = 1st firm use the limiter method to control chaos, upper or lower limit cannot be far away from the equilibrium point.

Fig. 6. i = 1st firm use the lower limit control method to make the output, cost, and profit stable to two-period orbit.

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From Section 3.1, we know when chaos occurs in the system due to all players’ fast response to the uncertain world, the output and profit of i = 1st firm and i = 2nd firm will have huge vibration, and their aggregate profits and aggregate sales revenues will decrease a lot. In this case, both firms will take certain measures. When values of parameters are: a = 10, b = 1, a1 = 0.44, a2 = 0.44, c1 = 1.1, c2 = 1, d1 = 1, d2 = 1, e1 = 1, e2 = 1.1, r = 0.3, the biggest Lyapunov exponent is 0.4140. In this case, the output, cost and profit display chaotic behavior, and both firms will restrict the outputs properly. Then we take the initial point as (1.76, 1.76). Using upper and lower limiter method, we let q01 ¼ 1:72; q11 ¼ 1:90; q02 ¼ 1:72; q12 ¼ 1:90 in system (13) and make it changing from stable to two-period orbit, as Fig. 7. If the player’s upper limit value is not above Nash equilibrium point (1.8228, 1.7089) or the lower limit value is not under (1.8228, 1.7089), it is controlled to one-period orbit. Using upper

Table 1 When a1 = 0.52, a2 = 0.10, i = 1st firm’s performance contrast between before the control and after the control with limiter method. Performance indices

Aggregate profits

Before the control After the control

Upper and lower limiter method

Upper limiter method

Lower limiter method

p¼1  0  q1 ¼ 1:83; q03 ¼ 2:10 p¼2 ðq01 ¼ 1:76; q11 ¼ 2:10Þ p¼1 ðq11 ¼ 1:80Þ p¼2 ðq11 ¼ 2:00Þ p¼1 ðq01 ¼ 1:84Þ p¼1 ðq01 ¼ 1:77Þ

Aggregate sales revenues (unit: 103)

Value

344.1878

1.0705

Value Increment rate

391.5609 13.76%

1.1944 11.57%

Value Increment rate

390.8940 13.57%

1.2003 12.13%

Value Increment rate

390.0620 13.33%

1.1761 9.87%

Value Increment rate

376.0858 9.27%

1.1314 5.69%

Value Increment rate

391.8272 13.84%

1.1994 12.04%

Value Increment rate

391.1082 13.63%

1.1990 12.00%

Fig. 7. i = 1st firm and i = 2nd firm use the upper limit control method to make the output, cost, and profit change from stable to two-period orbit.

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J. Du et al. / Applied Mathematics and Computation 217 (2010) 2370–2380 Table 2 The firms’ performance contrast between before and after the control when a1 = 0.44, a2 = 0.44 with limiter method. Performance indices Before the control

After the control

i = 1st firm i = 2nd firm Total p=2

Upper-lower limiter method ðq01 ¼ q02 ¼ 1:72; q11 ¼ q12 ¼ 1:9Þ

i = 1st firm i = 2nd firm Total

Upper limiter method ðq11 ¼ q12 ¼ 1:9Þ

i = 1st firm i = 2nd firm Total

Lower limiter method ðq01 ¼ q02 ¼ 1:72Þ

i = 1st firm i = 2nd firm Total

Value Increment Value Increment Value Increment Value Increment Value Increment Value Increment Value Increment Value Increment Value Increment

rate rate rate rate rate rate rate rate rate

Aggregate profits

Aggregate sales revenues (unit: 103)

333.7302 309.1284 642.8586

1.0457 0.9809 2.0266

387.5093 16.12% 364.2722 17.84% 751.7815 16.94% 393.7687 17.99% 358.1969 15.87% 751.9656 16.97% 387.0661 15.98% 362.3966 17.23% 749.4627 16.58%

1.1789 12.74% 1.1309 15.29% 2.3098 13.97% 1.1650 11.41% 1.0871 10.83% 2.2521 11.13% 1.1857 13.39% 1.1282 15.02% 2.3139 14.18%

limiter method, we let q11 ¼ 1:90; q12 ¼ 1:90 in system (14). It changes from stable to two-period orbit. If the player’s upper limit value is not above Nash equilibrium point (1.8228, 1.7089), it is controlled to one-period orbit. Using lower limiter method, we let q01 ¼ 1:72; q02 ¼ 1:72. The orbit which passes the initial point (1.76, 1.76) becomes convergent to two-period orbit quickly. When the players’ lower limit value is not under Nash equilibrium point, the above orbit becomes convergent to one-period orbit quickly. These three forms of the limiter method can control high-period orbit through decreasing the lower limit or increasing the upper limit. From Table 2, we know when chaotic state is caused by players together, both firms use no matter upper and lower limiter method, upper limiter method, or the lower limiter method, the aggregate profits and aggregate sales revenues will increase a lot. The numerical simulation finds these three forms of limiter method have no material difference in improving the firm’s performance. After the control, the aggregate profits and aggregate sales revenues’ improvement have relationship with value of the upper and lower limit. It is the variations of distance from player’s upper limit to equilibrium output or his lower limit to equilibrium output that results in the performance difference as Table 2. For example, in the upper and lower limiter method,, and the equilibrium output is. Obviously, i = 2nd firm’ output is more closely to the equilibrium output, so i = 2nd firm’s performance improvement is better than i = 1st firm. When players chooses the upper limit value and lower limit value, it is better for them to make their margin profit more closely to zero, that is, close to equilibrium output. In economic practice, due to the incomplete information, the player cannot know his own equilibrium output exactly. The player can take the former periods’ average value of output as the reference standard. If all the marginal profits of former periods are bigger than zero, the player can take the maximum value of former periods as the output lower limit; if all the marginal profits of former periods are smaller than zero, the player can take the least value of former periods as the output upper limit. In the later period, the player can adjust the output’s upper or lower limit properly according to the competition situation and the sign variety of marginal profit. 5. Conclusions This paper is concerned with the limiter method in multi-dimensional economic system. Actually, limiter method has many varieties in imperfect completive market. In this circumstance, whether a player should control the chaos or not and take performance measures before and after the control should be considered. The principle, feasibility and the practical operation process of the control method in an economic context also deserve to be discussed. The limiter method comply with the behavior characteristics of economic entities, also comply with the relationship of cooperation and competition among players in practice. In the game procedure of one player with other players, if the portfolio of the player has a big variety in certain time, and the aggregate profits and sales revenues decrease compared with other periods, he should take control measures. Three forms of limiter method illustrates that the output, cost and profit can become stable by proper choice of upper or lower limits. This method is very easy to operate because it only needs fix the upper or lower limit to control chaos in economic dynamic systems. In addition, this method only needs the player whose performance decreased caused by the chaotic state, and can improve the performance without the other player’s cooperation. This enhances the efficiency of limiter method. In practice, due to the limitation of economic resources, the player often restrains the output, advertisement expenses, research

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cost and so on, to reduce the range of these variables’ fluctuation. This shows the decision maker using this method unconsciously in practice, and this method can control chaos in some economic system under certain extents. Through the contrast between the performance before and after the control, it is found that the performance of the firm has improved a lot after the control. When there is chaos in dynamic output duopoly game, whichever forms of limiter method the economic object uses; its performance will have great improvement. Limiter method can be used in the other dynamic systems where players can restrain their state variables. Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grants 70731002, 70773051, 70571034, 70671053 and 70671055, and by the National Social Science Foundation of China under Grant 07CJL028, and by Jiangsu Planned Projects for Postdoctoral Research Funds under Grant 0601020C, and by Jiangsu University Advance Talent Foundation under Grant 06JDG025, and by Human Studies Foundation of Jiangsu Education Hall of China under Grant 08SJD6300005. This work was also sponsored by Qing Lan Project and Jiangsu University Top Talents Training Project. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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