The dynamics of a class of nonsymmetric quintic degenerate systems are investigated. According to the local stability analysis and Hopf bifurcation theory， the existence and stability of Hopf bifurcation are discussed. By analysing Melnikov function， we study the bifurcation curves for the little homoclinic orbit encircling one equilibrium and the big homoclinic orbit encircling all three equilibria points.Furthermore， by using PicardFuchs equations， it is proved that double limit cycle bifurcations occur between the degenerate Hopf and homoclinic bifurcation points. Moreover， a formula for calculating the bifurcation value of double limit cycle is derived. Finally， the complete bifurcation diagrams and associate phase portraits are obtained.

degenerate system; bifurcation; nonsymmetricity; limit cycle