The article presents the results of the study of nonlinear market prices dynamic, simulated using game theory model, maps and theory of bifurcation. Market pricing is presented as a two-dimensional map. Qualitative analyses of the rms' pricing system properties using the xed points, analysis of the trajectories near these xed points were ful lled Simulation of the market prices dynamic showed that the xed point of the map coincides with the local Nash equilibrium, so the analysis of the stability of Nash equilibrium was done based on the maps' xed points sustainability analysis. Numerical simulation results were visualized, bifurcations of the xed point were identi ed and transition from periodic to chaotic mode was demonstrated. Sustainability analysis was carried out with using Jacobian. The mechanism for the pricing decision support, allowing under a chaotic market dynamics to ensure maximum e ciency of the rms was proposed.
The last two decades the scienti c literature has been widely discussing the concept of deterministic chaos, chaotic dynamics occurring in di erent systems. Di erent approaches and methods for chaos control in theoretical and applied problem solution are suggested. At the same time a lot of attention is given to managing the impact of low power, meaning that the system has a number of characteristics, inherent properties and laws which allow to achieve expected results using weak management actions (without spending of signi cant resources).
In a number of studies of nonlinear systems dynamic (physical, chemical, biological, economic), it was found that the dynamic chaos mode is a typical phenomenon. Chaotic properties are manifested in a variety of systems, and if the chaos is not found, the reasons for this may be either the existence of chaos in a small area of the parameter space, or it is out of range of parameters.
Research associated with the problems of predictability of chaotic systems, control of system dynamics and stabilization of chaos is rapidly developing in di erent elds. Theoretical and applied studies in these elds have revealed an unexpected property of chaotic dynamical systems: they are controlled by external actions [3,13,18,21]. That is, by smaller impacts it is possible to signi cantly a ect the dynamics of chaotic systems, to stabilize their dynamics, transferring it from the chaotic mode to the required periodic mode.
In relation to economic systems the following studies in the eld of chaos control were conducted: [1, 2, 5{8, 11, 12, 14, 15, 19, 20]. Chaos control is proposed on the basis of production cost reduction [2], an increase in investment activity [11, 19], or control impacts are con rmed on the basis of revealed connection between the previous and current variables values, i. e. by e ective use of feedback [2, 19, 22]. 2
Phenomenon of Chaotic Market Dynamic: the Model Chaotic systems are a class of uncertain models that di ers from deterministic and stochastic systems in their properties [4]. In deterministic systems it is possible to build the future system trajectory from the initial state to in nite time interval. In stochastic systems it is possible to estimate the future system state into short time interval,determined by the prediction accuracy.
We consider a duopoly as a market model, when two rms-producers cooperate on the market, and the pricing process is controlled by price (which rms o er there di erential products to consumers) change. Further it will be illustrated that the pricing system is chaotic and then we will o er the decision support model for stabilization market prices dynamic.
The main precondition of the model: { The model of duopoly (model of price competition) with a di erentiated product in which the consumer demand function is given a utility function with constant elasticity of substitution is used; { Pricing dynamics is modeled as a map (recurrence relations); { Modeling of pricing decisions is realized with using the game theory and methods of nonlinear dynamic.
The model designations: i 2 I { producer's number; pi { price of the i-th producer ( rm); qi { the volume of sales of i-th producer; qi(p) { demand for the product of i-th producer; ci { unit costs for the product of i-th producer; i(p) = (pi ci) qi(p) { pro t of i-th producer.
Prices vector p = (pi ; i 2 I) is a local Nash equilibrium when it satis es the following conditions of local maxima conditional of the pro t function:
The strategy of i-th producer is a price changing, which is proportional to the change of its pro ts with some constant ki > 0 in order to maximize their e ectiveness, i. e. to achieve the global sustainable Nash equilibrium. The model of competitive interaction of rms described by two-dimensional system of differential equations was o ered in [16, 17] and represented as: 8 >< p1(t + 1) = p1(t) + k1 ( p12(t)p2(t)+2c1p1(t)p2(t)+c1p22(t)) ;
(p21(t)+p1(t)p2(t))2
>: p2(t + 1) = p2(t) + k2 ( p1(t)p22(t)+2c2p1(t)p2(t)+c2p12(t)) ;
(p22(t)+p1(t)p2(t))2
(
Nash equilibrium, characterized by a pair of prices p1 and p2, is the solution
of the following equations:
8
< p1 = c1 + c1
: p2 = c2 + c2
q
q
1 + Ap21 ;
1 + Ap12 :
(
From the economic point of view, this means that rms choose the mode (and parameters), which will lead to a change in the market, resulting in price levels will evolve predictable dynamic. If such control is permissible, then a transition to the market equilibrium will proceed, that ensure for each rm the maximum e ectiveness.
The xed point (p10; p20) of the system (
The decision of the latter system of equations will obviously be the same as
the solution of the system (
Mc =
Eigenvalues of this matrix are multiples of the map (
Mc = 1 61k1 30:5k1 19:8k2 + 4 > 0;
The solution of this system of inequalities that de ne the triangle of stability
for the xed point of the two-dimensional map prices (
Thus, the identi ed area of admissible values of adaptive price parameters k1 and k2 is highlighted in gray. To ensure the stability of the Nash price equilibrium (0.15; 0.23) in the conditions of the given cost values of both rms at 0.07 and 0.12 will allow the use of such rms adaptation strategy, which is based on a set of adaptation options combinations that are within the acceptable area.
As soon as the control parameters deviate from the permissible values, there occurs the equilibrium stability loosing and the system goes to another unstable mode { chaos. That is, in any initial price of rms if they use a pricing strategy in accordance with the values determined above, the rm de nitely reaches the Nash equilibrium. These pricing values are not able to change the Nash price equilibrium, so to increase the e ciency the rm should use the proposed decisions.
Figure 2 shows the chaotic attractors of system (
The chaotic nature of the prices system dynamics is due to the instability of the phase trajectories, the growth of small initial perturbations in time, mixing elements of the phase space and, as a consequence, leads to unpredictable system dynamic over long term.
In making pricing decisions in addition to the criterion of economic e ciency the rms must take into account the objective nature of the market dynamics as a whole and take into account the possibility of a chaotic regime of market dynamics.
Results and conclusions The described approach for rms pricing using the methods of game theory, nonlinear dynamics and bifurcation theory provides a new perspective on the dynamics of the process of competitive interaction of the rms and makes it possible to conduct a qualitative and visual analysis of the system properties with help of the singular points of the phase space ( xed points) and to analyze the systems trajectories near these xed points.
It is shown that the market pricing system has complex and diverse types of dynamics, so that the structure of the phase space and its dependence on the parameters of this structure are very complex. The phase space of the system is heterogeneous and has two basic types of system dynamic { stability and chaos.
Prices dynamics is modeled using a two-dimensional map; coordination of rms' pricing decisions is based on monitoring the stability of the Nash equilibrium. The analysis of the developed model shows that the Nash equilibrium coincides with the map xed point prices. Therefore, the analysis of Nash equilibrium stability is carried out on the basis of the analysis of the map xed points sustainability. Numerical simulations are demonstrated the existence of xed point bifurcations. Chaos in market pricing model means that when one rm change its price even slightly this can lead to unpredictable market prices changing of another producers and total market in long term. Therefore, all producers must have the tools of chaos control.
In order to avoid unexpected chaotic dynamics in market prices the mechanism for decision making support is o ered and is based on the price changing proportional to marginal pro t changing of each rm. These mechanism would ensure the stability of the Nash equilibrium, and therefore would balance the rms' economic interests, would coordinate price decisions and maintain maximum rms e ciency.
For the complex research and analysis of rms competitive interaction we use
the author program for modeling and visualization of nonlinear pricing dynamics
and decision support in rms price strategies. The program is designed to
simulate the strategic cooperation in the rms pricing process, use a four-parameter
map and form the optimal pricing policy in oligopoly, provided the e ective
control and decision making under prices chaotic dynamics. The program has the
following functions:
{ Assessment of local Nash equilibrium of prices;
{ Identi cation of bifurcations of xed price points in the map;
{ Identi cation of modes of stability and dynamical chaos in pricing;
{ Identi cation of transition scenarios to dynamical chaos;
{ Forming the pricing decisions, ensuring stable mode of market prices
dynamic.
12. Loskutov, A.Y.: Non-linear optimization of chaotic dynamics of the market.
Economics and Mathematical Methods 46(
Institute of Computer Science, Moscow { Izhevsk, Russia (2007) 14. Moon, F.: Chaotic Oscillations: Introductory Course for Scientists and Engineers.
Mir, Moscow, Russia (1990) 15. Neimark, Y.I., Ostrovsky, A.V.: On some models of pricing in the market economy.
Izvestiya VUZ. Applied Nonlinear Dynamics 6, 35{41 (1999)
16. Orlova, E.V.: Model for economic interests agreement in duopoly's making pricing
decision. Computer Research and Modeling 7(
Springer Berlin Heidelberg, Berlin, Heidelberg (2000) 21. Schuster, H.G., Just, W.: Deterministic chaos : an introduction. Wiley-VCH, Weinheim (2005) 22. Stachurski, J.: Economic Dynamics: Theory and Computation. MIT Press (2009)