LAUSR

In the paper we deal with a quintic Liénard system of the form ẋ=y−(a1x+a2x3+a3x5),ẏ=b1x+b2x3 with a Z2 -symmetry, where a1,a2,b1∈R and a 3 b 2 ≠ 0. A complete study of this system with b 2 > 0, called the saddle case, is finished, showing that the system exhibits at most two limit cycles, and the necessary and sufficient conditions are obtained on the existence of two limit cycles and a two-saddle heteroclinic loop. We also present a global bifurcation diagram and the corresponding phase portraits of this system, including Hopf bifurcation, Bautin bifurcation, two-saddle heteroclinic loop bifurcation and double limit cycle bifurcation.