Complex dynamics and chaos control of duopoly Bertrand model in Chinese air-conditioning market

• A dynamic duopoly Bertrand model with bounded rationality and quadratic cost function.
• In Chinese air-conditioning market the boundary equilibrium point is locally stable.
• The Lyapunov dimension of the chaos attractor is 1.9585.
• The adjustment speeds may cause a market structure to behave chaotically.
• The chaotic behavior can be controlled by decreasing the degree of substitutability.

A dynamic duopoly Bertrand model with quadratic cost function which is closer to reality and different from previous researches is discussed. The model is applied into air-conditioning market where the boundary equilibrium point is locally stable. Numerical simulations illustrate that the stability of Nash equilibrium strongly depends on the speed of adjustment of bounded rational player. The adjustment speeds and the degree of substitutability may undermine the stability of the equilibrium and cause a market structure to behave chaotically. The Lyapunov dimension of the chaos attractor is 1.9585 under some conditions. The stabilization of the chaotic behavior can be obtained by reducing the degree of substitutability. The results have an important theoretical and practical significance to Chinese air-conditioning market.