Uniqueness and hyperbolicity of limit cycles for autonomous planar systems with zero diagonal coefficient.

This paper investigates the uniqueness and hyperbolicity of the autonomous planar system with zero diagonal coefficient ẋ=p2(y)q2(x)y, ẏ=p3(y)q3(x)x+p4(y)q4(x)y and the generalized Liénard system ẋ=ϕ(z−F(x)), ż=−g(x). Some sufficient conditions that guarantee the uniqueness and hyperbolicity of limit cycles are established. The results of this paper generalize some previous results on this field.