This paper details the research of the Cournot– Bertrand duopoly model with the application of nonlinear dynamics theory.We analyze the stability of the fixed points by numerical simulation; from the result we found that there exists only one Nash equilibrium point. To recognize the chaotic behavior of the system, we give the bifurcation diagram and Lyapunov exponent spectrum along with the corresponding chaotic attractor. Our study finds that either the change of output modification speed or the change of price modification speed will cause the market to the chaotic state which is disadvantageous for both of the firms. The introduction of chaos control strategies can bring the market back to orderly competition.We exert control on the system with the application of the state feedback method and the parameter variation control method. The conclusion has great significance in theory innovation and practice.

Based on the researches of Szydlowski and Krawiec, we studied the inherent complexity of a chaotic business cycle with great excitements in non-resonant condition. First, we got the first-order and second-order pproximate solutions of the system by using multiple scale method. Then deduced the formulation reflecting the complex relations between vibration, phase, bifurcation parameter l and excite frequency X of first-order solution. As the great excitement F varied, the global changes of the system solutions were analyzed. We also explored the different paths leading the systems with different parameter combinations into catastrophe region, fuzzy region or chaos region. Finally, we discussed the evolution trends of business cycle models under the above-mentioned conditions. Hence, this paper has some theoretical and practical significance.

This paper is devoted to investigating a generalized nonlinear Fokker Planck diffusion equation with external force and absorption. We first investigate the integer nonlinear anomalous diffusion, and we obtain the corresponding exact solution expressed by q-exponential function. The solutions of nonlinear diffusion equation with one-fractional derivative and multi-fractional derivative are also studied in detail, and the solutions can have a compact behavior or a long tailed behavior.

According to the repeated game model in electric power duopoly, a triopoly outputs game model is presented. On the basis of some hypotheses, the dynamic characters are demonstrated with theoretical analysis and numerical simulations. The results show that the triopoly model is a chaotic system and it is better than the duopoly model in applications.

This paper studies Hopf bifurcation of a kind of complex economic systems with rich elasticity. The conditions for presence of bifurcation, the stability of periodic orbit before the emergence of Hopf bifurcation and the critical parameter value of the system are obtained. According to Taken s estimation the evolvement situation of the complex system is also given. Numerical examples are given to verify the validity of the present theory. The obtained result is of theoretical importance and has practical applications to exploring the inherence mechanism of the complicated continuous economic systems and establishing the reasonable macro control policy.