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This paper is devoted to the existence and bifurcations of positive periodic solutions of nonlinear differential equation related to the Liebau phenomenon in rigid pipe-tank flow configuration. We obtain some more general requirements on the existence of a positive harmonic solution which improves the results in literatures. It is revealed that the system undergoes saddle node bifurcation, period doubling bifurcation and Neimark-Sacker bifurcation generating various periodic solution of different stability types. The multiplicity of both the positive harmonic solution and positive 2-th order subharmonic solution is first detected in this equation by numerical bifurcation analysis. Moreover, this work reveals that the positive subharmonic solutions of the nonlinear differential equation will also lead to the Liebau phenomenon in the model, which will shed some new insights for future research.MSC—34B16; 34C25; 37G15.

The present article introduces the price transmission effect of establishing free trade areas with African countries from the perspective of trade facilitation. Studies on China-Africa economic cooperation, tariff reduction effects, trade facilitation effects, and price transmission effects in free trade areas were reviewed. The existing literature rarely incorporates tariffs, trade facilitation, and price transmission mechanisms into the research when studying the effects of free trade areas. The present article introduces trade facilitation into the research field of China-Africa cooperation. Taking the China-Mauritius free trade area as an example, the Global Trade Analysis Project model is used to study the price transmission effect after the establishment of the China-African free trade area. Create a model and motivation for other countries to establish free trade areas with African countries. The main conclusions are as follows: China-Mauritian free trade area has improved not only the terms of trade, GDP, and social welfare of the two countries, but also other African countries. This improvement effect mainly comes from the improvement of trade facilitation, and the contribution of zero tariff policy is relatively small. The domestic price changes in many industrial sectors of the two countries showed a significant increase, while the import prices generally showed a downward trend. Therefore, countries should actively participate in the process of global economic integration, and, in the negotiation of trade liberalization, both sides should adopt differentiated negotiation strategies and improve the weak aspects of trade facilitation, respectively, to achieve a win-win situation.

We study the effect of interest rate on phenomenon of business cycle in a Kaldor-Kalecki model. From the information of the People's Bank of China and the Federal Reserve System, we know the interest rate is not a constant but with remarkable periodic volatility. Therefore, we consider periodically forced interest rate in the model and study its dynamics. It is found that, both limit cycle through Hopf bifurcation in unforced system and periodic solutions generated by period doubling bifurcation or resonance in periodically forced system, can lead to cyclical economic fluctuations. Our analysis reveals that the cyclical fluctuation of interest rate is one of a key formation mechanism of business cycle, which agrees well with the pure monetary theory on business cycle. Moreover, this fluctuation can cause chaos in a business cycle system.

A damped Duffing equation with a singularity is considered in this paper, where the elastic restoring force g has a singularity at origin and satisfies superlinear condition at infinity. By applying the twist theorem of nonarea-preserving map, we obtain the equation has at least one period-mT solution, where the minimal period is mT. It is also shown by bifurcation analysis that for a explicit form, the equation undergoes fold bifurcation, period doubling bifurcation and Hopf bifurcation, which leads to different solutions, including harmonic solutions, subharmonic solutions and quasiperiodic solutions. At last, we give the phase portraits and correspond Poincare section of the solutions.

We investigate whether financing constraints affect the size distribution of Chinese industrial firms from 1998 to 2007. Although the firm size distribution does not follow Zipf distribution in China, it is approaching Zipf distribution over time. In general, financing constraints have a robustly negative effect on the size distribution of firms. Furthermore, firms in Western China grow significantly more equally after controlling for financing constraints. However, the effect of financing constraints in Central China is significantly negative, and the effect of financing constraints on firm size distribution in Eastern China is insignificant. We thus expect a non-linear relationship between financing constraints and firm size distribution.

This paper is devoted to the effect of periodic forcing on a system exhibiting a degenerate Hopf bifurcation. Two methods are employed to investigate bifurcations of periodic solution for the periodically forced system. It is obtained by averaging method that the system undergoes fold bifurcation, transcritical bifurcation, and even degenerate Hopf bifurcation of periodic solution. On the other hand, it is also shown by Poincare map that the system will undergo fold bifurcation, transcritical bifurcation, Neimark-Sacker bifurcation and flip bifurcation. Finally, we make a comparison between these two methods.

A three dimensional microbial continuous culture model with a restrained microbial growth rate is studied in this paper. Two types of dilution rates are considered to investigate the dynamic behaviors of the model. For the unforced system, fold bifurcation and Hopf bifurcation are detected, and numerical simulations reveal that the system undergoes degenerate Hopf bifurcation. When the system is periodically forced, bifurcation diagrams for periodic solutions of period-one and period-two are given by researching the Poincaré map, corresponding to different bifurcation cases in the unforced system. Stable and unstable quasiperiodic solutions are obtained by Neimark-Sacker bifurcation with different parameter values. Periodic solutions of various periods can occur or disappear and even change their stability, when the Poincaré map of the forced system undergoes Neimark-Sacker bifurcation, flip bifurcation, and fold bifurcation. Chaotic attractors generated by a cascade of period doublings and some phase portraits are given at last.

With the high speed development of China’s export, a series of problems have initiated. For example, the energy crisis and environmental pollution; more dependent on the export of labor-intensive product, the foreign trade industrial set-ups lifting problem; green trade barriers problem; labor problem and constant reinforce of the demand that enterprises of our country should fulfill social responsibility from trans-corporations, etc. Those problems have impeded our foreign trade competitiveness gravely. Corporation social responsibility (CSR) is one of the means in solving the above problems. The author thinks that export enterprises need to think of dealing the problems with CSR.

To achieve the "second pioneering" in high-tech development zone, it should vigorously develop high-tech industrial clusters to make a breakthrough. Only on the basis of industry clusters, by the establishment of an effective platform for science and technology innovation, through the vigorous support of technology with general character in industrial clusters, can improve the technological innovation capability of industrial clusters. Taking the industrial clusters in high-tech development zone in Jinan for example, the thesis has analyzed the inevitability of the spillover of scientific and technological knowledge. The results of analysis have supported the point of view that the strategy of scientific and technological innovation in the high-tech development zones should be based on the industrial clusters.