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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.

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] and [15]. As we know, economic systems are nonlinear and may display chaotic behavior (see [12], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29] and [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 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] and [15], Stoop and Wagner [10]. Zhang and Shen [14] and [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.

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 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.