Bifurcation Of Cubic Integrable System Under Cubic Perturbation

In this paper, we consider the question of bifurcation for cubic integrable system:where 0< |ε|<<1Using method of bifurcation theory and qualitative analysis, we obtain a complete bifurcation diagram in the neighbourhood of origin for the parameter plane. We obtain the following conclusion:1.In the parameter space (δ,L, μ), when δ = 0,L =-1 and μ varies from 0 to μ<0, where 0<-μ<<1,the system has one limit cycle; when δ=0,0<-μ<<1 and L varies from -1 to L<-1, 0<-(L + 1)<<1 ,the system has one limit cycle again; when 0 < -μ <<1,0 < -(L +1)<<1 and 8 varies from 0 to 0 < δ<<1, the system has one limit cycle again. Whenδ = 0,L = -1 and μ varies from 0 to μ>0, where 0<μ<<1, the system has no limit cycle; when δ = 0,0 < -μ<<1 and L varies from -1 to L>-1,0 < L +1<<1 ,the system has one limit cycle; when 0 < -μ <<1,0 < -(L +1)<<1 and δvaries from 0 to 0 <-δ <<1, the system has one limit cycle again.2, When μ =0, K>0, there is a heteroclinic loop passing through the singular points(0) of the system, and we give the phase portrait of heterclinic bifurcation in the parameter plane( δ, L); when K< 0, the system has no heteroclinic loop.3. When μ =0, we give the curve of the multiple orbit bifurcation in the parameter plane(δ, L). And we prove that the bifurcation of multiple loop is a double orbit bifurcation and the curve is convex.4. When μ =0, K>0, in the parameter plane ( δ, L) ,we point out the numbers of limit cycles of the system in the regions divided by bifurcation curve.