Small Perturbations Singular Successor Function Smooth Near The Closed Orbit

Considering a planar systemx = f(x,y) + εf₀(x,y,λ,ε)y = g(x,y)+εg₀(x,y,λ,ε)'where f,g,f₀,g₀∈C^k,ε∈R and 0≤ε≤1,λ∈R^m, and supposing that this planar system has a singular cycle when ε = 0. It is very important for successor function to judge the number, the stability and the relative place of the limit cycles of the perturbated system. In this paper, the smooth quality of successor function in the neighborhood of the singular cycle of the perturbated system has been studied by analyzing the smooth quality of Dulac mapping and regular mapping, and the normal form theory has been used to study the relation between the smooth quality of planar system and the order of week focus or week saddle point. We have got four main conclusions as following: 1. The successor function in the neighborhood of the singular cycle of the perturbated system is C^k-1; 2. The successor function in the neighborhood of the periodic orbit of the perturbated system is C^k; 3. The lowest upper of the order of week focus of theperturbated system is [k-1/2], meanwhile the lowest upper of the order of Hopf bifurcation is [k-1/2]; 4. The lowest upper of the orderof week saddle point of the perturbated system is [k-1/2] . Theseresults are concise and convenient, and that can reduce the smooth quality condition of planar system in the many theories.