Calculation of Limit Cycles and Their Stability and Bifurcationsfor Planar Quadratic Differential Systems

It is very helpful to determine the number,the function expression,their shapes and positions in the phase plane and the bifurcation curves in the parameter plane of the plane quadratic differentiation system in the fields of ecological,biological and applied sciences,e.g.nonlinear oscillations.The x coordinates of limit cycle phase portraits for planar quadratic polynomial differential systems are supposed as the generalized harmonic function.The approximate analytical expressions of y coordinates,frequency,periodic,stability index and bifurcation about the parameter are calculated by alternate method.The present will provide a way to solve the known as the Hilberts problem 16(second part as n=2).Am example with three limit cycles surrounding the singular point(0,0) is shown.