CALCULATION OF FOCAL VALUES FOR E_3~1 SYSTEMS

In this paper, we investigate the calculation method of focal values for E13systems. By means of Poincare method and Wu elimination, we design a special algorithm to calculate focal values for E13 systems and obtain some satisfactory results on analysing the number of limit cycles for E13 systems.

STUDYING THE FOCAL VALUE OF ORDINARY DIFFERENTIAL EQUATIONS BY NORMAL FORM THEORY

We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.

CENTER CONDITIONS AND BIFURCATION OF LIMIT CYCLES FOR A CLASS OF FIFTH DEGREE SYSTEMS

The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.

QUALITATIVE THEORY OF THE QUADRATIC DIFFERENTIAL SYSTEMS (II)-ERGODICITY OF LIMIT CYCLES

In this paper we study the variation of limit cycles around different foci when a coefficient in the equation of the quadratic differential system varies.

Complete Study on a Bi-Center Problem for the Z2-Equivariant Cubic Vector Fields

For the planar Z2-equivariant cubic systems having twoelementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Liapunov constants are completely discussed. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme （6 II 6） is proved.

Influences of the Three Gorges Project on seismic activities in the reservoir area

Reservoir-induced earthquakes related with the construction of the Three Gorges Project have attracted great concerns of the public. Since the first water impoundment on May 25, 2003, a number of earthquakes have occurred during the water storage stages, in which the largest was the Badong M5.1 earthquake on December 16, 2013. In this paper, the relationships between seismic activities, b value, seismic parameters, and reservoir water level fluctuations are studied. In addition, based on the digital seismic waveform data obtained since 2000, the focal depth changes and focal mechanism characteristics before and after the water impoundment are studied as well. These provide us important information to understand the earthquake mechanisms. The results show that these earthquakes are typical reservoir-induced earthquakes, which are closely related to water infiltration, pore pressure, and water level fluctuations.The majority of the micro and small earthquakes are caused by karst collapse, mine collapse, bank reformation, superficial unloading, and so on. The larger earthquakes are related to the fault structures to some extent. Due to the persistent effects of water impoundment on the seismic and geological environments around the reservoir and water infiltration into the rocks, the influences on the crustal deformation field, gravity field, seepage field, and fault medium-softening action may vary gradually from a higher strength to a weaker one. Therefore, it is possible that small earthquakes and few medium earthquakes(M≤5.5) will occur in the reservoir area in the future.

Center, Limit Cycles and Isochronous Center of a Z_4-equivariant Quintic System

In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.

Center and Isochronous Center Problems for Quasi Analytic Systems

This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved.