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This paper considers a mixed duopoly competition model, in which two enterprises produce heterogeneous commodities and compete for output in the same market. One enterprise separates the ownership from the management rights, and the enterprise owner entrusts a manager to manage the enterprise. The other enterprise considers consumer surplus. Through stability theory and numerical simulation, the local stability of each equilibrium point and the way the system performs bifurcation are analyzed. By using the basin of attraction, the global stability of the system is studied. The multistability is explained and the properties and some phenomena of the basin of attraction and the chaotic attractor are analyzed by using the knowledge of noninvertible map and critical curve. Finally, the influence of parameters on enterprise profit and objective is also analyzed, which is helpful for enterprises to make decisions in the process of game. • A mixed duopoly Cournot competition model is established based on heterogeneous products. • The complex dynamic behavior of the economic system is analyzed by means of chaos theory and numerical simulation. • The complex structures of the basin of attraction and chaotic attractor are studied by using noninvertible map. • The influence of the parameters on the average objective and the average profit of the two enterprises is analyzed. [ABSTRACT FROM AUTHOR]