Abstract: By using the Poincare ring domain theorem, Lyapunov-branch theorem, and constructing appropriate Poincare ring domain, it is shown that U(u,v,λ),V(u, v, λ) unanimously tends to 0 respectively as λ→0 in a bounded domain of the uov plane. When the parameter λ approaches the nonzero positive or positive infinity, the corresponding system has at least a stable or unstable limit cycle near the origin. For the two cases of the parameter λ approaches non-zero positive number and positive infinity, examples are given to verify the wide applicability of the conclusions.

Key words: analytic function, central stable focus, limit cycle