An Algorithm of Stability and Bifurcation of Limit Cycles for Cubic System

With a suitable parameter,the solution of the system is solved as this parameter equaled zero.This solution is taken as the initial value,and the parameter is given a small increment.The x coordinate of limit cycle phase portraits for planar cubic polynomial differential systems are supposed as the generalized harmonic function.And the y coordinate and the frequency of limit cycle are expanded as Fourier series.Corresponding to the increments of the parameter,the increment of the amplitude,eccentricity and the Fourier coefficients of y coordinate and the frequency of limit cycle are obtained.The linear algebra equations about these increments are got with harmonic balance.Solving these equations,these increments are obtained.The procedure is repeated with the initial value of the next step as the sum of the increments and the initial value,until the parameter is returned to original state.And then the approximate analytical expressions of frequency,periodic,stability index and bifurcation of limit cycles about the parameter are calculated.An example is shown at the end.