Oligopolies price game in fractional order system

Abstract

Abstract A new fractional discrete dynamical system of the price game model is proposed by considering long-term memory of price volatility based on the discrete fractional differentiation calculus. The complex dynamic behaviours are studied with various differential orders using bifurcation diagrams of price. The numerical simulation has indicated that long periods of price adjustment are needed to achieve a stable region as the fractional order decreases, whilst the dynamic behaviours of bifurcation and chaos become increasingly complex. The fractional system, that is more generalized than the integer-order, is stable on the low-price adjustment speed and generally chaotic on the fast-price adjustment speed. This study uses the parameter-dependent Lyapunov stability theory as basis to address the tracking errors for a fractional discrete dynamical system by controllers. When bifurcation and chaos exist in fractional discrete dynamical system, the controllers are presented to guarantee that the actual value prices converge to the expected value prices.