%0 Journal Article
%T Increase in Equilibrium Price by Fast Oscillations
%A Babar Ahmad
%A Khalid Iqbal Mahr
%J Journal of Nonlinear Dynamics
%D 2014
%R 10.1155/2014/593254
%X The dynamics of a market can be described by a differential equation. Using the concept of fast oscillation, the system (typical market) can also oscillate around a new equilibrium price, with an increase. Previously that increase was established by applying harmonic force. In present work, harmonic force is replaced by an arbitrary periodic force with zero mean. Hence the increase in equilibrium price can be controlled by varying the external arbitrary periodic force. 1. Introduction The statistical physics and nonlinear dynamics can be employed as tools in economics and social studies [1] to build up econophysics [2] and statistical finance [3]. Examples are trading and price formation [4], excess and clustering of stochastic volatility [5, 6], investigation of scaling [7] of the competitive equilibrium [8, 9], and role of noise to increase stability [10, 11] in many physical systems. Using Kapitza method [12], Landau and Lifshitz discussed the stability of the inverted pendulum under fast oscillation. He showed that when the suspension of a pendulum has vertical modulation with harmonic force, the position is always stable and is conditionally stable [13]. Using this approach in the market, Holyst and Wojciechowski have shown that due to fast oscillation a new equilibrium price can occur. Using external harmonic force, this new equilibrium price is proportional to the difference . Hence due to fast oscillation, the equilibrium price of the market will increase [14]. In 2009 Ahmad and Borisenok extended the idea of stability for arbitrary periodic force and stabilized the inverted pendulum with relatively low frequency. They used periodic kicking pulses in place of harmonic force. Then the conditional stable point is controlled by varying external periodic force [15]. In this paper, the dynamics of the market is studied along with external arbitrary periodic force, with zero mean. Then another equilibrium price can be established with an increase. This increase can be controlled by applying a particular periodic force. 2. Kapitza Method for Arbitrary Periodic Force A particle of mass is moving under a force due to time-independent potential : and a periodic fast oscillating force with zero mean. This fast oscillating force in Fourier expansion is Here and is the frequency of motion due to . The mean value of a function is denoted by bar and is defined as Also the Fourier coefficient is Since we are choosing a force with zero mean, then from (3) and (4) it follows that In (2), and are the Fourier coefficients, given by Due to (1) and (2) the equation
%U http://www.hindawi.com/journals/jndy/2014/593254/