Oligopolies price game in fractional order system

•Discrete fractional calculus is applied to the price game model.•The equilibrium points of the Caputo-like fractional dynamical system are evaluated.•The complex dynamic behaviours of the discrete fractional dynamic system can be obtained by the bifurcation diagrams.•When the bifurcation and chaos phenomena appear, the controllers can ensure the tracking errors converge to zero asymptotically.